On the abundance of k-fold semi-monotone minimal sets in bimodal circle maps

Abstract

Inspired by a twist map theorem of Mather. we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map g to the k-fold cover. For each irrational in the rotation set’s interior, the collection of the k-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$-dimensional ball with the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool used is a generalization of a construction of Hedlund and Morse that generates symbolic analogs of these k-fold well-ordered invariant sets.

1 Introduction

In a dynamical system a rotation number or vector measures the asymptotic speed and direction of an orbit. The rotation set collects all these together into a single invariant of the system. The natural question is how much this invariant tells you about the dynamics. Perhaps the first issue is whether for each rotation number there is a nice invariant set in which every point has that rotation number.

This question has been studied in a number of contexts, with the most complete answer available for maps of the circle, annulus, and two-dimensional torus. In these cases the basic question is enhanced by requiring that the invariant set of a given rotation vector has the same combinatorics as a rigid rotation. So, for example, for a continuous degree-one map g of the circle and a number $\omega $ in its rotation set, is there an invariant set $Z_\omega $ on which the action of g looks like the invariant set of rigid rotation of the circle by $\omega $ ? This is made clearer and more precise by lifting the dynamics to the universal cover ${\mathbb R}$ . The question then translates to whether the action of the lift $\tilde {g}:{\mathbb R}\rightarrow {\mathbb R}$ on the lift ${\tilde {Z}}_\omega $ is order-preserving. For this class of maps the answer is yes; such invariant sets always exist [18].

On the torus and annulus a general homeomorphism isotopic to the identity lacks the structure to force the desired invariant sets to be order-preserving, therefore topological analogs are used [11, 31, 41]. The monotone twist hypothesis provides the required additional structure in the annulus case. The celebrated Aubry–Mather theorem states that for each rational in the rotation set there is a periodic orbit and for each irrational a Denjoy minimal set, and the action of the map on these invariant sets is ordered in the circle factor like rigid rotation. These invariant sets are now called Aubry–Mather sets.

For an area-preserving monotone twist map the minimal set with a given irrational rotation number could be an invariant circle. When a parameter is altered and this circle breaks it is replaced by an invariant Denjoy minimal set. In [36] Mather investigated what additional dynamics this forces. He showed that in the absence of an invariant circle with a given irrational rotation number there are many other invariant minimal Cantor sets with the same rotation number and the dynamics on these sets is nicely ordered under the dynamics not in the base, but in finite covering spaces of the annulus.

More specifically, these invariant minimal sets are Denjoy minimal sets which are uniquely ergodic. Their collection is topologized using the weak topology on these measures. Mather showed that for a given irrational rotation number in the rotation set the collection of Denjoy minimal sets with that rotation number that are ordered in the k-fold cover contains a topological disk of dimension $k-1$ . In this paper we prove the analog of this result for a class ${\mathcal {G}}$ of bimodal degree-one maps of the circle. We also describe their periodic orbits which have nicely ordered lifts in the k-fold cover.

Mather’s proof use variational methods. The main methods here come from symbolic dynamics and use a construction that generalizes one due to Hedlund and Morse ([38] and [24, p. 111]). Such generalizations are a common tool in topological dynamics ([35], [3, pp. 234–241], and [12]). This Hedlund–Morse (HM) construction, used here for a rotation number $\omega $ and number k, generates the itineraries under rigid rotation by $\omega $ with respect to an address systems made from $2k$ intervals on the circle. The closure of this set of itineraries yields the symbolic analog of an invariant set that is nicely ordered in the k-fold cover. These sets are termed symbolic k-fold semi-monotone sets (symbolic kfsm sets). Varying the address system parameterizes the symbolic kfsm sets in both the Hausdorff and weak topologies.

A physical kfsm set (or just a kfsm set if the context is clear) is a g-invariant set Z that has a lift $Z'$ to the k-fold cover on which the lift $g_k$ of g acts like rigid rotation. Physical and symbolic kfsm sets are connected by a second tool.

The second tool uses addresses and itineraries, but this time to code orbits under the bimodal map g. Restricting to all orbits that land in the positive-slope region, we get an invariant set $\Lambda (g)$ which is coded by an order interval in the one-sided $2$ -shift $\Sigma _2^+$ . Because we study invariant sets ordered in the k-fold cover, we lift this coding to the orbits which stay in the positive-slope region under $g_k$ in the k-fold cover $S_k$ . This yields a $g_k$ -invariant set $\Lambda _k(g)$ which is then coded by a subshift ${\hat {\Lambda }}_k(g)\subset \Sigma _{2k}^+$ .

This result connects the physical kfsm sets in $\Lambda _k(g)$ , the symbolic kfsm sets in ${\hat {\Lambda }}_k(g)$ , and the symbolic sets generated by the HM construction. Part (c) will be explained below.

Theorem 1.1. For $g\in {\mathcal {G}}$ the following assertions are equivalent.

1. (a) $Z\subset \Lambda (g)$ is a recurrent kfsm set for g.

2. (b) The symbolic coding of Z via the itinerary map is constructable via the HM process.

3. (c) Z is a recurrent set of an interpolated semi-monotone map $H_{\vec {c}}$ in the k-fold cover.

Note that the result is restricted to recurrent kfsm sets. There are several reasons for this: first, the recurrent points are where the interesting dynamics occurs; second, invariant measures are always supported on recurrent sets; and finally, the HM construction produces recurrent sets. As is well known in Aubry–Mather theory, there are also non-recurrent kfsm sets which consist of a recurrent set and orbits homoclinic to that set. We also restrict to orbits that stay in the positive-slope region of g. Considering kfsm sets that also have points in the negative-slope region at most adds additional homoclinic orbits or shadow periodic orbits. See §13.2.

For each k, the HM construction depends on two parameters, a rotation number $\omega $ and a parameter $\vec {\nu }$ describing the address system on the circle. For a rational rotation number it produces a finite cluster of periodic orbits, while for irrationals it produces a semi-Denjoy minimal set. Since g is non-injective the analogs of Denjoy minimal sets have pairs of points that collapse in forward time, hence the ‘semi’ in their name.

Another main result is that the HM construction parameters $(\omega , \vec {\nu })$ yield a homeomorphic parameterization of the space of invariant measures on the recurrent symbolic kfsm sets with the weak topology. Via the itinerary map, this is pulled back to a parameterization of the space of invariant measures on the physical recurrent kfsm sets with the weak topology. It yields the following result in which $\rho (g)$ is the rotation interval of $g\in {\mathcal {G}}$ .

Theorem 1.2. Assume $g\in {\mathcal {G}}$ , $\alpha \not \in {\mathbb Q}$ , $\alpha \in \operatorname {\mathrm {Int}}(\rho (g))$ , and $k>0$ .

1. (a) In the weak topology there is a $(k-1)$ -dimensional disk of kfsm semi-Denjoy minimal sets with rotation number $\alpha $ .

2. (b) If $p_n/q_n$ is a sequence of rationals in lowest terms with $p_n/q_n \rightarrow \alpha $ , then the number of distinct kfsm periodic orbits of g with rotation number $p_n/q_n$ grows like $q_n^{k-1}$ .

Informally, a kfsm semi-Denjoy minimal set wraps k times around the circle with orbits moving at different average speeds in each loop. Lifting to the k-fold cover, these ‘speeds’ are given by the amount of the unique invariant measure present in a fundamental domain of $S_k$ : more measure means slower speed. The k-dimensional vector of these measures is called the skewness of the minimal set. The sum of the skewness components is one, and thus the collection of possible skewnesses contains a $(k-1)$ -dimensional ball. The skewness turns out to be an injective parameterization of the kfsm sets for a given irrational rotation number in the interior of the rotation set of a $g\in {\mathcal {G}}$ (see Remark 9.14).

The HM parametrization of kfsm sets with the Hausdorff topology is only lower semi-continuous. The points of discontinuity are given in Theorem 9.5.

We will on occasion use results derived from those of Aubry–Mather theory. While the context here is a bit different, the proofs are virtually identical and so are omitted. There are excellent expositions of Aubry–Mather theory; see, for example, [33], [30, Ch. 13], and [23, Ch. 2]. A version of Mather’s theorem on Denjoy minimal sets is given in [45] for monotone recursion maps.

We restrict attention here to a particular class of bimodal circle maps defined in §4.1. Using the Parry–Milnor–Thurston theorem for degree-one circle maps, the results can be transferred (with appropriate alterations) to general bimodal circle maps (see Remark 4.2).

Note that the results here immediately apply to a class of annulus homeomorphisms. This application can be done either via the Brown–Barge–Martin method using the inverse limit of $g\in {\mathcal {G}}$ [5, 14] or via the symbolic dynamics in annulus maps with good coding like a rotary horseshoe, for example, [12, 20, 25, 32].

Figure 1 illustrates the conceptual framework that inspired the results here. It shows the graph of a map $g \in {\mathcal {G}}$ lifted to the $3$ -fold cover. At three heights $(c_1, c_2, c_3) = \vec {c}$ the graph is cut off, yielding a semi-monotone circle map $H_{\vec {c}}$ . Such maps have a unique rotation number and well-understood recurrent sets which are of necessity semi-monotone sets. As $\vec {c}$ is varied, the rotation number $\rho (H_{\vec {c}}) = \rho (\vec {c})$ varies continuously. Thus one would expect that the level sets $\rho ^{-1}(\omega )$ provide a parameterization of the kfsm sets with rotation number $\omega $ . In particular, for irrational $\omega $ this level set should be a $(k-1)$ -dimensional disk as in Theorem 1.2(a). This is true for $g\in {\mathcal {G}}$ . Figure 5 below shows some level sets. It is worth noting that this figure is not a bifurcation diagram, but rather a detailed analysis of the dynamics present in a single map.

Figure 1 The lift of a $g\in {\mathcal {G}}$ to the 3-fold cover and an interpolated semi-monotone map.

While providing a valuable heuristic, this point of view is not as technically tractable as the HM construction and we content ourselves with just a few comments on it in §13. One of these adds item (c) to the list of equivalent conditions in Theorem 1.1.

The literature on bimodal circle map dynamics is vast and we briefly mention only two threads here. Symbolic dynamics for degree-one bimodal circle maps goes back at least to [6, 26, 27]. The interpolated ‘flat spot’ map trick for finding $1$ -fold semi-monotone sets was discovered and used by many people in the early 1980s; references include [10, 18, 29, 37, 40, 42–44]. The author learned the trick from G.R. Hall in spring 1983 and the idea of applying it in finite covers emerged in conversations with him.

There are many questions raised by this work; here we mention three. As is well known, the $1$ -fold symbolic semi-monotone sets generated by the HM construction are the much-studied Sturmian sequences. The general symbolic kfsm sets are thus a generalization of one property of the Sturmians to more symbols (there are many other generalizations). The Sturmians have many marvelous properties such as their connection to the Farey tree and substitutions: which properties are shared by symbolic kfsm sets?

The HM construction is an explicit parameterized way of getting well-controlled orbits that do not preserve the cyclic order in the base and thus in most cases force positive entropy as well as additional orbits. A second question is how the parameterization given by the the HM construction interacts with the forcing orders on orbits in dimension one and two (see, for example, [1, 13]).

A final question relates to the global parameterization of kfsm sets by the HM construction. Each bimodal map $\in {\mathcal {G}}$ corresponds to a specific set of parameters, namely, those that generate symbolic kfsm sets whose physical counterparts are present in g. What is the shape of this set of parameters?

After this work was completed the author became aware of the considerable literature on sets in the circle that are invariant and nicely ordered under the action of $z\mapsto z^d$ [8, 9, 17, 21, 22, 34]. While the exact relationship of that theory and what is contained in this paper is not clear, it is clear that the two areas share many basic ideas and methods. These include families of interpolated semi-monotone circle maps with flat spots, tight and loose gaps in invariant Cantor sets, parameterization of the sets using the position of the flat spots, and parameterization of the sets with irrational rotation number by an analog of skewness. Section 14 contains a few more comments on the relationship of the problems.

2 Preliminaries

2.1 Dynamics

Throughout this section X is a metric space and $g:X\rightarrow X$ is a continuous, onto map. Since the maps g we will be considering will usually not be injective, we will just be considering forward orbits, so $o(x,g) = \{x, g(x), g^2(x), \ldots \}$ .

A point x is recurrent if there exists a sequence $n_i\rightarrow \infty $ with $g^{n_i}(x)\rightarrow x$ . A g-invariant set Z is called recurrent if every point $z\in Z$ is recurrent. Note that a recurrent subset is usually different than the recurrent set, with the latter being the closure of all recurrent points. A compact invariant set Z is called minimal if every point $z\in Z$ has a forward orbit that is dense in Z.

The one-sided shift space on n symbols is $\Sigma ^+_n = \{0, \ldots , n-1\}^{\mathbb N}$ and that on ${\mathbb Z}$ symbols is $\Sigma ^+_{\mathbb Z} = {\mathbb Z}^{\mathbb N}$ . Occasionally we will write $\Sigma _\infty ^+$ for $\Sigma ^+_{\mathbb Z}$ . For clarity we note that in this paper $0\in {\mathbb N}$ . In every case we give one-sided shift spaces the lexicographic order and the left shift map is denoted by $\sigma $ , perhaps with a subscript to indicate the shift space upon which it acts. Maps between shifts and subshifts here will always be defined by their action on individual symbols, so, for example, $\varphi :\Sigma _n^+\rightarrow \Sigma _n^+$ defined on symbols by $s\mapsto \varphi (s)$ means that $\varphi (s_0 s_1 s_2\ldots ) = \varphi (s_0) \varphi (s_1) \varphi (s_2) \ldots .$ For a block $B = b_0 \ldots b_{N-1}$ in $\Sigma ^+_n$ its cylinder set is $[B] = \{ {\underline s}\in \Sigma _n^+\colon s_i = b_i, i = 0, \ldots , N-1\}$ . Note that all our cylinder sets start with index $0$ .

The space–map pairs $(X,f)$ and $(Y,g)$ are said to be topologically conjugate by h if h is a homeomorphism from X onto Y and $hf = gh$ .

We will frequently use the standard dynamical tool of addresses and itineraries. Assume $X = X_0\sqcup X_1\sqcup \cdots \sqcup X_{n-1}$ , with $\sqcup $ denoting disjoint union. Define the address map A as $A(x) = j$ when $x\in X_j$ and the itinerary map $\iota :X\rightarrow \Sigma _n^+$ by $\iota (x)_i = A(g^i(x))$ . It is immediate that $\sigma \circ \iota = \iota \circ g$ . In many cases here, $\iota $ will be continuous and injective, yielding a topological conjugacy from $(X,g)$ to a subset of $(\Sigma _n^+, \sigma )$ .

We will also encounter the situation where the $X_j$ are not disjoint, but intersect only in their frontiers $\operatorname {\mathrm {Fr}}(X_j)$ . In this case we define a ‘good set’ $G = \{x \colon o(x, g)\cap (\cup \operatorname {\mathrm {Fr}}(X_j)) = \emptyset \}$ . In this case the itinerary map is defined as $\iota :G\rightarrow \Sigma _n^+$ .

For $Z\subset X$ , its interior, closure and frontier are denoted by $\operatorname {\mathrm {Int}}(Z), \operatorname {\mathrm {Cl}}(Z),$ and $\operatorname {\mathrm {Fr}}(X)$ , respectively. The $\epsilon $ -ball about x is $N_\epsilon (x)$ . The Hausdorff distance between two sets is denoted $\operatorname {\mathrm {HD}}(X, Y)$ . For an interval I in ${\mathbb R}$ , $|I|$ denotes it length, and for a finite set Z, $\# Z$ is its cardinality. On an ordered k-tuple the map $\tau $ is the left cyclic shift, so $\tau (a_1, a_2, \ldots , a_k) = (a_2, a_3, \ldots , a_k, a_1)$ . On the circle $S^1$ , $\theta _1 < \theta _2$ is defined as long as $|\theta _1-\theta _2| < 1/2$ .

If $g:X\rightarrow X$ and $Y\subset X$ , then $h:X\rightarrow X$ is said to interpolate g on Y if $h(y) = g(y)$ for all $y\in Y$ , or in symbols, $g_{\vert Y} = h_{\vert Y}$ .

2.2 The circle, finite covers and degree-one circle maps

While the only compact manifold here will be a circle, it will clarify matters to use the language of covering spaces.

In general, if $\pi :{\tilde {Y}}\rightarrow Y$ is a covering space, and $Z\subset Y$ , a lift of Z is any set $Z'\subset {\tilde {Y}}$ with $\pi (Z') = Z$ . The full lift of Z is ${\tilde {Z}} = \pi ^{-1}(Z)$ . If $g:Y\rightarrow Y$ lifts to $\tilde {g}:{\tilde {Y}}\rightarrow {\tilde {Y}}$ and $Z\subset Y$ is g-invariant then the full lift ${\tilde {Z}}$ is $\tilde {g}$ -invariant, a property that is usually not shared by a lift $Z'$ .

The universal cover of the circle is ${\mathbb R}$ , with $T(x) = x+1$ generating the deck group. Thus the covering space is $\pi :{\mathbb R}\rightarrow S^1 = {\mathbb R}/T = {\mathbb R}/{\mathbb Z} = [0,1]/\mkern -3mu\sim $ . We will only study maps $g:S^1\rightarrow S^1$ whose lifts $\tilde {g}$ commute with the deck transformation, $\tilde {g} T = T \tilde {g}$ , or $\tilde {g}(x+1) = \tilde {g}(x) + 1$ . These circle maps are commonly termed degree one. Our given g will usually have a preferred lift $\tilde {g}$ and so all other lifts are obtained as $T^n \tilde {g}$ or $\tilde {g} + n$ .

Central to our study are the finite k-fold covers of the circle for each $k>0$ , $S_k = {\mathbb R}/T^k = {\mathbb R}/k{\mathbb Z} = [0,k]/\mkern -3mu\sim $ . A generator of the deck group is $T_k:S_k \rightarrow S_k$ induced by T on ${\mathbb R}$ and the covering space is $\pi _k:S_k\rightarrow S^1$ . A preferred lift $\tilde {g}$ of g to ${\mathbb R}$ induces a preferred lift $\tilde {g}_k:S_k\rightarrow S_k$ that commutes with $T_k$ . We also need the map from the universal cover to the k-fold cover treating $S_k$ as the base space $p_k:{\mathbb R}\rightarrow S_k$ .

A g-periodic point x is said to have rotation type $(p,q)$ with respect to the preferred lift $\tilde {g}:{\mathbb R}\rightarrow {\mathbb R}$ if x has period q and for some lift $x'\in {\mathbb R}$ , $\tilde {g}^q(x') = T^p x'$ . Note that there is no requirement here that p and q are relatively prime.

A central concern in this paper is how g-minimal sets in $S^1$ lift to k-fold covers.

Theorem 2.1. Let $g:S^1\rightarrow S^1$ be degree one and fix $1 < k< \infty $ .

1. (a) If $Z\subset S^1$ is a minimal set, then there exists an m which divides k so that the full lift of Z to $S_k$ satisfies

   (2.1) $$ \begin{align} {\tilde{Z}} = \bigsqcup_{j=1}^m Z_j' \end{align} $$
   with each $Z_j'$ minimal under $\tilde {g}_k$ , $\pi _k({\tilde {Z}}_j') = Z$ and $T_k(Z_j') = Z_{j+1}'$ with indices $\mathrm {mod}\ k$ .

2. (b) If $Z', Z"\subset S_k$ are $\tilde {g}_k$ minimal sets, we have $\pi _k(Z') = \pi _k(Z")$ if and only if $T^p_k(Z') = Z"$ for some p.

3. (c) If $x\in S^1$ is a periodic point with rotation type $(p,q)$ , let $m = \gcd (k,p)$ . There exist $x_j'\in \pi _k^{-1}(x)\subset S_k$ with $1 \leq j \leq m$ and

   (2.2) $$ \begin{align} \pi_k^{-1}(o(x,g)) = \bigsqcup_{j=1}^m o(x_j', \tilde{g}_k), \end{align} $$
   the period of each $x_j'$ under $\tilde {g}_k$ equal to $kq/m$ , and $T_k(x_j') = x_{j+1}' $ with indices $\mathrm {mod}\ k$ .

Proof. To prove (a) we begin with two preliminary facts with similar proofs. First, we show that for any $z'\in {\tilde {Z}}$ , $\pi _k({\operatorname {\mathrm {Cl}}(o(z',\tilde {g}_k))}) = Z$ . Let $z = \pi _k(z')$ and pick $y\in Z$ . By minimality there exists $g^{n_i}(z)\rightarrow y$ . Lifting and using the compactness of $ S_k$ , there are a subsequence $n_{i'}$ and a $y'\in S_k$ with $\tilde {g}_k^{n_{i'}}(z')\rightarrow y'$ . Thus $g_k^{n_{i'}}(z) = \pi _k(\tilde {g}_k^{n_{i'}}(z'))\rightarrow \pi _k(y')$ and so $y = \pi _k(y')$ .

Second, we show that for any $z',y'\in {\tilde {Z}}$ , there exists a p with $T^p(y')\in {\operatorname {\mathrm {Cl}}(o(z',\tilde {g}_k))}$ . Let $z=\pi _k(z')$ and $y = \pi _k(y')$ . By minimality again, we have $g^{n_i}(z)\rightarrow y$ . Lifting and passing to a subsequence, there are a subsequence $n_{i'}$ and a $y"\in S_k$ with $\tilde {g}_k^{n_{i'}}(z')\rightarrow y"$ . Thus $\pi _k(y") = y$ also, so there is a p with $y" = T_k^p(y')$ and so $T_k^p(y')\in {\operatorname {\mathrm {Cl}}(o(z',\tilde {g}_k))}$ .

Now for the main proof, pick $z'\in {\tilde {Z}}$ and let $Z_1' = {\operatorname {\mathrm {Cl}}(o(z',\tilde {g}_k))}$ , so by the first fact, $\pi _k(Z_1') = Z$ . We now show $Z_1'$ is minimal under $\tilde {g}_k$ . If not, there is a $y'\in Z_1'$ with ${\operatorname {\mathrm {Cl}}(o(y',\tilde {g}_k))} {\subsetneq } {\operatorname {\mathrm {Cl}}(o(z',\tilde {g}_k))}$ . By the second preliminary fact, there is some p with

$$ \begin{align*}\operatorname{\mathrm{Cl}}(o(T_k^p(z'), \tilde{g}_k)) \subset {\operatorname{\mathrm{Cl}}(o(y',\tilde{g}_k))} {\subsetneq} {\operatorname{\mathrm{Cl}}(o(z',\tilde{g}_k))}.\end{align*} $$

Acting by the homeomorphism $T_k^p$ and iterating the strict inclusions, we have

$$ \begin{align*} {\operatorname{\mathrm{Cl}}(o(z',\tilde{g}_k))} &= \operatorname{\mathrm{Cl}}(o(T_k^{pk}(z'), \tilde{g}_k)) {\subsetneq} \operatorname{\mathrm{Cl}}(o(T_k^{p(k-1)}(z'), \tilde{g}_k)) {\subsetneq} \cdots {\subsetneq} \operatorname{\mathrm{Cl}}(o(T_k^p(z'), \tilde{g}_k))\\ &{\subsetneq}\, {\operatorname{\mathrm{Cl}}(o(z',\tilde{g}_k))}, \end{align*} $$

a contradiction, so $\tilde {g}_k$ acting on $Z_1'$ is minimal. Thus since $\tilde {g}_k T_k = T_k \tilde {g}_k$ , $\tilde {g}_k$ acting on each $Z_j' := \tilde {g}_k^j(Z_1')$ is minimal. Now minimal sets either coincide or are disjoint, so there is a least m with $T_k^{m+1}Z_1' = Z_1'$ .

For the proof of (b), assume $\pi _k(Z') = \pi _k(Z")$ . Now $Z:= \pi _k(Z')$ is minimal under g and thus since $Z' \subset \pi _k^{-1}(Z)$ and minimal sets are always disjoint or equal, using (2.1) we have that $Z' = Z_j'$ for some j. Similarly, $Z" = Z_{j'}^{\prime }$ for some $j'$ , and thus $Z' = T^p{Z_1'}$ and $Z" = T^{p'}{Z_1'}$ , and so $Z" = T^{p'-p}(Z')$ as required.

Turning to (c), since the deck group of $S_k$ is ${\mathbb Z}_k$ there is a natural identification of $\pi _k^{-1}(x)$ with ${\mathbb Z}_k$ , with $x_1'$ identified with zero. Since $\tilde {g}_k^q({\tilde {x}}) = T^p{\tilde {x}}$ in ${\mathbb R}$ the induced action of $\tilde {g}_k^q$ on ${\mathbb Z}_k$ is $n \mapsto n+p\,\mathrm {mod}\ k$ . An easy elementary number theory argument yields that this action has exactly $\gcd (p,k)$ distinct orbits. Thus $\tilde {g}_k^q$ has exactly $\gcd (p,k)$ distinct orbits when acting on $\pi ^{-1}_k(x)$ . But $x_i', x_j'\in \pi ^{-1}_k(x)$ are on the same $\tilde {g}_k^q$ orbit if and only if they are on the same $\tilde {g}_k$ orbit and each orbit in $\pi ^{-1}_k(o(x,g))$ contains at least one point from $\pi ^{-1}_k(x)$ . Thus $\tilde {g}_k$ acting on $\pi ^{-1}_k(o(x,g))$ has exactly $\gcd (p,k)$ orbits. The rest of the form of (2.2) follows from part (a).

While it is not used here, a similar result holds for ${\mathbb Z}$ -covers and their cyclic quotients when the map on the base has a lift that commutes with the deck transformations.

Remark 2.2. Some special cases of (c) are worth noting. If $\gcd (p,k) = 1$ , then x lifts to a single period- $qk$ orbit in $S_k$ . If $p=k$ , then x lifts to k different period-q orbits in $S_k$ . When $k=2$ , there is a simple dichotomy. When p is odd, x lifts to one period- $2q$ orbit; when p is even, x lifts to a pair of period-q orbits.

2.3 Rotation number and interval

For $\tilde {g}:{\mathbb R}\rightarrow {\mathbb R}$ a fixed lift of a degree-one $g:S^1\rightarrow ~S^1$ , define the rotation number of $x'\in {\mathbb R}$ as

(2.3) $$ \begin{align} \rho(x', \tilde{g}) = \lim_{n\rightarrow\infty} \frac{\tilde{g}^n(x')-x'}{n} \end{align} $$

when the limit exists. Note that this value depend in a simple way on the choice of lift $\tilde {g}$ of g, namely, $\rho (x', \tilde {g} + m) = \rho (x', \tilde {g}) + m$ . In most cases below there will be a preferred lift of a given g that will be used in (2.3) and we define $\rho (x, g)=\rho (x', \tilde {g})$ where $x'$ is a lift of x. When g is understood we will just write $\rho (x)$ . If x is a periodic point of rotation type $(p,q)$ then $\rho (x) = p/q$ .

If Z is a g-invariant set, let

$$ \begin{align*} \rho(Z) = \{ \rho(x, g) : x\in Z\} \end{align*} $$

and $\rho (g) = \rho (S^1, g)$ . The latter set has been proved to be a closed interval [28, 39] and thus it is called the rotation interval of the map. We shall also have occasion to use $\rho (\tilde {g})$ with the obvious meaning.

There is a alternative way of computing the rotation interval using upper and lower maps that is now standard ([10, 18, 29, 37] and elsewhere). Given a lift of a degree-one circle map $\tilde {g}:{\mathbb R}\rightarrow {\mathbb R}$ , let $\tilde {g}_u(x) = \sup \{\tilde {g}(y)\colon y \leq x\}$ and $\tilde {g}_\ell (x) = \inf \{\tilde {g}(y)\colon y \geq x\}$ . If $g_u$ and $g_\ell $ are their descents to $S^1$ they are both semi-monotone maps and so each of their rotation sets is a single point (see Lemma 3.1 below). The rotation interval of g is

(2.4) $$ \begin{align} \rho(g) = [\rho(g_\ell), \rho(g_u)]. \end{align} $$

To define the rotation number of a g-invariant Borel probability measure $\mu $ , start by letting $\Delta _g:S^1\rightarrow {\mathbb R}$ be $\Delta _g(x) = \tilde {g}(x') - x'$ which is independent of the choice of lift $x'$ . Then

(2.5) $$ \begin{align} \rho(\mu) = \int \Delta_g\; d\mu. \end{align} $$

Note that when $\mu $ is ergodic by the pointwise ergodic theorem for $\mu $ -almost every x, $\rho (x,g) = \rho (\mu )$ .

For points, invariant sets and measures in the cyclic cover $S_k$ under the preferred lift $\tilde {g}_k$ , there are two ways to consider the rotation number. The most common will be to project to the base and define

(2.6) $$ \begin{align} \rho_k(x, g) = \rho(\pi_k(x), g). \end{align} $$

For $\mu $ a $\tilde {g}_k$ -invariant measure, let

(2.7) $$ \begin{align} \rho_k(\mu) = \rho((\pi_k)_* \mu ). \end{align} $$

Remark 2.3. The other way to work with rotation numbers in $S_k$ is to consider $\tilde {g}_k$ as a map of the circle itself. To work on the standard circle we first rescale $S_k$ via $D_k:S_k\rightarrow S^1$ via $D_k(\theta ) = \theta /k$ . Note that $D_k$ is not a covering map but rather a coordinate rescaling homeomorphism. For $Z\subset S_k$ , then $\rho (D_k Z, D_k\circ \tilde {g}_k\circ D_k^{-1})$ is the desired rotation number. These two methods are related simply by $\rho _k(x, g) = k\rho (D_k Z, D_k\circ \tilde {g}_k\circ D_k^{-1})$ .

3 Semi-monotone degree-one maps

3.1 Definition and basic properties

In this section we describe the basics of a small but crucial expansion of the class of circle homeomorphisms, namely, continuous maps whose lifts are semi-monotone. They share many of the properties of circle homeomorphisms and are a standard and important tool in circle dynamics.

Thus we consider continuous, degree-one $h:S^1 \rightarrow S^1$ whose lifts ${\tilde {h}}$ to ${\mathbb R}$ satisfy that $x_1' <x_2'$ implies ${\tilde {h}}(x_1') \leq {\tilde {h}}(x_2')$ . Note that this is independent of the choice of lift ${\tilde {h}}$ of h. We shall also call such maps weakly order-preserving. Note that in topology a monotone map is one with connected point inverses. In this sense a semi-monotone map ismonotone. On the other hand, considering the point of view of order relations, semi-monotone is contrasted with monotone. We adapt the latter viewpoint. Let ${\mathcal {H}}$ be the collection of all such maps with the $C^0$ -topology, and let $\tilde {{\mathcal {H}}}$ denote all their lifts.

A flat spot for a $h\in {\mathcal {H}}$ is a non-trivial closed interval J where $h(J)$ is a constant and for which there is no larger interval containing J on which h is constant. A given h can have at most a countable number of flat spots $J_i$ and we define the ‘positive-slope region’ of h as $P(h) = S^1\setminus (\cup \operatorname {\mathrm {Int}}{J_i})$ . The proof of the next result is standard. For the irrational case, see [4].

Lemma 3.1. Assume $h\in {\mathcal {H}}$ with preferred lift ${\tilde {h}}$ .

1. (a) The rotation number $\rho (x, h)$ exists and is the same for all $x\in S^1$ and so $\rho (h)$ is a single number.

2. (b) The map $\rho :\tilde {{\mathcal {H}}}\rightarrow {\mathbb R}$ is continuous.

3. (c) If ${\tilde {h}},{\tilde {h}}_1\in \tilde {{\mathcal {H}}}$ and ${\tilde {h}}_1 \leq {\tilde {h}}$ then $\rho ({\tilde {h}}_1)\leq \rho ({\tilde {h}})$ .

4. (d) If $\rho (h) = p/q$ in lowest terms then all periodic orbits have rotation type $(p,q)$ and the recurrent set of h consists of a union of such periodic orbits.

5. (e) If $\rho (h)= \alpha \not \in {\mathbb Q}$ then h has exactly one recurrent set which is a minimal set Z and it is wholly contained in $P(h)$ . Further, h is uniquely ergodic with the unique invariant measure supported on Z.

Definition 3.2. The minimal set in (e) above is called a semi-Denjoy minimal set. More generally, an abstract minimal set is called semi-Denjoy if it is topologically conjugate to the semi-Denjoy minimal set in a semi-monotone degree-one circle map.

Remark 3.3. A semi-Denjoy minimal set looks like a usual Denjoy minimal set, with the added feature that endpoints of gaps can collapse to a point under forward iteration. It is clear that any $h\in {\mathcal {H}}$ is a near-homeomorphism (the uniform limit of homeomorphisms). Thus, following from a theorem of Brown [15], the inverse limit ${\varprojlim } (h, S^1)$ is a circle and the natural extension is a circle homeomorphism. In particular, the inverse limit of a semi-Denjoy minimal set is a Denjoy minimal set. For example, in case of a single flat spot, the two endpoints of the flat spot form a gap in the minimal set and they have same forward orbit. Taking the inverse limit splits open this orbit into a forward invariant gap.

3.2 Finitely many flat spots

We next introduce a subclass of ${\mathcal {H}}$ which includes the semi-monotone maps considered in this paper. Let ${\mathcal {H}}_\ell $ consist of those $h\in {\mathcal {H}}$ that have exactly $\ell $ flat spots. In $P(h)$ we require that h is $C^1$ and $h'>1$ , where we have used a one-sided derivative at the end points of the flat spots.

Definition 3.4. Assume $h\in {\mathcal {H}}_\ell $ and $\rho (h)\not \in {\mathbb Q}$ . Thus h has a semi-Denjoy minimal set Z, and since $Z\subset P(h)$ , for any flat spot J, $Z\cap \operatorname {\mathrm {Int}}(J) = \emptyset $ . The flat spot J is called tight for h if $\operatorname {\mathrm {Fr}}(J)\subset Z$ , and otherwise the flat spot is called loose.

Lemma 3.5. Assume $h\in {\mathcal {H}}_\ell $ .

1. (a) If $\rho (h) = p/q$ in lowest terms then the number of $(p,q)$ -periodic orbits wholly contained in $P(h)$ is at least one and at most $\ell $ .

2. (b) If $\rho (h) \not \in {\mathbb Q}$ , a flat spot $J_i$ is loose if and only if there are an $n>0$ and an $i'$ with $h^n(J_i) \in J_{i'}$ . In particular, there is always at least one tight flat spot.

3. (c) If Z is the maximal recurrent set of h in $P(h)$ , then

   (3.1) $$ \begin{align} Z = S^1\setminus \bigcup_{n=0}^\infty\bigcup_{i=1}^\ell h^{-n}(\operatorname{\mathrm{Int}}(J_i)), \end{align} $$
   and so if $o(x,h)\subset P(h)$ then $h^n(x)\in Z $ for some $n\geq 0$ .

Proof. For part (a), since $\rho (h) = p/q$ in lowest terms, every periodic point has period q. By the conditions on the derivatives of $h\in {\mathcal {H}}_\ell $ , there are four classes of periodic points.

1. (1) x is unstable with $Dh^q(x)>1$ and $o(x,h)\subset \operatorname {\mathrm {Int}}(P(h))$ .

2. (2) x is superstable with $Dh^q(x) =0$ and $o(x, h) \cap (\bigcup _{i=1}^\ell \operatorname {\mathrm {Int}}(J_i)) \not = \emptyset $ while $o(x, h) \cap (\bigcup _{i=1}^\ell \operatorname {\mathrm {Fr}}(J_i)) = \emptyset $ .

3. (3) x is superstable with $Dh^q(x) =0$ and $o(x, h) \cap (\bigcup _{i=1}^\ell \operatorname {\mathrm {Int}}(J_i)) \not = \emptyset $ while $o(x, h) \cap (\bigcup _{i=1}^\ell \operatorname {\mathrm {Fr}}(J_i)) \not = \emptyset $ and $o(x,h)$ contains both left and right endpoints of flat spots.

4. (4) x is semi-stable with $Dh^q(x) =0$ from one side and $Dh^q(x)>1$ from the other and $o(x, h) \cap (\bigcup _{i=1}^\ell \operatorname {\mathrm {Int}}(J_i)) = \emptyset $ while $o(x, h) \cap (\bigcup _{i=1}^\ell \operatorname {\mathrm {Fr}}(J_i)) \not = \emptyset $ and $o(x,h)$ contains only left or only right endpoints of flat spots.

This implies that all periodic points are isolated so there are finitely many of them.

For $i=1, \ldots , 4$ let $n_i$ be the number of periodic orbits of type (1). Using the fixed point index on $h^q$ , $n_1 = n_2 + n_3$ . Each orbit of type (3) hits two flat spots and each of type (2) and (4) at least one and a flat spot cannot contain multiple periodic orbits, and so $n_1 + 2 n_3 + n_4 \leq \ell $ . Thus the total number of periodic orbits wholly contained on $P(h)$ is $n_1 + n_3 + n_4 = n_2 + 2 n_3 + n_4 \leq \ell $

For part (b) assume first that $h^n(J_i) \cap J_{i'} = \emptyset $ for all $n>0$ and $i'$ . If $J_i$ was loose, there would exist $z_1, z_2\in Z$ with $z_1\leq J_i \leq z_2$ with at least one inequality strict and $Z\cap (z_1, z_2) = \emptyset $ . Thus $h((z_1, z_2))$ is a non-trivial interval with $h^n((z_1, z_2)) \subset P(h)$ for all $n>0$ . This is impossible since h is expanding in $P(h)$ and so $J_i$ must not be loose.

For the converse, say $h^n(J_i) \in J_{i'}$ for some $n>0$ and first note $i=i'$ is impossible since that would imply h has a periodic point. Since $h^n(J_i)$ is a point there exists a non-trivial interval $[z_1, z_2]$ properly containing $J_i$ with $h^n([z_1, z_2]) = J_{i'}$ and so $(z_1, z_2)\cap Z = \emptyset $ , and so $J_i$ is a loose flat spot.

Finally, since $h^n(J_i) \cap J_{i} = \emptyset $ and there are finitely many flat spots there is at least one $J_{i}$ with $h^n(J_i) \cap J_{i'} = \emptyset $ for all $n>0$ and $i'$ .

For (c), assume y is such that $o(y,h)\subset P(h)$ and $o(y,h)\cap Z = \emptyset $ . Let $x,x'\in Z$ with $y\in (x, x')$ and $(x, x')\cap Z = \emptyset $ . Because of the uniform expansion in $P(h)$ there are a flat spot J and an $n\geq 0$ so that $J\subset h^n([x, x'])$ . If $\rho (h)\not \in {\mathbb Q}$ , then by (c) for some $n'$ , $h^{n+n'}([x, x'])$ is a tight flat spot and so $h^{n + n' + 1}(y) \in Z$ , where Z is the semi-Denjoy minimal set given in Lemma 3.1(c).

Now assume $\rho (h) = p/q$ . In this case x and $x'$ are periodic orbits and so $J \subset h^{n+w q}([x, x'])$ for all $w\geq 0$ , and so $ h^{n + wq} (y) \in h^{n+w q}([x, x'])\setminus \operatorname {\mathrm {Int}}(J)$ using the assumption that $o(y,h)\subset P(h)$ . But from (a), $ h^{n + wq} (J) \subset J$ . Thus by monotonicity, $ h^{n + wq} (y)$ is either always in the left component of $[x, x'] \setminus \operatorname {\mathrm {Int}}(J)$ or in the right component. This violates the expansion in $P(h)$ and so, for some $j> 0$ , $h^j(y)\in Z$ , which yields (3.1).

4 A class of bimodal circle maps and their positive-slope orbits

4.1 The class ${\mathcal {G}}$

We introduce the class of bimodal, degree-one maps of the circle that will be the focus here. The class is defined using properties of their lifts. We say that a lift $\tilde {g}:{\mathbb R}\rightarrow {\mathbb R}$ is piecewise smooth if it is continuous and there are $0\leq x_0 \leq \cdots \leq x_n \leq 1$ so that $\tilde {g}$ is $C^2$ in each interval $(x_i, x_{i+1})$ and the right- and left-hand derivatives exist at each $x_i$ .

Definition 4.1. Let $\tilde {{\mathcal {G}}}$ be the class of all $\tilde {g}:{\mathbb R}\rightarrow {\mathbb R}$ with the following properties.

1. (a) $\tilde {g}$ is piecewise smooth and $\tilde {g}(x'+1) = \tilde {g}(x')+1$ .

2. (b) There are a pair of points $0 = {x_{\min }} < {x_{\max }} < 1$ so that $\tilde {g}'>1$ in $[{x_{\min }}, {x_{\max }}]$ and $\tilde {g}$ is monotone decreasing in $[{x_{\max }}, {x_{\min }} + 1]$ .

3. (c) ${x_{\min }} \leq \tilde {g}({x_{\min }}) < \tilde {g}({x_{\max }}) \leq {x_{\max }} + 1$ .

The class ${\mathcal {G}}$ consists of all $g:S^1\rightarrow S^1$ which have a lift in $\tilde {{\mathcal {G}}}$ .

Note that without loss of generality we have assumed that ${x_{\min }} = 0$ . Also by assumption, ${x_{\min }}$ and ${x_{\max }}$ are a non-smooth local minimum and maximum, respectively. It follows from (2.4) that $g\in {\mathcal {G}}$ implies $\rho (g)\subset [0,1]$ .

Standing assumption. From this point on g denotes a given element of ${\mathcal {G}}$ and its preferred lift is the one with $\tilde {g}\in \tilde {{\mathcal {G}}}$ .

Remark 4.2. Assume h is a general bimodal map with $\rho (h)\subset (0,1)$ and $\rho (h)$ not equal to a single point. The Parry–Milnor–Thurston theorem for degree-one circle maps yields a semi-conjugacy to a map $g\in {\mathcal {G}}$ which is PL^†. Point inverses of the semi-conjugacy are either points or a closed interval. Thus, using standard results from one-dimensional dynamics and under various hypotheses, most of the results of the paper can be transferred with appropriate alterations to a general bimodal map.

4.2 The model map

We will use a model map $f_m$ as a specific example throughout the paper. We shall see that, in a sense, it is the largest map in the class ${\mathcal {G}}$ and all other maps $g\in {\mathcal {G}}$ may be considered subsystems.

Define $\tilde {f}_m:{\mathbb R}\rightarrow {\mathbb R}$ on $[0,1]$ as

$$ \begin{align*} \tilde{f}_m(x) = \begin{cases} 3 x & \text{for}\ 0 \leq x \leq 1/2,\\ -x + 2 & \text{for}\ 1/2 \leq x \leq 1, \end{cases} \end{align*} $$

and extend it to ${\mathbb R}$ to satisfy $\tilde {f}_m(x+1) = \tilde {f}_m(x) + 1$ . Let $f_m$ be the projection of $\tilde {f}_m$ to $S^1$ . See Figure 2. Thus, ${x_{\min }} = 0$ , ${x_{\max }} = 1/2$ , and $\rho (f_m) = [0,1]$ .

Figure 2 The model map $f_m$ in the $3$ -fold cover.

4.3 Positive-slope orbits

Given $g\in {\mathcal {G}}$ with preferred lift $\tilde {g}$ , let $\Lambda _\infty (g)$ be the points $x'\in {\mathbb R}$ whose orbits under $\tilde {g}$ stay in the closed region where $\tilde {g}$ has positive slope, so

$$ \begin{align*} \Lambda_\infty(g) = \left\{x'\in {\mathbb R} : o(x',\tilde{g})\subset \bigcup_{j=-\infty}^\infty [j, j+{x_{\max}}]\right\}\hspace{-2pt}. \end{align*} $$

We give $\Lambda _\infty (g)$ the total order coming from its embedding in ${\mathbb R}$ . Note that it is both $\tilde {g}$ - and T-invariant.

Now we treat the k-fold cover as $S_k = [0,k]/\mkern -3mu\sim $ and let $\Lambda _k(g)$ be the orbits that stay in the positive-slope region of $\tilde {g}_k:S_k\rightarrow S_k$ , so

$$ \begin{align*} \Lambda_k(g)= \left\{x'\in S_k : o(x',\tilde{g}_k)\subset \bigcup_{j=0}^{k-1} [j, j+{x_{\max}}]\right\}\!. \end{align*} $$

Alternatively, $\Lambda _k(g) = p_k(\Lambda _\infty (g))$ or $\Lambda _k(g) = \pi _k^{-1}(\Lambda _1(g))$ .

We discuss the restriction to positive-slope orbits in §13.2.

Standing assumption. Unless otherwise specified, the terminology ‘physical kfsm set’ or just ‘kfsm set’ carries the additional restriction that it is contained in the positive-slope region of some $g\in {\mathcal {G}}$ .

5 Symbolic description of positive-slope orbits

For a map $g\in {\mathcal {G}}$ we develop in this section a symbolic coding for the orbits in $\Lambda _k$ for $k = 1, \ldots , \infty $ .

5.1 The itinerary maps

We work first in the universal cover or $k=\infty $ . Since $g\in {\mathcal {G}}$ , we may find points ${z_{\max }}$ and ${z_{\min }}$ with $0 = {x_{\min }} < {z_{\max }} < {z_{\min }} < {x_{\max }}$ and $\tilde {g}({z_{\max }}) = {x_{\max }}$ and $\tilde {g}({z_{\min }}) = {x_{\min }} + 1$ . For $j\in {\mathbb Z}$ define a collection of intervals $\{I_j\}$ on ${\mathbb R}$ by

(5.1) $$ \begin{align} \begin{aligned} I_{2j} &= [j, {z_{\max}} + j], \\ I_{2j + 1} &= [{z_{\min}} + j, {x_{\max}} + j]. \end{aligned} \end{align} $$

See Figure 3. Note that since $\tilde {g}([{z_{\max }}, {z_{\min }}]) = [{x_{\max }}, {x_{\max }}+1]$ we have that

$$ \begin{align*} \Lambda_\infty(g) = \left\{x'\in {\mathbb R}: o(x',\tilde{g})\subset \bigcup_{j=-\infty}^\infty I_j\right\}\hspace{-2pt}. \end{align*} $$

Using $\{I_j\}$ as an address system with the dynamics $\tilde {g}$ , let the itinerary map be $\iota _\infty :\Lambda _\infty (g) \rightarrow \Sigma _{\mathbb Z}^+$ . Note that $\Lambda _\infty $ is the good set and, using expansion and the disjointness of the address intervals, $\iota _\infty $ is a homeomorphism onto its image.

Figure 3 The address intervals in the $2$ -fold cover.

Now passing to the k-fold cover, to code the positive-slope orbits $\Lambda _k(g)$ , treat $S_k = [0,k]/\mkern -3mu\sim $ and use the dynamics $\tilde {g}_k$ with the address system $\{I_0, I_1, \ldots , I_{2k-2}, I_{2k-1}\}$ . This yields an itinerary map $\iota _k:\Lambda _k\rightarrow \Sigma _{2k}^+$ which is also a homeomorphism onto its image.

Example: The model map. For the model map $f_m$ we have ${z_{\max }} = 1/6$ and ${z_{\min }} = 1/3$ and so $I_{2j} = [j, 1/6 + j]$ and $I_{2j + 1} = [1/3 + j, 1/2 + j]$

5.2 Symbolic analogs of covering spaces

This section develops the necessary machinery for the complete description of the image of the various itinerary maps. We will need the symbolic analogs of the covering spaces and maps described in §2.2.

Definition 5.1. Define a subshift $\Omega _\infty \subset \Sigma _{\mathbb Z}^+$ by its allowable transitions

(5.2) $$ \begin{align} 2j \rightarrow 2j, \ 2j \rightarrow 2j+1, \ 2j+1\rightarrow 2j + 2, \ 2j + 1 \rightarrow 2j+3. \end{align} $$

For $k <\infty $ let $\Omega _k$ be the subshift of $\Sigma _{2k}^+$ with allowable transitions as in (5.2) for $j = 0, \ldots , 2k-1$ and indices reduced $\mathrm {mod}\ 2k$ .

Since for $g\in {\mathcal {G}}$ we have $\tilde {g}(I_{2j})\subset I_{2j}\cup I_{2j+ 1}$ and $\tilde {g}(I_{2j+1})\subset I_{2j+2}\cup I_{2j+ 3}$ , we have the following lemma.

Lemma 5.2. For $g\in {\mathcal {G}}$ and $k = 1, \ldots , \infty $ , $\iota _k(\Lambda _k(g))\subset \Omega _k$ .

Under the itinerary maps the spaces ${\mathbb R}$ , $S_k$ , and $S^1$ will correspond to the shift spaces $\Omega _\infty $ , $\Omega _k$ , and $\Omega _1 = \Sigma _{2}^+$ . The dynamics on the ‘physical spaces’ induced by g will correspond to left shifts on the symbol spaces. The shift spaces will also have the analogs of the covering projections and deck transformations. These maps will be indicated by a hat and defined using the action on individual symbols as follows.

The analogs of the generator of the group of covering translations are ${\hat {T}}_\infty :\Omega _\infty \rightarrow \Omega _\infty $ given by $s\mapsto s+2$ for all $s\in {\mathbb Z}$ and ${\hat {T}}_k:\Omega _k \rightarrow \Omega _k$ given by $s\mapsto s+2\,\mathrm {mod}\ 2k $ for all $s\in {\mathbb Z}$ , while the analogs of the covering maps are ${\hat {p}}_k:\Omega _\infty \rightarrow \Omega _k$ by $s\mapsto s\,\mathrm {mod}\ 2k$ and ${\hat {\pi }}_k:\Omega _k\rightarrow \Sigma _{2}^+$ by $s\mapsto s\,\mathrm {mod}\ 2$ . In the latter we allow $k=\infty $ under the convention that $2\infty = {\mathbb Z}$ , yielding ${\hat {\pi }}_\infty :\Omega _\infty \rightarrow \Sigma _{2}^+$ . Note then that ${\hat {\pi }}_\infty = {\hat {p}}_1$ . A lift and the full lift are defined as usual with, for example, a lift of $Y\subset \Omega _1 = \Sigma _2^+$ to $\Omega _k$ being a set $Y'\subset \Omega _k$ with ${\hat {\pi }}_k(Y') = Y$ . Note that ${\hat {T}}_k, {\hat {\pi }}_k$ , and $p_k$ are all continuous.

The roles of the maps g, $\tilde {g}_k$ , and $\tilde {g}$ in §2.2 are played by the various shift maps on the sequence spaces. For clarity we use a subscript to indicate which space the shift is acting on: $\sigma _k:\Omega _k\rightarrow \Omega _k$ . We again allow $k=\infty $ . All the various maps satisfy the same commutativity relations as their unhatted analogs. So, for example, ${\hat {\pi }}_k{\hat {T}}_k = \pi _k$ , $\sigma _k {\hat {T}}_k = {\hat {T}}_k \sigma _k$ , and ${\hat {\pi }}_k\sigma _k = \sigma _1 {\hat {\pi }}_k$ . The itinerary maps $\iota _k:\Lambda _k(g)\rightarrow \Omega _k $ act naturally by transforming the spaces and maps of §2.2 to their symbolic analogs as in part (b) of the next lemma.

Lemma 5.3. For $k = 1, \ldots , \infty $ , the following assertions hold.

1. (a) $\Omega _k = {\hat {p}}_k(\Omega _\infty )$ .

2. (b) ${\hat {\pi }}_k \iota _k = \iota _1 \pi _k$ .

3. (c) If ${\underline s},{\underline t}\in \Omega _k$ and ${\hat {\pi }}_k({\underline s}) = {\hat {\pi }}_k({\underline t})$ , then there exists an n with ${\underline s} = {\hat {T}}_k^n{\underline t}$ .

Proof. Parts (a) and (b) are easy to verify. For (c) we prove the case $k=\infty $ which implies the $k < \infty $ cases. Assume ${\hat {\pi }}_\infty ({\underline s}) = {\underline w}$ . The transitions in (5.2) coupled with the structure of ${\underline w}$ imply that once $s_0$ is determined the parity structure of ${\underline s}$ determines all of ${\underline w}$ . Similarly, once $t_0$ is determined all of ${\underline t}$ is determined. Once again (5.2) implies that if $s_0 - t_0 = 2n$ then for all i, $s_i = t_i + 2n$ .

Remark 5.4. It would perhaps seem more natural that $\Sigma ^+_{\mathbb Z}$ should act as the symbolic universal cover of $\Sigma _2^+$ , but the crucial covering space property expressed by (c) would not hold in this case. For example, if ${\underline s} =.1 3 1^\infty $ and ${\underline t} = .1 5 1^\infty $ then ${\hat {\pi }}_\infty ({\underline s}) = {\hat {\pi }}_\infty ({\underline t})$ but ${\hat {T}}^n({\underline s}) \not = {\underline t} $ for all n.

5.3 Rotation numbers and sets

We give the analogs of the definitions in §2.3 for the symbolic case. For ${\underline s}\in \Sigma _2^+$ let

(5.3) $$ \begin{align} {\hat{\rho}}({\underline s}) = \lim_{n\rightarrow\infty} \frac{1}{n+1} \sum_{i=0}^n s_i \end{align} $$

when the limit exists. For ${\hat {\mu }}$ a shift-invariant measure on $\Sigma _2^+$ , let ${\hat {\rho }}({\hat {\mu }}) = {\hat {\mu }}([1])$ . When ${\hat {\mu }}$ is ergodic, by the pointwise ergodic theorem, for ${\hat {\mu }}$ -almost every ${\underline s}$ , ${\hat {\rho }}({\underline s}) = {\hat {\rho }}({\hat {\mu }})$

For ${\hat {Z}}\subset \Omega _k$ let ${\hat {\rho }}_k({\hat {Z}}) = {\hat {\rho }}({\hat {\pi }}_k({\hat {Z}}))$ , and for ${\hat {\mu }}$ a $\sigma _k$ -invariant measure on $\Omega _k$ let ${\hat {\rho }}_k({\hat {\mu }}) = {\hat {\rho }}(({\hat {\pi }}_k)_*({\hat {\mu }}))$ .

6 Topological conjugacies and the image of the itinerary maps

In this section we develop the analog of kneading invariants for the symbolic coding of the positive-slope orbits for $g\in {\mathcal {G}}$ .

Recall that $\Sigma _2^+$ is given the lexicographic order. Assume ${\underline \kappa }_0, {\underline \kappa }_1\in \Sigma _2^+$ satisfy

(6.1) $$ \begin{align} {\underline \kappa}_0 \leq o({\underline \kappa}_i, \sigma) \leq {\underline \kappa}_1 \end{align} $$

for $i = 0,1$ . The corresponding dynamical order interval is

$$ \begin{align*}\langle {\underline \kappa}_0, {\underline \kappa}_1 \rangle = \{ {\underline s}\colon {\underline \kappa}_0 \leq o({\underline s}, \sigma) \leq {\underline \kappa}_1\}. \end{align*} $$

Returning to $g\in {\mathcal {G}}$ , note that $\tilde {g}(I_0) \subset [{x_{\min }}, {x_{\max }}]$ and $\tilde {g}(I_1) \subset [{x_{\min }}, {x_{\max }}]$ , while $\tilde {g}([{z_{\max }}, {z_{\min }}]) = [{x_{\max }}, {x_{\min }} + 1]$ . This implies that $\Lambda _1(g) \subset [{x_{\min }}, {x_{\max }}]$ . Since $\Lambda _1(g)$ is compact we may define ${\underline \kappa }_0 = {\underline \kappa }_0(g) =\iota _1(\min (\Lambda _1))$ and ${\underline \kappa }_1 = {\underline \kappa }_1(g) = \iota _1(\max (\Lambda _1))$ . By construction these ${\underline \kappa }$ s satisfy (6.1).

We showed above that $\iota _k(\Lambda _k(g))\subset \Omega _k$ . The next theorem says that the image is constrained by the dynamical order interval $\langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ . Accordingly, for $k=1, \ldots , \infty $ we define ${\hat {\Lambda }}_k(g) = \Omega _k \cap {\hat {\pi }}_k^{-1}(\langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle )$ and note that this is a $\sigma _k$ -invariant set.

Theorem 6.1. Assume $g\in {\mathcal {G}}$ and construct $\kappa _0$ and $\kappa _1$ from g as above. Then for $k = 1, \ldots , \infty $ the itinerary map $\iota _k$ is a topological conjugacy from $(\Lambda _k(g), (\tilde {g}_k)_{\vert \Lambda _k(g)})$ to $({\hat {\Lambda }}_k(g), \sigma _k)$ . Further, $\iota _\infty $ is order-preserving.

Proof. We first prove the first assertion for $k=1$ or that $\iota _1(\Lambda _1(g)) = \langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ . Let $\ast $ be an arbitrary symbol and define a map $\chi :[0,{x_{\max }}]\sqcup \{\ast \}\rightarrow [0,{x_{\max }}]\sqcup \{\ast \}$ by

$$ \begin{align*} \chi(x) = \begin{cases} \tilde{g}(x) & \text{for} \ x\in I_0,\\ \ast\ & \text{for} \ x\in ({z_{\min}}, {z_{\max}})\sqcup\{\ast\},\\ \tilde{g}(x) -1 & \text{for} \ x\in I_1. \end{cases} \end{align*} $$

It easily follows that

$$ \begin{align*} \Lambda_1(g) = \{x\in [0, {x_{\max}}]\colon \chi^n(x)\not= \ast \ \text{for all} \ n>0\}, \end{align*} $$

and if we use the dynamics of $\chi $ with the address system $I_0, I_1$ the resulting itinerary map $\Lambda _1\rightarrow \Sigma _2^+$ is exactly $\iota _1$ . Now since g is expanding on $I_0 \cup I_1$ and $I_0 \cap I_1 = \emptyset $ , $\iota _1$ is an order-preserving conjugacy from $(\Lambda _1,g)$ to $(\iota _1(\Lambda _1),\sigma _1)$ . Finally, since $\min \Lambda _1(g) \leq o(x,g) \leq \max \Lambda _1(g)$ for all $x\in \Lambda _1(g)$ we have that $\kappa _0 \leq o({\underline s},\sigma ) \leq \kappa _1$ for all ${\underline s}\in \iota _1(\Lambda _1)$ , and further that for any such ${\underline s}$ there is an $x\in \Lambda _1(g)$ with $\iota _1(x) = {\underline s}$ . Thus $\iota _1(\Lambda _1(g)) = \langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ .

We now show that

(6.2) $$ \begin{align} \iota_k(\Lambda_k(g)) = \Omega_k \cap {\hat{\pi}}_k^{-1}(\langle {\underline \kappa}_0, {\underline \kappa}_1 \rangle). \end{align} $$

We already know from Lemma 5.2 that the left-hand side is in $\Omega _k$ . Next, since $\pi _k(\Lambda _k(g)) = \Lambda _1(g)$ using Lemma 5.3(b) and the first paragraph of the proof, we have

(6.3) $$ \begin{align} \langle {\underline \kappa}_0, {\underline \kappa}_1 \rangle = \iota_1(\Lambda_1(g)) = \iota_1(\pi_k(\Lambda_k(g))) = {\hat{\pi}}_k\iota_k(\Lambda_k(g)) \end{align} $$

so the left-hand side of (6.2) is also in ${\hat {\pi }}_k^{-1}(\langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle )$ .

Now assume that ${\underline s}$ is in the right-hand side of (6.2). Certainly then ${\hat {\pi }}_k({\underline s})\in \langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ and so there is an $x\in \Lambda _1(g)$ with $\iota _1(x) = {\hat {\pi }}_k({\underline s})$ . Pick a lift $x'\in \Lambda _k(g)$ with $\pi _k(x') = x$ . Again using Lemma 5.3(b),

(6.4) $$ \begin{align} {\hat{\pi}}_k({\underline s}) = \iota_1(x) = \iota_1\pi_k(x') = {\hat{\pi}}_k\iota_k(x'). \end{align} $$

Thus, using Lemma 5.3(c), there is an n with $\iota _k(x') = {\hat {T}}_k^n({\underline s})$ and so

$$ \begin{align*} \iota_k{\hat{T}}_k^{-n} x' = {\hat{T}}_k^{-n}\iota_k x' = {\underline s} \end{align*} $$

and $ {\hat {T}}_k^{-n} x'\in \Lambda _k$ . Thus ${\underline s}\in \iota _k(\Lambda _k)$ as required.

For $\iota _\infty $ as with $\iota _1$ , since the $I_j$ are disjoint and the $\tilde {g}_{\vert I_j}$ are expanding, we have that $\iota _\infty $ is an order-preserving homeomorphism onto its image. The fact that it is a semi-conjugacy follows because it is an itinerary map.

Example. (The model map)

For the model map $f_m$ we have $\kappa _0 = .0^\infty $ and $\kappa _1 = .1^\infty $ and so in this case ${\hat {\Lambda }}_k(\langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle )$ is the entire subshift $\Omega _k$ .

Remark 6.2

1. (a) ${\hat {\rho }}\circ \iota _k = \rho $ (when defined) and ${\hat {\rho }}_k\circ \iota _k = \rho _k$

2. (b) When $\mu $ a g-invariant measure supported in $\Lambda _1(g)$ , we have $\rho (\mu ) = \mu (I_1)$ .

7 k-fold semi-monotone sets

While our eventual interest is in invariant sets in the circle, it is convenient to first give definitions in the universal cover ${\mathbb R}$ and the cyclic covers $S_k$ .

7.1 Definitions

The next definition makes sense for any degree-one map, but for concreteness we restrict to $g\in {\mathcal {G}}$ .

Definition 7.1. Let $g\in {\mathcal {G}}$ have preferred lift $\tilde {g}:{\mathbb R}\rightarrow {\mathbb R}$ .

1. (a) A $\tilde {g}$ -invariant set $Z'\subset {\mathbb R}$ is kfsm if $T^k(Z') = Z'$ and $\tilde {g}$ restricted to $Z'$ is weakly order-preserving, or for $z_1', z_2'\in Z'$ ,

   $$ \begin{align*} z_1' < z_2' \quad \text{implies} \quad \tilde{g}(z_1') \leq \tilde{g}(z_2') \end{align*} $$

2. (b) A $\tilde {g}_k$ -invariant set $Z\subset S_k$ is kfsm if it has a $\tilde {g}$ -invariant lift $Z'\subset {\mathbb R}$ which is.

These definitions are independent of the choice of lift $\tilde {g}$ . Note that the same terminology is used for sets in the universal and cyclic covers and that implicit in being a kfsm set is the fact that the set is invariant.

When $k=1$ the lift $Z'$ in the definition must satisfy $T(Z') = Z'$ and $\pi (Z') = Z$ and so $Z' = \pi ^{-1}(Z)$ , the full lift to ${\mathbb R}$ .

7.2 Interpolation

To say that $Z\subset S_k$ is kfsm means roughly that it is semi-monotone, treating $S_k$ as the usual circle. To formalize this as in Remark 2.3 it will be useful to rescale $S_k$ to $S^1$ using $D_k:S_k\rightarrow S^1$ and consider the map $ D_k\circ \tilde {g}_k\circ D_k^{-1}$ .

Lemma 7.2. The following assertions are equivalent.

1. (a) The $\tilde {g}_k$ -invariant set $Z\subset S_k$ is kfsm.

2. (b) $D_k Z$ is $1$ -fold semi-monotone under $ D_k\circ \tilde {g}_k\circ D_k^{-1}$ and there exists a semi-monotone circle map h defined on $S_k$ which interpolates $\tilde {g}_k$ acting on Z.

3. (c) The lift $Z'\subset {\mathbb R}$ of Z in Definition 7.1(b) has the property that there is a continuous $H:{\mathbb R}\rightarrow {\mathbb R}$ that interpolates $\tilde {g}$ acting on ${\tilde {Z}}^*$ , is weakly order-preserving, and satisfies $H(x + k) = H(x) + k$ .

We now restrict to positive-slope orbits as in §4.3 and collect together kfsm invariant sets in $S_k$ and their invariant measures. We will comment on kfsm sets which intersect the negative-slope region in §13.2. We also restrict attention to invariant sets that are recurrent.

Definition 7.3. Given $g\in {\mathcal {G}}$ , let ${\mathcal {B}}_k(g)$ be all compact, recurrent kfsm sets in $\Lambda _k(g)\subset S_k$ with the Hausdorff topology and ${\mathcal {N}}_k(g)$ be all $\tilde {g}_k$ -invariant, Borel probability measures whose support is a $Z\in {\mathcal {B}}_k(g)$ with the weak topology.

Remark 7.4. A standard argument from Aubry–Mather theory yields that the collection of all kfsm sets is compact in the Hausdorff topology. Since $\Lambda _k(g)$ is compact, the collection of positive-slope kfsm sets is also compact. However, since ${\mathcal {B}}_k(g)$ contains just the recurrent kfsm sets, it is not compact (see §§9.1 and 13.2). We show shortly that ${\mathcal {N}}_k(g)$ is compact.

7.3 Symbolic k-fold semi-monotone sets and the map g

As with kfsm sets in the ‘physical’ spaces $S_k$ and ${\mathbb R}$ , we define their symbolic analogs in the symbol spaces $\Omega _k$ and $\Omega _\infty $ where we give the symbol spaces the lexicographic order.

Definition 7.5

1. (1) A $\sigma _\infty $ -invariant set ${\hat {Z}}'\subset \Omega _\infty $ is symbolic k-fold semi-monotone if ${\hat {T}}_\infty ^k({\hat {Z}}') = {\hat {Z}}'$ and $\sigma _\infty $ restricted to ${\hat {Z}}'$ is weakly order-preserving, or for ${\underline s}, {\underline t}\in {\hat {Z}}'$ ,

   $$ \begin{align*} {\underline s} < {\underline t} \quad \text{implies} \quad \sigma_\infty({\underline s}) \leq \sigma_\infty({\underline t}). \end{align*} $$

2. (2) A $\sigma _k$ -invariant set ${\hat {Z}}\subset \Omega _k$ is symbolic k-fold semi-monotone if there is a $\sigma _\infty $ -invariant lift ${\hat {Z}}'$ to $\Omega _\infty $ (that is, ${\hat {p}}_k({\hat {Z}}') = {\hat {Z}}$ ) which is kfsm.

Everything has been organized thus far to ensure that k-fold semi-monotone sets are preserved under the itinerary maps.

Theorem 7.6. Given $g\in {\mathcal {G}}$ , for $k = 1, 2, \ldots , \infty $ , a $\tilde {g}_k$ -invariant set $Z\subset \Lambda _k(g)$ is kfsm if and only if $\iota _k(Z)\subset {\hat {\Lambda }}_k(g)$ is.

Proof. We prove the $k=\infty $ case; the $k<\infty $ case follows. Theorem 6.1 shows that $\iota _\infty $ is an order-preserving bijection. Since $\iota _\infty T_\infty ^k = {\hat {T}}_\infty ^k \iota _\infty $ , we have that $T_\infty ^k(Z) = Z$ if and only if ${\hat {T}}^k_\infty \iota _\infty (Z) = \iota _\infty (Z)$ . Using the additional fact that $\iota _\infty \tilde {g} = \sigma _\infty \iota _\infty $ we have that $\tilde {g}$ is weakly order-preserving on Z if and only if $\sigma _\infty $ is weakly order-preserving on $\iota _\infty (Z)$

In analogy with Definition 7.3 we collect together the various symbolic kfsm sets and their invariant measures.

Definition 7.7. For $k<\infty $ , given $g\in {\mathcal {G}}$ , let ${\hat {{\mathcal {B}}}}_k(g)$ be all compact, invariant, recurrent symbolic kfsm sets in ${\hat {\Lambda }}_k(g)$ with the Hausdorff topology and ${\hat {{\mathcal {N}}}}_k(g)$ be all g-invariant, Borel probability measures with the weak topology whose support is a ${\hat {Z}}\in {\hat {{\mathcal {B}}}}_k(g)$ .

Lemma 7.8. For $k < \infty $ , the following assertions hold.

1. (a) The map $\iota _k:\Lambda _k(g) \rightarrow {\hat {\Lambda }}_k(g)$ induces homeomorphisms ${\mathcal {B}}_k(g)\rightarrow {\hat {{\mathcal {B}}}}_k(g)$ and ${\mathcal {N}}_k(g)\rightarrow {\hat {{\mathcal {N}}}}_k(g)$ .

2. (b) The spaces ${\mathcal {N}}_k(g)$ and ${\hat {{\mathcal {N}}}}_k(g)$ are compact.

Proof. For part (a) we know that $\iota _k$ is a conjugacy that takes kfsm sets to kfsm sets, which yields that ${\mathcal {N}}_k(g)\rightarrow {\hat {{\mathcal {N}}}}_k(g)$ is a homeomorphism. By hypothesis any $g\in {\mathcal {G}}$ is $C^2$ in $P(g)$ and so there is some $M>1$ with $g'< M$ on $P(g)$ and thus on all address intervals $I_j$ . It is standard that this implies that $\iota _k$ is Hölder with exponent $\nu = \log 2k/\log M$ . This then implies that $\iota _k$ preserves Hausdorff convergence and so ${\mathcal {B}}_k(g)\rightarrow {\hat {{\mathcal {B}}}}_k(g)$ is a homeomorphism.

For part (b), since the space of all $\tilde {g}_k$ -invariant Borel probability measures is compact metric, it suffices to show that ${\mathcal {N}}_k(g)$ is closed, and so assume $\mu _n\in {\mathcal {N}}_k(g)$ and $\mu _n\rightarrow \mu $ weakly with $X_n := \operatorname {\mathrm {supp}}(\mu _n)$ a recurrent kfsm set.

As noted in Remark 7.4 the collection of all kfsm sets in $\Lambda _k$ is compact in the Hausdorff topology and so there exist a kfsm set X and $n_i\rightarrow \infty $ with $X_{n_i}\rightarrow X$ . A standard argument which we give here shows that $\operatorname {\mathrm {supp}}(\mu ) \subset X$ . If this inclusion does not hold, there exists an $x\in \operatorname {\mathrm {supp}}(\mu ) \cap X^c$ , then let $\epsilon = d(x, X)$ . Since the atoms of $\mu $ are countable, we may find an $\epsilon _1 < \epsilon /4$ so that, letting $U = N_{\epsilon _1}(x)$ , we have that $\mu (\operatorname {\mathrm {Fr}}(U)) = 0$ . Thus U is a continuity set for $\mu $ . It then follows, using the fact that $x\in \operatorname {\mathrm {supp}}(\mu )$ and a standard result (pp. 16–17 of [7]), that $\mu _{n_i}(U) \rightarrow \mu (U)>0$ . Thus for large enough i, with $m = n_i$ we have $X_m\subset N_{\epsilon /4}(X)$ and so $\emptyset = U \cap X_m = U\cap \operatorname {\mathrm {supp}}(\mu _m)$ with $\mu _m(U)>0$ a contradiction. Thus $\operatorname {\mathrm {supp}}(\mu ) \subset X$ . Now any invariant measure supported on X must be supported on its recurrent set and so $\mu \in {\mathcal {N}}_k(g)$ , as required. The compactness of ${\hat {{\mathcal {N}}}}_k(g)$ follows from part(a).

Example: The model map. For the model map $f_m$ , ${\hat {\Lambda }}_k(f_m) = \Omega _k$ , and so the set ${\hat {{\mathcal {B}}}}_k(f_m)$ is the collection of all symbolic recurrent kfsm sets in $\Omega _k$ . Thus while the definition of symbolic kfsm set is abstract and general by Theorems 7.6 and 6.1, symbolic kfsm sets share all the properties of ‘physical’ kfsm sets.

7.4 Rotation numbers and sets

For $Z\in {\mathcal {B}}_k(g)$ recall from §2.3 that $\rho _k(Z) = \rho (\pi _k(Z), g)$ .

Lemma 7.9. Assume $Z\in {\mathcal {B}}_k(g)$ . Then the following assertions hold.

1. (a) $\rho _k(Z)$ exists and is a single number.

2. (b) If $\rho _k(Z) = \omega \not \in {\mathbb Q}$ then Z is a semi-Denjoy minimal set.

3. (c) If $\rho _k(Z) = p/q$ with $\gcd (p,q) = 1$ , then Z consists of at least one and at most k periodic orbits all with the same rotation number and period equal to $q k/\gcd (p,k)$ .

4. (d) $\rho _k:{\mathcal {B}}_k(g)\rightarrow {\mathbb R}$ and ${\hat {\rho }}_k:{\hat {{\mathcal {B}}}}_k(g)\rightarrow {\mathbb R}$ are continuous

Proof. By Theorem 7.2 there exists a continuous, semi-monotone $H:S_k\rightarrow S_k$ which interpolates the action of $\tilde {g}_k$ on Z. Rescaling to the standard circle, let $H_k:S^1\rightarrow S^1$ be defined as $H_k:= D_k\circ H\circ D_k^{-1}$ . By Lemma 3.1(a), $\rho (H_k) = \omega $ is a single number and, since $\rho _k(Z) = k \rho (D Z, H_k)$ , (a) follows. If $\rho _k(Z)\not \in {\mathbb Q}$ then $\rho (D Z, H_k)\not \in {\mathbb Q}$ and so, by Lemma 3.1(e), $D_kZ$ and thus Z is a semi-Denjoy minimal set, yielding (b).

Now assume $\rho _k(Z) = p/q$ in lowest terms and so $\rho (D_k Z, H_k) = p/(qk)$ . Written in lowest terms,

$$ \begin{align*}\frac{p}{qk} = \frac{p/\gcd(p,k)}{kq/\gcd(p,k)}.\end{align*} $$

But since $H_k$ is semi-monotone, its recurrent set is a collection of periodic orbits and its rotation number in lowest terms has their period as its denominator which is thus $q k{/}\kern-1pt\gcd (p,k)$ . Since by assumption, $Z\kern1pt{\subset}\kern1pt \Lambda _k\kern-0.5pt(g)$ we may choose H to have k flat spots. Thus, using Lemma 3.5, Z consists of at least one and at most k periodic orbits, finishing (c).

It is standard from Aubry–Mather theory that $\rho $ is continuous on the collection of all kfsm sets and thus it is continuous restricted to the recurrent kfsm sets. As for measures, since $\rho (\mu ) = \int \Delta _g\; d\mu $ using definition (2.5) with $\Delta _g$ continuous, continuity of ${\hat {\rho }}_k$ follows from the definition of weak convergence.

Definition 7.10. If $Z\in {\mathcal {B}}_k$ and it consists of a finite collection of periodic orbits, then it is called a cluster.

Remark 7.11

1. (a) For the case of general recurrent symbolic kfsm ${\hat {Z}}$ , as we commented at the end of the last subsection, we may consider ${\hat {Z}}\in {\hat {\Lambda }}_k(f_m) = \Omega _k$ with $f_m$ the model map. Using the itinerary map $\iota _k:\Lambda _k(f_m) \rightarrow {\hat {\Lambda }}_k(f_m)$ we have from Theorem 7.6 that $(\iota _k)^{-1}({\hat {Z}})$ is kfsm for $f_m$ and then all the conclusions of the previous theorem hold for it. Then using Theorem 6.1, the conclusions of the previous theorem hold with the obvious addition of hats in the appropriate places.

2. (b) We shall need this implication of the symbolic case below. If ${\hat {Z}}\subset \Omega _k$ with $\rho _k({\hat {Z}}) = \alpha \not \in {\mathbb Q}$ , then there exists a continuous, onto $\phi :{\hat {Z}}\rightarrow S_k$ which is weakly order-preserving, $\phi \sigma _k = R_\alpha \phi $ , and $\#\phi ^{-1}(x) = 1$ for all but a countable number of $R_\alpha $ -orbits on which $\#\phi ^{-1}(x) = 2$ .

3. (c) Using Lemma 3.1, a measure in ${\mathcal {N}}_k(g)$ is either the unique measure on a semi-Denjoy minimal set or a convex combination of measures supported on the periodic orbits in a cluster.

4. (d) A $Z\kern0.5pt{\in}\kern0.5pt {\mathcal {B}}_k\kern-0.7pt(g\kern-0.7pt)$ is minimal if and only if it is uniquely ergodic, and similarly for $Z\kern0.5pt{\in}\kern0.5pt {\hat {{\mathcal {B}}}}_k\kern-0.7pt(g\kern-0.7pt)$

8 The HM construction

At this point for a given $g\in {\mathcal {G}}$ we have reduced the identification of its positive-slope kfsm sets to a question in symbolic dynamics. In this section we answer this symbolic question via a generalization of the HM procedure. The generalization constructs all symbolic kfsm recurrent sets for each k.

Since a linear order is essential to the notion of semi-monotonicity we will again begin working on the line and then project to cyclic covers.

8.1 Definition and basic properties

Fix an integer $k>0$ , a real number $\omega \in (0,1)$ , and a vector $\vec {\nu } = (\nu _1, \ldots , \nu _k)$ with $\nu _i\geq 0$ and $\sum \nu _i =k -k\omega $ . Such a pair $(\omega , \vec {\nu })$ is called allowable. Start with the intervals defined for $0 \leq j \le k-1$ by

(8.1) $$ \begin{align} \begin{aligned} X_{2j} &= \bigg(\sum_{i=1}^j \nu_i + j\omega, \sum_{i=1}^{j+1} \nu_i + j\omega\bigg),\\ X_{2j+1} &= \bigg(\sum_{i=1}^{j+1} \nu_i + j\omega, \sum_{i=1}^{j+1} \nu_i + (j+1)\omega\bigg), \end{aligned} \end{align} $$

and then extend for $\ell \in {\mathbb Z}$ and $0 \leq m \le 2k-1$ as $X_{\ell k + m} = X_m + \ell k$ . Thus each $X_{2j}$ has width $\nu _{j+1}$ and each $X_{2j+1}$ has width $\omega $ and the entire structure yields a $T^k$ -invariant address system under the dynamics $R_\omega (x) = x + \omega $ on ${\mathbb R}$

The good set G depends on k, $\omega $ , and $\vec {\nu }$ and is given by

$$ \begin{align*} G = \{ x'\in {\mathbb R}: o(x', R_\omega)\cap \partial X_i = \emptyset \text{ for all}\ i\}. \end{align*} $$

Note that G is dense, $G_\delta $ and has full Lebesgue measure. The itinerary map with respect to the given address system is denoted by $\zeta _\infty :G\rightarrow \Sigma ^+_{{\mathbb Z}}$ .

Definition 8.1. Let $A_{k}(\omega ,\vec {\nu }) = \operatorname {\mathrm {Cl}}(\zeta _\infty (G))$ .

Remark 8.2. By construction, $A_{k}(\omega ,\vec {\nu })$ is $\sigma _\infty $ - and ${\hat {T}}_\infty ^k$ -invariant. In addition, since for all j, $R_\omega (X_{2j}) \subset X_{2j} \cup X_{2j+1} $ and $R_\omega (X_{2j+1}) \subset X_{2j+2} \cup X_{2j+3}$ , we have $\zeta _\infty (G_\infty )\subset \Omega _\infty $ .

8.2 Cyclic covers

We now return to the compact quotients where the recurrent dynamics takes place and introduce measures into the HM construction.

For fixed $k>0$ and allowable $(\omega , \vec {\nu })$ treat $\{X_0, \ldots , X_{2k-1}\}\subset S_k = [0,k]/\mkern -3mu\sim $ as an address system under the dynamics given by $R_\omega (x) = x + \omega \,\mathrm {mod}\ k$ . Define the good set $G_{k\omega \vec {\nu }}$ and on it the itinerary map $\zeta _{k\omega \vec {\nu }}$ . We will often suppress the dependence of these quantities on various of the subscripted variables when they are clear from the context.

Definition 8.3. Given k and an allowable $(\omega , \vec {\nu })$ , define the itinerary map $\zeta _k:G_k\rightarrow \Sigma ^+_{2k}$ as above. Let $B_{k}(\omega , \vec {\nu }) = \operatorname {\mathrm {Cl}}(\zeta _k(G_{k}))\subset \Sigma _{2k}^+$ and $\unicode{x3bb} _{k}(\omega , \vec {\nu }) = (\zeta _k)_*(m)/k$ where m is the measure on $S_k$ induced by Lebesgue measure on ${\mathbb R}$ .

Remark 8.4

1. (1) By construction, ${\hat {p}}_k(A_{ k}(\omega ,\vec {\nu })) = B_{k}(\omega , \vec {\nu })\subset \Omega _k$ , and so $\rho _k(B_k(\omega , \vec {\nu })) = \omega $ .

2. (2) Let $W_k = \{(x, \omega , \vec {\nu }) \colon x\in G_{k\omega \vec {\nu }}\}.$ It is easy to check that the map $(x,\omega , \vec {\nu })\mapsto \zeta _{k\omega \vec {\nu }}(x)$ is continuous on $W_k$ .

The next theorem describes the structure of the $B_k(\omega , \vec {\nu })$ and shows that all symbolic kfsm sets are constructed by the HM procedure with $\omega $ equal to their rotation number.

Theorem 8.5

1. (a) For $\alpha \not \in {\mathbb Q}$ , $B_k(\alpha , \vec {\nu })$ is a semi-Denjoy minimal set with unique invariant probability measure $\unicode{x3bb} _k(\omega ,\vec {\nu })$ .

2. (b) For $p/q\in {\mathbb Q}$ , $B_k(p/q, \vec {\nu })$ is a finite collection of periodic orbits each with rotation number $p/q$ and period $qk/\gcd (p,k)$ , and $\unicode{x3bb} _k(p/q, \vec {\nu })$ is a convex combination of the measures supported on the periodic orbits.

3. (c) A ${\hat {Z}}\subset \Omega _k$ is a recurrent symbolic kfsm set with $\rho _k(Z) = \omega $ if and only if ${\hat {Z}} = B_k(\omega , \vec {\nu })$ for some allowable $\vec {\nu }$ . Thus the collection of invariant probability measures supported on symbolic recurrent kfsm sets is exactly the collection of $\unicode{x3bb} _k(\omega , \vec {\nu })$ for all allowable $(\omega , \vec {\nu })$ .

Proof. We begin by proving portions of (a) and (b). For part (a) we first show that $B_k(\alpha , \vec {\nu })$ is minimal using a characterization usually attributed to Birkhoff. If $f:X\rightarrow X$ is a continuous function of a compact metric space and $x\in X$ , then $\operatorname {\mathrm {Cl}}(o(x,f))$ is a minimal set if and only if for all $\epsilon>0$ there exists an N so that for all $n\in {\mathbb N}$ there is a $0<i\leq N$ with $d(f^{n+i}, x) < \epsilon $ . Pick x in the good set G. Since $(S^1, R_\alpha )$ is minimal, $o(x, R_\alpha )$ has the given property. Since $\zeta _k$ restricted to G is a homeomorphism and $\zeta _k R_\alpha = \sigma _k \zeta _k$ , $o(\zeta _k(x), R_\alpha )$ has the desired property and, further, $o(\zeta _k(x), R_\alpha )$ is dense in $\zeta _k(G)$ and thus in $B_k(\alpha , \vec {\nu }) = \operatorname {\mathrm {Cl}}(\zeta _k(G))$ . Thus $B_k(\alpha , \vec {\nu })$ is minimal under $\sigma _k$ .

For part (b) note first that since $R_{p/q}$ is of finite order and there are finitely many address intervals, $B_k(p/q, \vec {\nu })$ must consist of finitely many periodic orbits. The other properties in (a) and (b) will follow from (c) (proved using just these two partial results on (a) and (b)) and Theorem 7.9 using Remark 7.11(a).

For part (c), we first show that $B_k(\omega , \vec {\nu })$ is a recurrent symbolic kfsm set. By parts (a) and (b) we know that $B_k(\omega , \vec {\nu })$ is recurrent, and by Remark 8.4 that $B_k(\omega , \vec {\nu })\subset \Omega _k$ and $\rho _k(B_k(\omega , \vec {\nu }) = \omega $ . We show that $B_k(\omega , \vec {\nu })$ is a symbolic kfsm set by showing that its full lift $A_k(\omega , \vec {\nu })$ to $\Omega _\infty $ is as required by Definition 8.1. As noted in Remark 2.2, ${\hat {T}}^k_\infty (A_{k}(\omega ,\vec {\nu })) = A_{k}(\omega ,\vec {\nu })$ so we need to show that ${\hat {\sigma }}_\infty $ is semi-monotone on $A_k(\omega , \vec {\nu })$ .

The first step is to show that $\zeta _\infty $ is weakly order-preserving. Assume $x_1', x_2'\in G$ with $x_1' < x_2'$ . It could happen (when $\omega $ is rational) that $\zeta _\infty (x_1') = \zeta _\infty (x_2')$ , but if there exists a least n with $(\zeta _\infty (x_1'))_n \not = (\zeta _\infty (x_2'))_n$ , then since $I_m < I_{m+1}$ for all m and $R_\omega $ is order-preserving, certainly $(\zeta _\infty (x_1'))_n < (\zeta _\infty (x_2'))_n$ , and so $\zeta _\infty $ is weakly order-preserving.

We now show that ${\hat {\sigma }}_\infty $ is semi-monotone on $A_k(\omega , \vec {\nu })$ . Let G be the good set for $\zeta _\infty $ and assume ${\underline s}, {\underline t}\in \zeta _\infty (G)$ with ${\underline s} < {\underline t}$ . Then there exist $x_1', x_2'\in G$ with $\zeta _\infty (x_1') = {\underline s}$ and $\zeta _\infty (x_2') = {\underline t}$ and, of necessity, $x_1' < x_2'$ and so $R_\omega (x_1') < R_\omega (x_2')$ . Since $\zeta _\infty R_\omega = \sigma _\infty \zeta _\infty $ , we have

$$ \begin{align*} \sigma_\infty({\underline s}) = \sigma_\infty \zeta_\infty(x_1') = \zeta_\infty R_\omega(x_1) \leq \zeta_\infty R_\omega(x_2) = \sigma_\infty \zeta_\infty(x_2') = \sigma_\infty({\underline t}). \end{align*} $$

Thus $\sigma _\infty $ is weakly order-preserving on $\zeta _\infty (G)$ and so on $A_k(\omega , \vec {\nu })$ . We have that $A_k(\omega , \vec {\nu })$ satisfies all the conditions of the lift in Definition 7.5 and thus $B_k(\omega , \vec {\nu })$ is symbolic kfsm.

Now for the converse assume that ${\hat {Z}}\subset \Omega _k$ is symbolic recurrent kfsm with ${\hat {\rho }}({\hat {Z}}) = \omega $ . Let ${\hat {Z}}'\subset \Omega _\infty $ be the lift that satisfies Definition 7.5. The proof splits into the two cases when $\omega $ is rational and irrational.

First assume $\omega = p/q$ with $\gcd (p,q) = 1$ . We know from Lemma 7.9 and Remark 7.11 that ${\hat {Z}}$ consists of at most k distinct periodic orbits each with period $kq/d$ with $d=\gcd (p,k)$ . We assume for simplicity that ${\hat {Z}}$ is a single periodic orbit. The case of multiple periodic orbits is similar but with more elaborate indexing.

For $i = 0, \ldots , kq/d-1$ let $P_i = (2 i + 1)d/2q\subset S_k$ and ${\mathcal {P}}= \{P_i\}$ . Since ${\hat {Z}}$ is a kfsm periodic orbit with $\rho _k$ -rotation number $p/q$ we may find an order-preserving bijection $\phi :{\hat {Z}}\rightarrow {\mathcal {P}}$ with $\phi \sigma _k = R_{p/q} \phi $ on ${\hat {Z}}$ . Thus $\phi \sigma _k\phi ^{-1}$ acts on ${\mathcal {P}}$ as $P_i \mapsto P_{i+p/d}$ reducing indices mod $kq/d$ .

For $j = 0, \ldots , k-1$ , let $X^{\prime }_j = \phi ({\hat {Z}} \cap [j])$ where recall that $[j]$ is the length-one cylinder set in $\Sigma ^+_{2k}$ . Since $\phi $ is order-preserving, each $X_j'$ consists of a collection of adjacent points from ${\mathcal {P}}$ . If $X_j' = \{ P_{n(j)}, \ldots , P_{m(j)}\} \not = \emptyset $ , let $X_j = [P_{n(j)} - d/(2q), P_{m(j)} + d/(2q)]$ and when $X_j'=\emptyset $ let $X_j=\emptyset $ . We now claim that $\{X_j\}$ is an address system as used in the HM construction where $\vec {\nu }$ is defined by $\nu _{j+1} = |X_{2j}|$ and that $|X_{2j+1}| = p/q$ for $j = 0, \ldots , k-1$ , yielding ${\hat {Z}}' = B_k(p/q, \vec {\nu })$ .

Letting $\zeta $ be the itinerary map for the address system $\{X_j\}$ , for ${\underline s}\in {\hat {Z}}$ we have by construction that $\zeta \phi ({\underline s}) = {\underline s}$ . In addition, for all $x\in [\phi ({\underline s})-d/(2q), \phi ({\underline s}) + d/(2q)]$ we also have $\zeta (x) = \zeta \phi ({\underline s}) = {\underline s}$ . Thus for any point x in the good set G, $\zeta (x) = {\underline s}$ for some ${\underline s}\in {\hat {Z}}$ . This shows that ${\hat {Z}} = \operatorname {\mathrm {Cl}}(\zeta (G))$ . The last step needed to show that ${\hat {Z}} = B_k(p/q, \vec {\nu })$ is to check that the address system is of the type used in the HM construction.

We need only check that $|X_{2j+1}| = p/q$ , and for this it suffices to show that $\# X_{2j+1} = p/d$ . Assume first that $\# X_{2j+1} < p/d$ . Recalling that $\phi \sigma _k \phi ^{-1}$ acts on the $X_i'$ like $i \mapsto i + p/d$ , we see that there will be some $P_m\in X_{2j}'$ and $P_{m+p/d}\in X_{2j+2}'$ . Thus using $\phi ^{-1}$ there is a ${\underline s}\in {\hat {Z}}$ with $s_0 = 2j$ and $s_1 = 2j+2$ , a contradiction to the fact that ${\hat {Z}}\subset \Omega _k$ and thus its allowable transitions are given by (5.1). On the other hand, if $\# X_{2j+1}> p/d$ we have some $P_m\in X_{2j+1}'$ and $P_{m+p}\in X_{2j+1}'$ , again yielding a contradiction to ${\hat {Z}}\subset \Omega _k$ .

The irrational case is basically a continuous version of the rational one. By Remark 7.11(b) we have a continuous, onto $\phi :{\hat {Z}}\rightarrow S_k$ which is weakly order-preserving, $\phi \sigma _\infty = R_\alpha \phi $ , and $\#\phi ^{-1}(x) = 1$ for all but a countable number of $R_\alpha $ -orbits on which $\#\phi ^{-1}(x) = 2$ .

For $j= 0, \ldots , k-1$ , let $X_j = \phi ([j])$ . Thus $X_j$ is a closed interval (perhaps empty) with $\cup X_j = {\mathbb R}$ , $X_j \leq X_{j+1}$ and adjacent intervals intersect only in their single common boundary point. We use $\{ X_j\}$ as an address system with dynamics $R_\alpha $ , good set G, and itinerary map $\zeta $ . By construction, if ${\underline s}\in {\hat {Z}}$ with $\phi ({\underline s})\in G$ , then ${\underline s} = \zeta \phi ({\underline s})$ and so $\phi ^{-1}(G) = \zeta \phi (\phi ^{-1}(G)) = \zeta (G)$ . Since ${\hat {Z}}$ is a Cantor set and $\phi ^{-1}(G)$ is ${\hat {Z}}$ minus a countable set of $\sigma _k$ -orbits we have that $\phi ^{-1}(G)$ is dense ${\hat {Z}}$ . Thus, taking closures, ${\hat {Z}} = \operatorname {\mathrm {Cl}}(\zeta (G))$ .

To finish we must show that $\{ X_j\}$ is the type of address system allowable in the HM construction. We just need $|X_{2j+1}| = \alpha $ for all j. The proof is similar to the rational case. If $|X_{2j+1}| < \alpha $ then ${\hat {Z}}$ has a transition $2j \rightarrow 2j+2$ , and if $|X_{2j+1}|> \alpha $ then ${\hat {Z}}$ has a transition $2j+1 \rightarrow 2j+1$ . Either is a contradiction to ${\hat {Z}}\subset \Omega _k$ . Thus letting $\nu _{j+1} = |X_{2j}|$ for $j = 0, \ldots , k-1$ , we have ${\hat {Z}} = \operatorname {\mathrm {Cl}}(\zeta (G)) = B_k(\alpha , \vec {\nu })$ .

The last sentence in (c) follows from the construction of $\unicode{x3bb} _k(\omega , \vec {\nu })$ .

Remark 8.6. In §9.3 below we shall see that for the irrational case $\rho ({\hat {Z}}) = \omega \not \in {\mathbb Q}$ there is a unique $\vec {\nu }$ with ${\hat {Z}} = B_k(\omega , \vec {\nu })$ and for rational $p/q$ there are, in general, many $\vec {\nu }$ with $Z = B_k(p/q, \vec {\nu })$ . But note that if ${\hat {Z}}$ is a single periodic orbit then the proof above produces what we show is the unique $\vec {\nu }$ with ${\hat {Z}} = B_k(p/q, \vec {\nu })$ .

9 Parameterization of ${\mathcal {B}}_k(g)$ and ${\mathcal {N}}_k(g)$ by the HM construction

We know from Theorem 8.5(c) that the HM construction yields a correspondence between sets $B_k(\omega , \vec {\nu })$ and symbolic kfsm sets in $\Omega _k$ . In addition, for a map $g\in {\mathcal {G}}$ , using Theorem 7.6, we get a bijection from kfsm sets in $\Lambda _k(g)$ to those in ${\hat {\Lambda }}_k(g)\subset \Omega _k$ . Thus the HM construction provides a parameterization of ${\mathcal {B}}_k(g)$ . In this section we examine this parameterization in detail as well as that of ${\mathcal {N}}_k(g)$ .

9.1 Resonance and holes

As remarked above, the collection of all kfsm sets is closed in the compact metric space consisting of all compact g-invariant sets with the Hausdorff topology. Thus the collection of all kfsm sets is complete. We have restricted attention here to recurrent kfsm sets or ${\mathcal {B}}_k(g)$ . This is because the recurrent ones are the most dynamically interesting and carry the invariant measures, but also, as shown in Theorem 8.5, they are what is parameterized by the HM construction. As a consequence our primary space of interest ${\mathcal {B}}_k(g)$ is not complete, but rather has holes at points to be specified. What happens roughly is that as one takes the Hausdorff limit of recurrent kfsm sets the resulting kfsm set can have homoclinic points that are not recurrent and so the limit is not recurrent and thus not any $B_k(\omega , \vec {\nu })$ . This is a phenomenon well known in Aubry–Mather theory. Another point of view on these ‘holes’ is given in §13.2 using the family of interpolated semi-monotone maps.

In the HM construction fix $0<k < \infty $ . For a given allowable $(\omega , \vec {\nu })$ , recall that the address intervals are $X_j = X_{j\vec {\nu }}$ for $j = 0, \ldots , 2k-1$ . Define $\ell _j = \ell _{j\vec {\nu }}$ and $r_j = r_{j\vec {\nu }}$ by $[\ell _j, r_j] := X_j$ . Note that $r_{j+1} = \ell _j$ with indices reduced mod $2k$ .

Definition 9.1. The pair $(\omega , \vec {\nu })$ is called resonant if $R_\omega ^n(\ell _j) = \ell _{j'}$ for some $n>1$ and $j,j'$ . A pair that is not resonant is called non-resonant.

Remark 9.2. Note that for a rational $\omega =p/q$ all $(p/q, \vec {\nu })$ are resonant, as are all $(\omega , \vec {\nu })$ when some $\nu _i = 0$ . Also, for all $(\omega , \vec {\nu })$ and j,

(9.1) $$ \begin{align} R_\omega(\ell_{2j-1}) = \ell_{2j}, \end{align} $$

which is the reason why n is restricted to $n>1$ in the definition.

The next lemma locates the ‘holes’ in the space of all symbolic kfsm sets and thus in any ${\hat {{\mathcal {B}}}}(g)$ .

Lemma 9.3

1. (a) Assume parameters $(\alpha , \vec {\nu })$ with $\alpha \not \in {\mathbb Q}$ resonant. There exist a sequence $\vec {\nu }^{(i)}\rightarrow \vec {\nu }$ and a non-recurrent kfsm Z with $B_k(\alpha , \vec {\nu }^{(i)})\rightarrow Z$ in the Hausdorff topology on all compact subsets of $\Sigma ^+_{2k}$ .

2. (b) Assume parameters $(p/q, \vec {\nu })$ with $p/q\in {\mathbb Q}$ . There exists a sequence $\omega ^{(i)}\rightarrow p/q$ and a non-recurrent kfsm Z with $B_k(\omega ^{(i)}, \vec {\nu })\rightarrow Z$ in the Hausdorff topology on all compact subsets of $\Sigma ^+_{2k}$ .

Proof. We suppress the dependence on k to simplify notation. For (a), the resonance hypothesis implies that there are odd a and b with $R_{\alpha }^n(X_{a\vec {\nu }}) = X_{b\vec {\nu }}$ for some $n>0$ where we may assume $a<b$ . Since $R_\alpha ^n(r_{a\vec {\nu }}) = r_{b\vec {\nu }}$ , by shrinking some $\nu _j$ for $a < j < b$ we obtain a $\vec {\nu }^{\;\prime }$ and $x < r_{a\vec {\nu }}$ and arbitrarily close to it with $x\in G_{\alpha \vec {\nu }^{\;\prime }}$ and $R_{\alpha }^n(x) \in X_{b+1, \vec {\nu }^{\;\prime }}$ . In this way we can obtain sequences $\vec {\nu }^{(i)}\rightarrow \vec {\nu }$ and $x_i\nearrow r_{a\vec {\nu }}$ with $x_i\in G_{\alpha , \vec {\nu }^{(i)}}$ and $R_{\alpha }^n(x_i) \in X_{b+1, \vec {\nu }^{(i)}}$ . Thus

$$ \begin{align*} \zeta_{\alpha,\vec{\nu}^{(i)}}(x_i) = .a\ldots (b+1) \zeta_{\alpha,\vec{\nu}^{(i)}}(R_{\alpha}^{n+1}(x_i). \end{align*} $$

To simplify matters, assume that $R_{\alpha }(r_{b\vec {\nu }}) \in G_{\alpha , \vec {\nu }}$ ; more complicated resonances are similar. Since $R_\alpha ^{n+1}(x_i)\rightarrow R_\alpha ^{n+1}(r_{a\vec {\nu }}) = R_{\alpha }(r_{b\vec {\nu }})$ using Remark 8.4(b),

$$ \begin{align*} \zeta_{\alpha,\vec{\nu}^{(i)}}(x_i) \rightarrow .a\ldots (b+1) \zeta_{\alpha,\vec{\nu}}(R_{\alpha}(r_{b\vec{\nu}}) := {\underline s}. \end{align*} $$

Passing to a subsequence if necessary, by the compactness of the collection of symbolic kfsm sets there is a kfsm Z with $B_k(\alpha , \vec {\nu }^{(i)})\rightarrow Z$ in the Hausdorff topology, and by its construction ${\underline s}\in Z$ . But ${\underline s}$ cannot be recurrent since by the resonance any length- $(n+1)$ block in $\zeta _{\alpha ,\vec {\nu }}(R_{\alpha } (r_{b\vec {\nu }})$ must start with a and end in b.

The argument for (b) is similar, but now the perturbation must be in the parameter $\omega $ . This is because if $\omega = p/q$ is fixed, then $R_{\omega }^n(X_a) = X_a$ with $n = qk/\gcd (p,k)$ for all $\vec {\nu }$ . Fix an a and so $R_{p/q}^n(r_a) = r_a$ . By increasing $\omega $ incrementally we may find sequences $\omega ^{(i)}\searrow p/q$ and $x_i\nearrow r_a$ with $x_i\in G_{\omega ^{(i)}, \vec {\nu }}$ so that the initial length- $(n+1)$ block of $\zeta _{\omega ^{(i)} \vec {\nu }}(x_i)$ is $a \ldots a+1$ . Thus if $\zeta _{p/q\vec {\nu }}(r_a + \epsilon ) = P^\infty $ for small $\epsilon $ then

$$ \begin{align*} \zeta_{\omega^{(i)},\vec{\nu}}(x_i) \rightarrow .a\ldots (a+1) P_2 P_3 \ldots P_{n-1} P^\infty := {\underline t} \end{align*} $$

where $P = (a+1) P_2 P_3 \ldots P_{n-1}$ . As in the proof of (a) passing to a subsequence if necessary, there is a kfsm Z with $B_k(\omega ^{(i)}, \vec {\nu })\rightarrow Z$ in the Hausdorff topology, and by its construction ${\underline t}\in Z$ . But ${\underline t}$ cannot be recurrent since any length- $(n+1)$ block in $P^\infty $ must start and end with a.

9.2 Continuity and injectivity

In the HM construction the explicit dependence of $A_k$ and $ B_k$ on the pair $(\omega , \vec {\nu })$ was included. However, note that the elements of the pair have the interdependence $\sum \nu _i = k(1-\omega )$ . Thus when we treat $A_k$ and $B_k$ as functions it is sometimes better to eliminate the interdependence and treat them as functions of $\vec {\nu }$ alone. Nonetheless, the two variable version will also continue to be useful. Thus we sometimes overload the function $A_k$ and write

$$ \begin{align*} A_k(\vec{\nu}) = A_k\left(1-\sum\nu_i/k, \vec{\nu}\right)\hspace{-2pt}, \end{align*} $$

and similarly for $B_k$ and the measure-valued map $\unicode{x3bb} _k$ . The collection of allowable parameters for each k is then

$$ \begin{align*} {\mathcal{D}}_k = \left\{ \vec{\nu}\in{\mathbb R}^k\colon \nu_i\geq 0, \sum_{i=1}^k\nu_i \leq k \right\}. \end{align*} $$

The set of HM parameters corresponding to symbolic kfsm sets for $g\in {\mathcal {G}}$ is defined as

$$ \begin{align*}\operatorname{\mathrm{HM}}_k(g) = \{\vec{\nu}\in{\mathcal{D}}_k\colon B_k( \vec{\nu})\subset {\hat{{\mathcal{B}}}}_k(g)\}.\end{align*} $$

Remark 9.4. By Theorem 8.5, $B_k:\operatorname {\mathrm {HM}}(g)\rightarrow {\hat {{\mathcal {B}}}}_k(g)$ is surjective and so $\iota _k^{-1}B_k:\operatorname {\mathrm {HM}}(g)\rightarrow {\mathcal {B}}_k(g)$ provides a parameterization of the positive-slope kfsm recurrent sets of $g\in {\mathcal {G}}$ and $(\iota _k^{-1})_* \unicode{x3bb} _k:\operatorname {\mathrm {HM}}(g)\rightarrow {\mathcal {N}}_k(g)$ their invariant measures.

Example: The model map. For the model map $f_m$ , $\operatorname {\mathrm {HM}}_k(f_m) = {\mathcal {D}}_k$ since ${\hat {\Lambda }}_k(f) = \Omega _k$ .

The first issue in what the HM construction tells us about ${\mathcal {B}}_k(g)$ and ${\mathcal {N}}_k(g)$ is to understand the nature of the maps $B_k$ and $\unicode{x3bb} _k$ . Lemma 9.3 showed that for $B_k$ there is an essential distinction between the resonance and non-resonance cases.

Theorem 9.5. Assume $g\in {\mathcal {G}}$ , for each $k>0$ ,

1. (a) The map $(\iota _k)^{-1}\circ B_k:\operatorname {\mathrm {HM}}_k(g)\rightarrow {\mathcal {B}}_k(g)$ is onto, and further it is continuous at non-resonant values and discontinuous at resonant values.

2. (b) The map $(\iota _k)^{-1}_*\circ \unicode{x3bb} _k:\operatorname {\mathrm {HM}}_k(g)\rightarrow {\mathcal {N}}_k(g)$ is a homeomorphism and thus $\operatorname {\mathrm {HM}}_k(g)$ is compact.

Proof. Since we know from Lemma 7.8 that $\iota _k$ and $(\iota _k)_*$ are homeomorphisms we only consider $B_k$ and $\unicode{x3bb} _k$ . While these are functions of $\vec {\nu }$ alone, for the proof it is clearer to resort to the two-variable versions with the proviso that $\omega = 1 - \sum \nu _i/k$ . Note that we have already shown in Theorem 8.5 that $\unicode{x3bb} _k$ and $B_k$ are onto ${\hat {{\mathcal {N}}}}_k(g)$ and ${\hat {{\mathcal {B}}}}_k(g)$ , respectively. We will often need to include the explicit dependence of various objects on the variables, for example, $\ell _j(\omega , \vec {\nu })$ , and we often suppress the dependence on k.

We prove (b) first. We first show that $\unicode{x3bb} _k$ is continuous. For each $j = 1, \ldots , 2k-1$ and $i\in {\mathbb N}$ , let $\ell _j^{(i)}(\omega , \vec {\nu }) = R_{\omega }^{-i}(\ell _j(\omega , \vec {\nu }))$ . The first observation from the HM construction is that

(9.2) $$ \begin{align} |\ell_j^{(i)}(\omega, \vec{\nu}) - \ell_j^{(i)}(\omega_0, \vec{\nu}_0)| \leq \|(\omega, \vec{\nu}) - (\omega_0, \vec{\nu}_0)\|_1. \end{align} $$

For a length-N block $B = b_0 \ldots b_{N-1}$ in $\Omega _k$ , let

$$ \begin{align*} Y_B(\omega, \vec{\nu}) = \bigcap_{i=0}^{N-1} R^{-i}_{\omega} (\operatorname{\mathrm{Int}}(X_{b_i}(\vec{\nu}))), \end{align*} $$

and so $x\in G_{\omega , \vec {\nu }} \cap Y_B(\omega , \vec {\nu })$ implies that $\zeta _{\omega \vec {\nu }} (x)$ begins with the block B. Also by the HM construction, $\unicode{x3bb} _k(\omega , \vec {\nu }) ([B]) = m( Y_B(\omega , \vec {\nu }))$ , with m being Lebesgue measure on the circle.

Recall that the weak topology on $\Sigma _{2k}^+$ is generated by the metric

$$ \begin{align*} d(\mu, \mu') = \sum_{i=1}^\infty \frac{|\mu([B_i]) - \mu'([B_i])|}{2^i} \end{align*} $$

where $\{B_i\}$ is some enumeration of the blocks in $\Sigma _{2k}^+$ . Since each $Y_B(\omega , \vec {\nu })$ is a (perhaps empty) interval with endpoints some $\ell _i^{(j)}(\omega , \vec {\nu })$ , inequality (9.2) implies that

$$ \begin{align*} |m(Y_B(\omega, \vec{\nu})) - m(Y_B(\omega_0, \vec{\nu}_0))| \leq 2 \|(\omega, \vec{\nu}) - (\omega_0, \vec{\nu}_0)\|_1. \end{align*} $$

Thus, summing over blocks,

$$ \begin{align*} d(\unicode{x3bb}_k(\omega, \vec{\nu}), \unicode{x3bb}_k(\omega_0, \vec{\nu}_0)| \leq 2 \|(\omega, \vec{\nu}) - (\omega_0, \vec{\nu}_0)\|_1, \end{align*} $$

so $\unicode{x3bb} _k$ is continuous.

Since by definition in the HM construction, $\unicode{x3bb} _k(\omega , \vec {\nu })([2j]) = \nu _{j+1}$ , $\unicode{x3bb} _k$ is injective. Recall now that for the model map, $\operatorname {\mathrm {HM}}(f_m) = {\mathcal {D}}_k$ which is compact. So $\unicode{x3bb} _k:\operatorname {\mathrm {HM}}(f_m)\rightarrow {\hat {{\mathcal {N}}}}_k(f_m)$ is a homeomorphism whose image is the set of all measures on recurrent symbolic kfsm sets in $\Omega _k$ . Thus, since $\operatorname {\mathrm {HM}}_k(g)\subset {\mathcal {D}}_k$ we have that $\unicode{x3bb} _k:\operatorname {\mathrm {HM}}_k(g)\rightarrow {\hat {{\mathcal {N}}}}_k(g)$ is also a homeomorphism. The compactness of ${\hat {{\mathcal {N}}}}_k(g)$ was proved in Lemma 7.8.

The proof of (a) is based on the following claim: $B_k$ is continuous at $(\omega _0, \vec {\nu }_0)$ if and only if for all N there exists $\delta>0$ so that $\|(\omega , \vec {\nu }) - (\omega _0, \vec {\nu }_0)\| < \delta $ implies that for all blocks B of length up to N we have $Y_B(\omega _0, \vec {\nu }_0)$ non-empty exactly when $Y_B(\omega , \vec {\nu })$ is non-empty.

To prove the claim, first note that continuity is equivalent to the following: given $\epsilon> 0$ , there exists $\delta>0$ so that $\| (\omega , \vec {\nu }) - (\omega _0, \nu _0)\| < \delta $ implies that for each ${\underline s}\in \zeta _{\omega \vec {\nu }}(G_{\omega \vec {\nu }})$ there is a ${\underline t}\in \zeta _{\omega _0\vec {\nu }_0}(G_{\omega _0 \vec {\nu }_0})$ with $d({\underline s}, {\underline t}) < \epsilon /2$ and for each ${\underline t}\in \zeta _{\omega _0 \vec {\nu }_0}(G_{\omega _0 \vec {\nu }_0})$ there is an ${\underline s}\in \zeta _{\omega \vec {\nu }}(G_{\omega \vec {\nu }})$ with $d({\underline s}, {\underline t}) < \epsilon /2$ . This implies that $\operatorname {\mathrm {HD}}(\zeta _{\omega \vec {\nu }}(G_{\omega \vec {\nu }}), \zeta _{\omega _0 \vec {\nu }_0}(G_{\omega _0 \vec {\nu }_0})) < \epsilon /2$ and thus $\operatorname {\mathrm {HD}}(B_k(\omega ,\vec {\nu }), B_k(\omega _0,\vec {\nu }_0)) < \epsilon $ . Since $d({\underline s}, {\underline t})$ is small exactly when ${\underline s}$ and ${\underline t}$ agree in a long prefix block B and $Y_B(\omega , \vec {\nu }) = \zeta _{\omega \vec {\nu }}^{-1}([B])$ , the claim follows.

We show that $B_k$ satisfies the condition in the claim when $(\omega , \vec {\nu })$ is non-resonant. Given N for $j = 1, \ldots , 2k-1$ and $i = 0, \ldots , N$ , consider again $\ell _j^{(i)}(\omega , \vec {\nu }) = R_{\omega }^{-i}(\ell _j(\omega , \vec {\nu })$ . By the HM construction we have $\ell _{2m}^{(n+1)}(\omega , \vec {\nu }) = \ell _{2m-1}^{(n)}(\omega , \vec {\nu })$ for all n, m, and $(\omega , \vec {\nu })$ . By non-resonance at $(\omega _0, \vec {\nu }_0)$ , all the other $\ell _j^{(i)}(\omega _0, \vec {\nu }_0)$ are disjoint. Since by (9.2) each $\ell _j^{(i)}(\omega , \vec {\nu })$ depends continuously on $(\omega , \vec {\nu })$ and the endpoints of each $Y_B(\omega , \vec {\nu })$ are some $\ell _j^{(i)}(\omega , \vec {\nu })$ , we may find a $\delta $ so that $\| (\omega , \vec {\nu }) - (\omega _0, \vec {\nu }_0)\| < \delta $ implies that the $\ell _j^{(i)}(\omega , \vec {\nu })$ are ordered around $S_k$ in the same way and with the same gaps between them as the $\ell _j^{(i)}(\omega _0, \vec {\nu }_0)$ . This implies that for each block B of length $B\leq N$ , $Y_B(\omega _0, \vec {\nu }_0)$ is non-empty exactly when $Y_B(\omega , \vec {\nu })$ is non-empty and so $B_k$ is continuous.

For the discontinuity, since the sets Z in Lemma 9.3(ab) are not recurrent, they are not equal to $B_k(\omega _0, \vec {\nu }_0)$ .

Remark 9.6

1. (a) The parameter space ${\mathcal {D}}_k$ is $(k-1)$ -dimensional. Assuming $\omega \not \in {\mathbb Q}$ , for a fixed $n>1$ and $j,j'$ , the collection of all $\vec {\nu }\in {\mathcal {D}}_k$ which yield $R_\omega ^n(\ell _j) = \ell _{j'}$ is a $(k-2)$ -dimensional affine subspace. Thus the set of resonance parameters is a countable dense collection of codimension- one affine subspaces and so the non-resonance case is a full measure and dense $G_\delta $ set.

2. (b) One can show that $B_k$ is lower semi-continuous [12]; in particular, if $(\omega ^{(i)}, \vec {\nu }^{(i)}) \rightarrow (\omega , \vec {\nu })$ and some subsequence of $B_k(\omega ^{(i)}, \vec {\nu }^{(i)})$ converges to Z in the Hausdorff topology, then $ B_k(\omega , \vec {\nu }) \subset Z$ . The semi-continuity lemma (see p. 114 of [19]) yields that a lower semi-continuous set-valued function is continuous on a dense $G_\delta $ set. In the case of $(\iota _k)^{-1}\circ B_k$ the last theorem exactly identifies this continuity set as the non-resonant $(\omega , \vec {\nu })$ .

9.3 Slices and skewness

Recall that the rotation number functions $\rho :{\mathcal {B}}_k(g)\rightarrow {\mathbb R}$ and ${\hat {\rho }}:{\mathcal {B}}_k(g)\rightarrow {\mathbb R}$ are continuous on the various spaces, as are their measure-theoretic analogs. Thus we may define closed slices with a given rotation number as follows.

Definition 9.7. For $g\in {\mathcal {G}}$ let ${\mathcal {B}}_{k\omega }(g) = \{Z\in {\mathcal {B}}_k(g)\colon \rho (Z) = \omega \}$ and ${\hat {{\mathcal {B}}}}_{k\omega }(g) = \{Z\in {\hat {{\mathcal {B}}}}_k(g)\colon {\hat {\rho }}(Z) = \omega \}$ ; the restriction of $\iota _k$ to ${\mathcal {B}}_{k\omega }(g)$ is denoted by $\iota _{k\omega }$ . The slices of invariant measures ${\mathcal {N}}_{k\omega }(g)$ and ${\hat {{\mathcal {N}}}}_{k\omega }(g)$ are defined similarly. The $\omega $ -slice of HM parameters is $\operatorname {\mathrm {HM}}_{k\omega }(g) = B_k^{-1}({\mathcal {B}}_{k\omega }(g)) = \unicode{x3bb} _k^{-1}({\mathcal {N}}_{k\omega }(g))$ .

Definition 9.8. For $p/q\in {\mathbb Q}$ an allowable parameter $\vec {\nu }$ is called pure if $B_k(p/q, \vec {\nu })$ consists of a single periodic orbit. The collection of $p/q$ pure parameters is denoted by $\operatorname {\mathrm {Pure}}_{k, p/q}\subset {\mathcal {D}}_{k,p/q}$ and it will be shown in Lemma 12.4 to be an affine lattice. For a $g\in {\mathcal {G}}$ its pure parameters are $ \operatorname {\mathrm {Pure}}_{k, p/q}(g) = \operatorname {\mathrm {HM}}_{k, p/q}(g)\cap \operatorname {\mathrm {Pure}}_{k, p/q}$ .

Remark 9.9. For a given symbolic kfsm $p/q$ -periodic orbit P, by Theorem 8.5(c) there is some $\vec {\nu }$ with $B_k(p/q, \vec {\nu }) = P$ . Since a periodic orbit is uniquely ergodic and $\unicode{x3bb} _k$ is injective this $\vec {\nu }$ is unique. Thus there is a bijection between symbolic kfsm $p/q$ -periodic orbits and $\operatorname {\mathrm {Pure}}_{k, p/q}$ .

Lemma 9.10. Assume $g\in {\mathcal {G}}$ . Then the following assertions hold.

1. (a) For all $\omega $ , $(\iota _{k\omega }^{-1})_*\circ \unicode{x3bb} _{k\omega }:\operatorname {\mathrm {HM}}_{k\omega }(g)\rightarrow {\mathcal {N}}_{k\omega }(g)$ is a homeomorphism.

2. (b) When $\alpha \not \in {\mathbb Q}$ , $\iota _{k\alpha }^{-1}\circ B_{k\alpha }:\operatorname {\mathrm {HM}}_{k\alpha }(g)\rightarrow {\mathcal {B}}_{k\alpha }(g)$ is injective as well as continuous at non-resonant $(\alpha , \vec {\nu })$ and discontinuous at resonant $(\alpha , \vec {\nu })$ .

3. (c) When $p/q\in {\mathbb Q}$ , $\iota _{k p/q}^{-1}B_{k p/q}:\operatorname {\mathrm {HM}}_{k p/q}(g)\rightarrow {\mathcal {B}}_{k p/q}(g)$ is injective on $\operatorname {\mathrm {Pure}}_{k, p/q}$ .

Proof. Since $\iota _k$ restricts to a homeomorphism on slices we only consider $B_{k\omega }$ and $\unicode{x3bb} _{k\omega }$ . Part (a) follows immediately from Theorem 9.5.

For (b), when $\alpha \not \in {\mathbb Q}$ the assignment of a semi-Denjoy kfsm set with rotation number $\alpha $ to its unique invariant measure yields a bijection ${\mathcal {B}}_{k\alpha }(g)\rightarrow {\mathcal {N}}_{k\alpha }(g)$ and ${\hat {{\mathcal {B}}}}_{k\alpha }(g)\rightarrow {\hat {{\mathcal {N}}}}_{k\alpha }(g)$ . Since, by (a), $\unicode{x3bb} _{k\alpha }$ is injective, we have that $B_{k\alpha }$ is also. Continuity of $B_{k\alpha }$ at non-resonant values on irrational slices follows directly from (a). Discontinuity at resonant values on irrational slices follows from Lemma 9.3(a).

For (c), when $p/q \in{\mathbb Q}$ the assignment of the single periodic orbit $B_k(p/q, {\vec{\nu}})$ to its unique invariant measure yields the injectivity using (a) as in the proof of (b).

Remark 9.11. Since ${\mathcal {B}}_{k p/q}(g)$ is a finite set, the continuity of $\iota _{k p/q}^{-1}B_{k p/q}:\operatorname {\mathrm {HM}}_{k p/q}(g)\rightarrow {\mathcal {B}}_{k p/q}(g)$ is not particularly interesting, but we will remark on it in §12.4.

The skewness $\gamma (\mu )$ of a $\tilde {g}_k$ -invariant measure in $S_k$ equals the amount of measure in each fundamental domain. When its jth component is large, its $\tilde {g}_k$ -orbits are moving slowly through $[j-1,j)$ . When we project to the base $S^1$ in the next section the skewness thus indicates how quickly orbits are moving in the jth loop of the kfsm set.

Definition 9.12. Assume $g\in {\mathcal {G}}$ . Then the following assertions hold.

1. (a) For $\eta \in {\mathcal {N}}_k(g)$ , $\gamma (\eta ) = (\eta ([0,1)), \eta ([1,2)), \ldots , \eta ([k-1,k)))$ .

2. (b) For ${\hat {\eta }}\in {\hat {{\mathcal {N}}}}_k(g)$ , ${\hat {\gamma }}(\eta ) = ({\hat {\eta }}([0]\cup [1]), {\hat {\eta }}([2]\cup [3]) \ldots , {\hat {\eta }}([2k-2]\cup [2k-1]))$ .

Note that the skewness takes values in the unit simplex $\sum a_i = 1$ , $a_i \geq 0$ , and contains no information about the rotation number.

Lemma 9.13. Assume $g\in {\mathcal {G}}$ . Then the following assertions hold.

1. (a) ${\hat {\gamma }} \circ (\iota _k)_* = \gamma $ .

2. (b) $\gamma (\unicode{x3bb} _k(\omega , \vec {\nu })) = (\omega + \nu _1, \omega + \nu _2, \ldots , \omega + \nu _k)/k$ .

3. (c) For $\eta \in {\mathcal {N}}_{k\omega }$ , is inverse to $(\iota _k)_*^{-1} \circ \unicode{x3bb} _k$ and so it is a homeomorphism.

4. (d) $\gamma $ is a homeomorphism from ${\mathcal {N}}_{k\omega }(g)$ onto its image. as is ${\hat {\gamma }}$ from ${\hat {{\mathcal {N}}}}_{k\omega }(g)$ onto its image.

Remark 9.14. The last lemma formalizes the description in the Introduction of the parametrization of the weak disks of semi-Denjoy minimal sets by their speed in each ‘loop’ around the circle. For rational pure parameters the skewness counts the number of elements in each fundamental domain and this thus yields a discrete parametrization of the kfsm $p/q$ -periodic orbits.

10 kfsm sets in $S^1$ and $\Omega _1$

10.1 In $S^1$

We now return to our central concern, g-invariant sets in $S^1$ that have a lift to $S_k$ that is semi-monotone. Once again the definition makes sense for any degree-one circle map, but we restrict to the class ${\mathcal {G}}$ .

Definition 10.1. Given $g\in {\mathcal {G}}$ , a compact g-invariant set $Z\subset S^1$ is kfsm if it has a $\tilde {g}$ -invariant lift $Z'\subset {\mathbb R}$ which is kfsm, or equivalently, Z has a $\tilde {g}_k$ -invariant lift $Z^*\subset S_k$ which is kfsm. Let ${\mathcal {C}}_k(g)$ be all compact, invariant, recurrent kfsm sets in $\Lambda _1(g)$ with the Hausdorff topology and ${\mathcal {O}}_k(g)$ be all g-invariant, Borel probability measures supported on $Z\in {\mathcal {C}}_k(g)$ with the weak topology

Thus when Z is kfsm, it has a lift to $S_k$ which is semi-monotone under the action of $\tilde {g}_k$ on its lift.

To make contact with the usual definitions in Aubry–Mather theory, assume that $x\in S^1$ is such that $o(x,f)$ is kfsm. This happens exactly when there is a point $x'\in {\mathbb R}$ with $\pi _\infty (x') = x$ and for all positive integers $\ell , m,n$ ,

$$ \begin{align*} \tilde{g}^\ell(x') < T^{k m} \tilde{g}^n(x')\quad \text{implies} \quad \tilde{g}^{\ell+1}(x') \leq T^m \tilde{g}^{n+1}(x') \end{align*} $$

In Aubry–Mather theory one would write $x_j = \tilde {g}^j(x')$ .

Remark 10.2

1. (a) $\pi _k:S_k\rightarrow S^1$ induces continuous onto maps ${\mathcal {B}}_k(g)\rightarrow {\mathcal {C}}_k(g)$ and ${\mathcal {N}}_k(g)\rightarrow {\mathcal {O}}_k(g)$ .

2. (b) $Z^*\subset S_k$ is kfsm if and only if $\pi _k(Z^*)\subset S^1$ is.

3. (c) If $Z\subset S^1$ is k-fold semi-monotone then it is also $\ell k$ -fold semi-monotone for any $\ell> 0$ .

4. (d) If P is a periodic orbit of g of type $(p,q)$ (which are perhaps not relatively prime) then P has a lift $P'$ to ${\mathbb R}$ with $T^p(P') = P'$ and is monotone since $g\in {\mathcal {G}}$ implies $\tilde {g}(x') \geq x'$ and so P is automatically p-fold semi-monotone.

5. (e) Using Lemma 7.6 a recurrent kfsm set in $S^1$ is either a collection of periodic orbits all with the same rotation number (a cluster) or else a semi-Denjoy minimal set. A minimal kfsm set in $S^1$ is either a single periodic orbit or else a semi-Denjoy minimal set.

6. (f) A collection of periodic orbits all with the same rotation number that individually are kfsm when considered as a set is not of necessity a kfsm set (that is, a cluster).

10.2 Symbolic kfsm sets in $\Omega _1$

We now consider symbolic kfsm sets in the symbolic base $\Omega _1 = \Sigma _2^+$ .

Definition 10.3. A $\sigma _1$ -invariant set ${\hat {Z}}\subset \Omega _1 = \Sigma _{2}$ is kfsm if there is a $\sigma _\infty $ -invariant lift ${\hat {Z}}'$ (that is, ${\hat {p}}_\infty ({\hat {Z}}') = {\hat {Z}}$ ) which is kfsm or equivalently, ${\hat {Z}}$ has a $\sigma _k$ -invariant lift ${\hat {Z}}^*\subset \Omega _k$ which is kfsm. Given $g\in {\mathcal {G}}$ , let ${\hat {{\mathcal {C}}}}_k(g)$ be all recurrent kfsm sets in ${\hat {\Lambda }}_1(g)$ with the Hausdorff topology and ${\hat {{\mathcal {O}}}}_k(g)$ be all g-invariant, Borel probability measures supported on ${\hat {Z}}\in {\hat {{\mathcal {C}}}}_k(g)$ with the weak topology

Using Theorem 7.6, we connect kfsm sets in $\Lambda _1(g)$ to their symbolic analogs in ${\hat {\Lambda }}_1(g)$ and obtain the following corollary.

Corollary 10.4. A g-invariant set $Z\subset \Lambda _1(g)$ is kfsm if an only if $\iota _1(Z)\subset {\hat {\Lambda }}_1(g)$ is. Further, $\iota _1$ induces homeomorphisms ${\mathcal {C}}_k(g)\rightarrow {\hat {{\mathcal {C}}}}_k(g)$ and ${\mathcal {O}}_k(g)\rightarrow {\hat {{\mathcal {O}}}}_k(g)$ .

Remark 10.5. All the comments in Remark 10.2 hold mutatis mutandis for symbolic kfsm sets.

10.3 The HM construction and its symmetries

We bring the HM construction back into play and take the projections from $\Omega _k$ to $\Omega _1$ .

Definition 10.6. Let $C_{k}(\omega , \vec {\nu }) = {\hat {\pi }}_k(B_{k}(\omega , \vec {\nu }))$ and $\mu _{k}(\omega , \vec {\nu }) = ({\hat {\pi }}_k)_*(\unicode{x3bb} _{k}(\omega , \vec {\nu }))$

We know from Theorem 9.5 that the HM construction provides a parameterization of ${\mathcal {B}}_k(g)$ and ${\mathcal {N}}_k(g)$ . The goal now is to get a parameterization of the kfsm sets and their invariant measures in $S^1$ , that is, of ${\hat {{\mathcal {C}}}}_k(g)$ and ${\hat {{\mathcal {O}}}}_k(g)$ . For this we need to understand the symmetries inherent in the HM construction.

Recall that the left shift on the parameter $\nu $ is $\tau (\nu _1, \ldots , \nu _k) = (\nu _2, \ldots , \nu _k, \nu _1)$ . There are two types of symmetries to be considered. The first is when different $\vec {\nu }$ give rise to the same $C_{k}(\omega , \vec {\nu })$ . For minimal $C_{k}(\omega , \vec {\nu })$ this happens if and only if the $\vec {\nu }$ s are shifts of each other as is stated in parts (a) and (d) in the lemma below. The second sort of symmetry happens when some $C_k(\omega , \vec {\nu })$ is also a $C_j(\omega , \vec {\nu }^{\;\prime })$ for some $j<k$ , which is to say the map ${\hat {\pi }}_k:B_k(\omega , \vec {\nu }) \rightarrow C_k(\omega , \vec {\nu })$ is not one-to-one. In the minimal case this happens if and only if $\tau ^j(\vec {\nu }) = \vec {\nu }$ as is stated in parts (b) and (c) below.

Lemma 10.7. Fix $k>0$ and assume $\vec {\nu }$ is allowable for $\omega $ .

1. (a) For all j, $B_k(\omega ,\tau ^j(\vec {\nu })) = {\hat {T}}_k^j( B_k( \omega ,\nu ))$ and so $C_k(\omega ,\tau ^j(\vec {\nu })) = C_k( \omega ,\nu )$ .

2. (b) If $\tau ^j(\vec {\nu }) = \vec {\nu }$ for some $0 < j < k$ then

   (10.1) $$ \begin{align} B_k(\omega,\vec{\nu}) = {\hat{T}}_k^j( B_k( \omega,\vec{\nu})) \end{align} $$
   and $C_k(\omega ,\vec {\nu }) = C_j(\omega ,\vec {\nu }^{\;\prime })$ where $\vec {\nu }^{\;\prime } = (\nu _1, \ldots , \nu _j)$ .

3. (c) If $B_k(\omega ,\vec {\nu })$ is minimal and (10.1) holds then $\vec {\nu } = \tau ^j(\vec {\nu })$ . If $\vec {\nu } \not = \tau ^j(\vec {\nu })$ for all $0 < j < k$ , then ${\hat {\pi }}_k:B_k(\omega ,\vec {\nu }) \rightarrow C_k(\omega ,\vec {\nu })$ is a homeomorphism.

4. (d) If $B_k(\omega , \vec {\nu })$ and $B_k(\omega , \vec {\nu }^{\;\prime })$ are minimal and $C_k(\omega , \vec {\nu })=C_k(\omega , \vec {\nu }^{\;\prime })$ , then for some j, $\vec {\nu }^{\;\prime } = \tau ^j(\vec {\nu })$ .

Proof. The fact that $B_k(\omega ,\tau ^j(\vec {\nu })) = {\hat {T}}_k^j( B_k( \omega ,\nu ))$ is an easy consequence of the HM construction, and since ${\hat {\pi }}_k {\hat {T}}^k = {\hat {\pi }}_k$ we have $C_k(\omega ,\tau ^j(\vec {\nu })) = C_k( \omega ,\nu )$ , proving (a) The first part of (b) follows directly from (a), using the given fact that $\tau ^j(\vec {\nu }) = \vec {\nu }$ .

For the second part of (b), first note that if $\{X_i\}$ is the address system for k and $(\omega ,\vec {\nu })$ then since $\tau ^j(\vec {\nu }) = \vec {\nu }$ , we have $T_k^j(X_i) = X_{i + 2j}$ . This implies that under the quotient $S_k\rightarrow S_j$ , $\{X_i\}$ descends to an allowable HM address system on $S_j$ using $(\omega ,\vec {\nu }^{\;\prime })$ . Thus using the dynamics $R_\omega $ on both address systems, the corresponding entries of $B_k(\omega , \vec {\nu })$ and $B_j(\omega , \vec {\nu }^{\;\prime })$ are equal $\mathrm {mod}\ 2$ and so $C_k(\omega , \vec {\nu }) = C_j(\omega , \vec {\nu }^{\;\prime })$ .

To prove the first part of (c), as remarked in Remark 7.11, if $B_k( \omega ,\nu )$ is minimal it is uniquely ergodic. Thus if (10.1) holds, then $\unicode{x3bb} _k(\omega ,\vec {\nu }) = \unicode{x3bb} _k( \omega ,\tau ^j(\vec {\nu }))$ , and since $\unicode{x3bb} _k$ is injective by Theorem 9.5, $\vec {\nu } = \tau ^{j}(\vec {\nu })$ . Now for the second part of (c), certainly ${\hat {\pi }}_k:B_k(\omega ,\vec {\nu }) \rightarrow C_k(\omega ,\vec {\nu })$ is continuous and onto, so assume it is not injective. Then there exist ${\underline s}, {\underline t}\in B_k(\omega ,\vec {\nu })$ with ${\underline s}\not ={\underline t}$ and ${\hat {\pi }}_k({\underline s}) = {\hat {\pi }}_k({\underline t})$ . Thus for some $0 < j' < k$ , ${\underline t} = {\hat {T}}_k^{j'}({\underline s})$ , and so if $j = k-j'$ , then $B_k(\omega ,\vec {\nu }) \cap T_k^j B_k(\omega ,\vec {\nu }) \not =\emptyset $ . But by assumption $B_k(\omega ,\vec {\nu })$ is minimal and so $B_k(\omega ,\vec {\nu }) = T_k^j B_k(\omega ,\vec {\nu })$ and so $\vec {\nu } = \tau ^j(\vec {\nu })$ , a contradiction. Thus ${\hat {\pi }}_k:B_k(\omega ,\vec {\nu }) \rightarrow C_k(\omega ,\vec {\nu })$ is injective, as required.

For part (d), $C_k(\omega , \vec {\nu })=C_k(\omega , \vec {\nu }^{\;\prime })$ implies that

$$ \begin{align*} \bigcup_{i=1}^k T^i( B_k(\omega, \vec{\nu})) ={\hat{\pi}}^{-1}( C_k(\omega, \vec{\nu})) ={\hat{\pi}}^{-1}(C_k(\omega, \vec{\nu}^{\;\prime})) = \bigcup_{i=1}^k T^i( B_k(\omega, \vec{\nu}^{\;\prime})). \end{align*} $$

Since each of $T^i( B_k(\omega , \vec {\nu }))$ and $T^i( B_k(\omega , \vec {\nu }^{\;\prime }))$ is minimal, for some j, $T^j( B_k(\omega , \vec {\nu }))= B_k(\omega , \vec {\nu }^{\;\prime })$ , and so by part (c), $\tau ^j(\vec {\nu }) = \vec {\nu }^{\;\prime }$ .

Remark 10.8. It is possible that if $B_k(p/q,\vec {\nu })$ is a cluster of periodic orbits, $\pi _k$ could be injective on some of them and not on others.

10.4 Continuity and injectivity

Let $\overline {\operatorname {\mathrm {HM}}_k(g)} = \operatorname {\mathrm {HM}}_k(g)/\tau $ with equivalence classes denoted by $[\vec {\nu }]$ . Note that $\tau ^j(\vec {\nu })\in {\mathcal {D}}_k$ for some j is resonant if and only if $\vec {\nu }$ is, so we may call $[\vec {\nu }]$ resonant or non-resonant.

Since the $\tau $ -action preserves slices, we define $\overline {\operatorname {\mathrm {HM}}}_{k\omega } = \operatorname {\mathrm {HM}}_{k\omega }(g)/\tau $ . The $\omega $ -slices of ${\mathcal {C}}_k(g)$ and ${\mathcal {O}}_k(g)$ are defined in the obvious way. If $(p/q, \vec {\nu })$ is a pure parameter so is $\tau ^j(\vec {\nu })$ for any j and so we define $\overline {\operatorname {\mathrm {Pure}}}(k, p/q) = \operatorname {\mathrm {Pure}}(k,p/q)/\tau $ . Note that $\overline {\operatorname {\mathrm {Pure}}}(k,p/q)$ is all $[\vec {\nu }]$ such that $B_k(p/q, \vec {\nu })$ is a single periodic orbit, it is not all $[\vec {\nu }]$ such that $C_k(p/q, \vec {\nu }) = {\hat {\pi }}_k B_k(p/q, \vec {\nu })$ is a single periodic orbit.

Definition 10.9. Lemma 10.7 implies that $(\iota _1)^{-1} \circ C_k$ induces a map $\theta _k:\overline {\operatorname {\mathrm {HM}}}_k(g)\rightarrow {\mathcal {C}}_k(g)$ and that $(\iota _1)^{-1}_* \circ \mu _k$ induces a map $\beta _k:\overline {\operatorname {\mathrm {HM}}}_k(g)\rightarrow {\mathcal {O}}_k(g)$ . The induced maps on slices are $\theta _{k\omega }:\overline {\operatorname {\mathrm {HM}}}_{k\omega }(g)\rightarrow {\mathcal {C}}_{k\omega }(g)$ and $\beta _{k\omega }:\overline {\operatorname {\mathrm {HM}}}_{k\omega }(g)\rightarrow {\mathcal {O}}_{k\omega }(g)$ .

Theorem 10.10. Assume $g\in {\mathcal {G}}$ , for each $k>0$ .

1. (a) The map $\theta _k$ is onto, continuous at non-resonant values and discontinuous at resonant values. Restricted to an irrational slice, it is injective, continuous at non-resonant values, and discontinuous at resonant values. Restricted to a rational slices, it is injective on the pure lattice.

2. (b) The map $\beta _k$ is a homeomorphism when restricted to irrational slices and pure rational lattices.

Proof. By construction we have the following commuting diagram:

(10.2)

The vertical maps are all onto and continuous, while by definition the composition of the bottom horizontal maps is $\theta _k$ . In a slight abuse of notation, the map $C_k$ in the diagram denotes the map induced on equivalence classes in $\overline {\operatorname {\mathrm {HM}}}_k(g)$ by $C_k$ . Since ${\iota _k}$ and $\iota _1$ are homeomorphisms we need only consider $C_k$ and $\mu _k$ . The fact that these are continuous follows from Lemma 9.10 and the just stated properties of the diagram, as do the various continuity assertions in the theorem. We prove the discontinuity result for $C_k$ on irrational slices. The other discontinuity assertions follow similarly.

Assume $(\alpha , \vec {\nu })$ is resonant with $\alpha \not \in {\mathbb Q}$ . From Lemma 9.3 and its proof we have a sequence $(\alpha , \vec {\nu }^{(i)}) \rightarrow (\alpha , \vec {\nu })$ so that $B_k(\alpha , \vec {\nu }^{(i)}) \rightarrow Z$ , and an ${\underline s}\in Z\setminus B_k(\alpha , \vec {\nu })$ with ${\underline s}$ non-recurrent. In the quotients, $[\vec {\nu }^{(i)}] \rightarrow [\vec {\nu }]$ and $C_k(\alpha , \vec {\nu }^{(i)}) \rightarrow {\hat {\pi }}_k(Z)$ by continuity. We need to show that ${\hat {\pi }}_k(Z)\not = C_k(\alpha , \vec {\nu })$ Now if $\pi _k({\underline s})\in {\hat {\pi }}_k(Z)\setminus C_k(\alpha , \vec {\nu })$ we are done, so assume $\pi _k({\underline s})\in C_k(\alpha , \vec {\nu })$ . Thus for some ${\underline t}\in B_k(\alpha , \vec {\nu })$ , $\pi _k({\underline s}) = \pi _k({\underline t})$ , and so by Lemma 5.3(e), for some j, ${\hat {T}}^j_k({\underline s}) = {\underline t}$ . This implies that the action of $\sigma _k$ on $\operatorname {\mathrm {Cl}}(o({\underline s}, \sigma _k))$ is conjugated to that on $\operatorname {\mathrm {Cl}}(o({\underline t}, \sigma _k))$ by ${\hat {T}}_k^j$ . But by Theorem 8.5, $\operatorname {\mathrm {Cl}}(o({\underline t}, \sigma _k))$ is a minimal set and thus so is $\operatorname {\mathrm {Cl}}(o({\underline s}, \sigma _k))$ , and so ${\underline s}$ is recurrent, a contradiction, yielding the discontinuity.

To show $C_k$ is injective on the sets indicated, assume $C_k(\omega , \vec {\nu }) = C_k(\omega , \vec {\nu }^{\;\prime })$ with either $\omega = p/q$ and $\vec {\nu }, \vec {\nu }^{\;\prime }$ in the pure lattice or $\omega \not \in {\mathbb Q}$ . In either case $B_k(\omega , \vec {\nu })$ and $B_k(\omega , \vec {\nu }^{\;\prime })$ are minimal and since ${\hat {\pi }}_k$ is a semi-conjugacy, $C_k(\omega , \vec {\nu })$ and $C_k(\omega , \vec {\nu }^{\;\prime })$ are also. Thus by Lemma 10.7(d), for some j, $\vec {\nu } = \tau ^{j}\vec {\nu }^{\;\prime }$ and so $[\vec {\nu }] = [\vec {\nu }^{\;\prime }]$ .

Now for part (b), there is a diagram similar to (10.2) for $\beta _k$ . Since Denjoy minimal sets and individual periodic orbits are uniquely ergodic, the injectivity asserted for $\mu _k$ follows from that of $C_k$ just proved. Continuity and surjectivity follow from the diagram and Lemma 9.10.

Remark 10.11. We remark on the relationship of pure parameters to $C_k$ and $B_k$ . As a shorthand we indicate symbolic periodic orbits by their repeating block. A simple computation shows that $B_2(2/5, (3/5, 3/5)) = 01223\ \cup \ 00123$ and so $C_2(2/5, (3/5, 3/5)) = 01001$ . Note that, as required, ${\hat {T}}_2( 01223) = 00123$ and $01001$ is the $2/5$ -Sturmian (as defined in the next section). Now $B_2(2/5, (4/5, 2/5)) = 00123$ and so $C_2(2/5, (4/5, 2/5)) = 01001 = C_2(2/5, (3/5, 3/5))$ . A further computation shows that both $\mu _2(2/5, (4/5, 2/5))$ and $\mu _2(2/5, (3/5, 3/5))$ are the unique invariant measure on $01001$ and thus $\mu _2$ is not injective on rational slices of $\overline {\operatorname {\mathrm {HM}}}_k$ despite the fact that it is injective on rational slices of $\operatorname {\mathrm {HM}}_k$ . The underlying explanation is that being a pure parameter requires $B_k$ to be a single periodic orbit, not that $C_k$ be one.

Definition 10.12. Let $\mathcal {Q}_k = P_k/\tau $ where $P_k\subset {\mathbb R}^{k+1}$ is the standard k-dimensional simplex and $\tau $ is the shift. Equivalence classes in $\mathcal {Q}_k$ are denoted $[\cdot ]$ . For ${\hat {\eta }}\in {\hat {{\mathcal {O}}}}_k(g)$ with $\rho ({\hat {\eta }}) = \omega $ from Theorem 10.10 we may find an $\vec {\nu }$ with ${\hat {\eta }} = \mu _k(\omega , \vec {\nu })$ . The skewness of ${\hat {\eta }}$ is defined as $\overline {\gamma }({\hat {\eta }}) := [\gamma (\unicode{x3bb} _k(\omega , \vec {\nu }))]$ . Note that by Lemma 10.7 this is independent of the choice of $\mu _k(\omega , \vec {\nu })$ . And also for $\eta \in {\mathcal {O}}_k(g)$ via $\overline {\gamma }(\eta ) = \overline {\gamma }((\iota _k)_*(\eta ))$ .

Remark 10.13. On an irrational quotient slice ${\mathcal {O}}_{k, \alpha }$ , let . Then $\overline {\gamma }_1$ is the inverse of $\beta _k$ and may be viewed as a parameterization of ${\hat {{\mathcal {O}}}}_{k, \omega }$ by skewness as in Remark 9.14. Also as in that remark, skewness also provides a parameterization of the quotient of the pure parameters.

10.5 Sturmian minimal sets, the case $k=1$

We will need the special and much-studied case of symbolic kfsm sets for $k=1$ . When $k=1$ there is only one allowable choice for $\nu $ , namely $\nu = 1-\omega $ , and so we write $C_1(\omega )$ for $C_1(\omega , 1-\omega ) = B_1(\omega , 1-\omega )$ . When $\omega $ is rational $C_1(\omega )$ is a single periodic orbit and when $\omega $ is irrational it is a semi-Denjoy minimal set. These minimal sets (and associated sequences) have much historical importance and an abundance of literature (see [2] for a survey). Their main importance here is as an indicator of when a given number is in the rotation set.

Definition 10.14. The minimal set $C_1(\omega )\subset \Sigma _2^+$ is called the Sturmian minimal set, with rotation number $\omega $ which may be rational or irrational.

Because there are many definitions in the literature, to avoid confusion we note that here ‘Sturmian’ refers to a minimal set and not a sequence, and it is subset of the one-sided shift $\Sigma _2^+$ . The next result is standard and we remark on one proof in Remark 13.6.

Lemma 10.15. $\omega \in \rho (\langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle )$ if and only if $C_1(\omega )\subset \langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ . If $0 \leq \omega _1 < \omega _2 \leq 1$ , then in $\Sigma _2^+$ ,

$$ \begin{align*} \min C_1(\omega_1) < \min C_1(\omega_2) < \max C_1(\omega_1) < \max C_2(\omega_2). \end{align*} $$

Definition 10.16. For a fixed k, let $\vec {\nu }_s(\omega )$ be defined by $(\vec {\nu }_s(\omega ))_i = 1-\omega $ for $i = 1, \ldots , k$ .

Remark 10.17. Since $\tau (\vec {\nu }_s) = \vec {\nu }_s$ it follows directly from Lemma 10.7 that for any k, $C_k(\omega , \vec {\nu }_s) = C_1(\omega )$ , the Sturmian minimal set with rotation number $\omega $ .

11 Structure of $\operatorname {\mathrm {HM}}_k(g)$

One obvious property of $\operatorname {\mathrm {HM}}_k(g)$ is the symmetry $\tau (\operatorname {\mathrm {HM}}_k(g)) = \operatorname {\mathrm {HM}}_k(g)$ for all k. The full structure of $\operatorname {\mathrm {HM}}_k(g)$ for a general $g\in {\mathcal {G}}$ is quite complicated and will be saved for future papers. Here we focus on the structure near the diagonal in ${\mathcal {D}}_k$ .

11.1 Irrationals on the diagonal

We parameterize the diagonal $\Delta _k\subset {\mathcal {D}}_k$ by $\omega $ using $\vec {\nu }_s(\omega )\in \Delta _k$ as defined in the previous section, and so

$$ \begin{align*} \Delta_k = \{ \vec{\nu}_s(\omega) \colon 0 \leq \omega \leq 1\}. \end{align*} $$

For $g\in {\mathcal {G}}$ the next result asserts that for each irrational $\alpha \in \operatorname {\mathrm {Int}}(\rho (g))$ there is some $\delta = \delta (\alpha )$ so that the neighborhood $N_{\delta }(\vec {\nu }_s(\alpha ))\subset \operatorname {\mathrm {HM}}_k(g)$ . It gives the proof of Theorem 1.2(a).

Theorem 11.1. Assume $g\in {\mathcal {G}}$ and $k>0$ .

1. (a) $\operatorname {\mathrm {HM}}_k(g) \cap \Delta _k = \{\vec {\nu }_s(\omega )\colon \omega \in \rho (g)\}$ .

2. (b) If $\alpha \not \in {\mathbb Q}$ with $\alpha \in \operatorname {\mathrm {Int}}(\rho (g))$ , there exists a $\delta>0$ so that $N_\delta (\vec {\nu }_s(\alpha ))\subset \operatorname {\mathrm {HM}}_k(g)$ .

3. (c) If $\alpha \in \operatorname {\mathrm {Int}}(\rho (g))\setminus {\mathbb Q}$ , then ${\mathcal {O}}_k(g)$ contains a $(k-1)$ -dimensional topological disc consisting of unique invariant measures each supported on a member of a family of kfsm semi-Denjoy minimal sets with rotation number $\alpha $ .

Proof. Assume ${\hat {\Lambda }}_1(g) = \langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ . For (a) $C_k(\omega , \vec {\nu }_s(\omega )) = C_1(\omega )$ , the Sturmian minimal set with rotation number $\omega $ , and from Lemma 10.15, $C_1(\omega ) \subset \langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ if and only if $\omega \in \rho (\langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle ) = \rho (g)$ .

For (b) note that the pair $(\alpha , \vec {\nu }_s)$ is non-resonant. We will first show that if $\alpha \in \operatorname {\mathrm {Int}}(\rho (g))$ then there exists an $\epsilon>0$ , so that $\operatorname {\mathrm {HD}}(C_k(\alpha , \vec {\nu }_s(\alpha )), C_k(\omega , \vec {\nu })) < \epsilon $ implies $ C_k(\omega , \vec {\nu }) \subset \langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ . Pick $\alpha _1,\alpha _2\in \operatorname {\mathrm {Int}}(\rho ( g))$ with $\alpha _1< \alpha <\alpha _2$ . Thus by Lemma 10.15, in $\Sigma _2^+$ ,

$$ \begin{align*} {\underline \kappa}_0 < \min C_1(\alpha_1) < \min C_1(\alpha) < \max C_1(\alpha) < \max C_1(\alpha_2) < {\underline \kappa}_1, \end{align*} $$

and let

$$ \begin{align*} \epsilon = \min\{d(\min C_1(\alpha_1), \min C_1(\alpha)), d(\max C_1(\alpha), \max C_1(\alpha_2))\}. \end{align*} $$

Thus $\operatorname {\mathrm {HD}}(C_k(\alpha , \vec {\nu }_s(\alpha )), C_k(\omega , \vec {\nu })) =\operatorname {\mathrm {HD}}(C_1(\alpha ), C_k(\omega , \vec {\nu })) < \epsilon $ implies that the compact, invariant set $C_k(\omega , \vec {\nu })$ satisfies $\min C_1(\alpha _1) < C_k(\omega , \vec {\nu }) < \max C_1(\alpha _2)$ and so $ C_k(\omega , \vec {\nu }) \subset \langle {\underline \kappa }_0, {\underline \kappa }_1 \rangle $ .

Using the continuity of $C_k$ at non-resonant irrationals from Theorem 10.10(a), there is a $\delta>0$ so that $\|(\omega ,\vec {\nu }) - (\alpha , \vec {\nu }_s)\| < \delta $ implies $C_k(\omega ,\vec {\nu }) \subset N_\epsilon (C_k(\alpha , \vec {\nu }_s))$ , and so $(\omega , \vec {\nu })\in \operatorname {\mathrm {HM}}(g)$ .

Since $\tau (N_\delta (\vec {\nu }_s(\alpha ))) = (N_\delta (\vec {\nu }_s(\alpha )))$ , the neighborhood descends to one in $\overline {\operatorname {\mathrm {HM}}}_k(g)$ and $\beta _k$ is a homeomorphism on irrational slices of $\overline {\operatorname {\mathrm {HM}}}_k(g)$ (Theorem 10.10(b)), yielding (c).

12 Rational slices

In this section we study rational slices in the HM parameter and in ${\mathcal {B}}_k(g)$ and ${\mathcal {C}}_k(g)$ . As proved in Theorem 8.5, each $B_k(p/q, \vec {\nu })$ is a collection of periodic orbits. They each have period $qk/\gcd (p,k)$ and all have rotation number $p/q$ . Collectively as a set they are kfsm. Note that this is stronger than each periodic orbit being individually kfsm. The invariant measure $\unicode{x3bb} _k(p/q, \vec {\nu })$ is a convex combination of the unique measures supported on each periodic orbit.

12.1 Periods in $\Omega _1$

The next lemma examines how the periods of $B_k$ can change after projection to $C_k$ via ${\hat {\pi }}_k$ .

Lemma 12.1. Fix $k>0$ and $p/q\in {\mathbb Q}$ and assume $\vec {\nu }$ is allowable for $p/q$ . If $\tau ^j(\vec {\nu }) = \vec {\nu }$ with $0 < j \leq k$ and it is the least such j, then the period of $C_k(p/q, \vec {\nu })$ is $jq/\gcd (j,p)$ .

Proof. Recall from Theorem 8.5 that the period of $B_k(p/q, \vec {\nu })$ is $kq/\gcd (k,p)$ . If $j=k$ by Lemma 10.7(c), ${\hat {\pi }}_k: B_k(p/q, \vec {\nu }) \rightarrow C_k(p/q, \vec {\nu })$ is injective, and since $\sigma _1{\hat {p}}_k = {\hat {\pi }}_k \sigma _k$ , $B_k(p/q, \vec {\nu })$ and $C_k(p/q, \vec {\nu })$ have the same period. Now if $j<k$ by Lemma 10.7(b), $C_k(p/q, \vec {\nu }) = C_j(p/q, \vec {\nu }^{\;\prime })$ where $\vec {\nu }^{\;\prime } = (\nu _1, \ldots , \nu _j)$ . Since j is the least such, ${\hat {\pi }}_j:B_j(p/q, \vec {\nu }^{\;\prime })\rightarrow C_j(p/q, \vec {\nu }^{\;\prime })$ is injective and $C_j(p/q, \vec {\nu }^{\;\prime })$ has period $jq/\gcd (j,p)$ .

12.2 The rational structure theorem

The theorem in this section describes in more detail how the measures on $p/q$ -kfsm sets vary with the parameter.

In the HM construction fix k, $0 < p/q < 1$ with $\gcd (p,q) = 1$ , and an allowable $\vec {\nu }$ . We often suppress dependence on these choices and so $R = R_{p/q}$ , etc. Let $N =qk/\gcd (p,k)$ so N is the period of R acting on $S_k$ . Recall that the address intervals are $X_j = [\ell _j, r_j]$ for $j = 0, \ldots , 2k-1$ and so $r_j = \ell _{j+1}$ . The good set is G and the itinerary map is $\zeta $ . When we write $\zeta (x)$ it is implicitly assumed that $x\in G$ .

The orbit of $0$ , $o(0, R)$ , partitions $S_k$ into N pieces, each of width $k/N = \gcd (p,k)/q$ . Thus $J = [k-\gcd (p,k)/q, 1)$ is a fundamental domain for the action of R on $S_k$ in the sense that $S_k = \bigcup _{i=0}^{N-1} R^i(J)$ as a disjoint union. Thus for each $0 \leq p \leq 2k-1$ there is a unique $0 \leq m < N$ with $\ell _{p}\in R^{m}(J)$ , and then let $d_{p} = R^{-m}(\ell _{p})$ . Note that since $|X_{2j+1}| = p/q$ , $d_{2j} = d_{2j-1}$ , and that all $d_{2j+1}$ as well as both endpoints of J are not in G. Finally, for $j = 0, 1, \ldots , 2k-1$ and $x\in J\cap G$ , let $M_j(x) = \{0 \leq i < N\colon \zeta (x)_i = j\} = \{i \colon R^i(x) \in X_j\}$ .

Lemma 12.2. Assume $x, x'\in J\cap G$ .

1. (a) $\zeta (x) = \zeta (x')$ if and only if $M_{2j+1}(x) = M_{2j+1}(x')$ for all $j = 0, \ldots , k-1$ .

2. (b) For each j, $M_{2j+1}(x) = M_{2j+1}(x')$ if and only if $ x$ and $x'$ are in the same component of $J-\{d_{2j+1}\}$ .

3. (c) For each k, $\# M_{2j}(x) = \#M_{2j}(x')$ if and only if x and $x'$ are in the same component of $\Sigma \setminus \{d_{2j-1}, d_{2j+1} \}$ where $\Sigma $ is the circle $\Sigma = J/\mkern -3mu\sim $ with $(k-1/N)\sim k$ .

Proof. First note that both endpoints of J are not in G so they are out of consideration for x and $x'$ in what follows.

For (a) one implication is obvious. For the other, it suffices to show that the collection of $M_{2j+1}(x)$ determines ${\underline s} = \zeta (x)$ . By Remark 7.11 we know that ${\underline s}\in \Omega _k$ and so its one-step transitions are governed by (5.1). If $s_i = 2j+1$ then $s_{i+1} = 2j+2 $ or $2j+3$ and we know which depending on whether $i+1\in M_{2j + 3}$ or not. Similarly, if $s_i = 2j$ then $s_{i+1}$ is determined by whether $i+1\in M_{2j + 1}$ or not. Thus ${\underline s}$ is determined, completing the proof of (a).

For (b), first note that $|X_{2k-1}| = p/q$ and $(p/\gcd (p,k)) (k/N) = p/q$ . Thus $X_{2k-1}$ is exactly filled with $p/\gcd (p,k)$ iterates of J with disjoint interiors. Thus $M_{2k-1}(x) = M_{2k-1}(x')$ for all $x\in J\cap G$ . Thus we only consider $0 \leq j < k-1$ . If i is such that $R^i(J)\subset X_{2j+1}$ , then $i\in M_{2j+1}(x)$ for all $x\in J$ , and if $R^i(\operatorname {\mathrm {Int}}(J))\cap X_{2j+1} = \emptyset $ then $i\not \in M_{2j+1}(x)$ for all $x\in J\cap G$ . If $\ell _{2j+1}\in R^i(\operatorname {\mathrm {Int}}(J))$ , then for $x>d_{2j+1}$ in J, we have $i\in M_{2j+1}(x)$ and for $x<d_{2j+1}$ , $i\not \in M_{2j+1}(x)$ . The last situation to consider is $r_{2j+1}\in R^{i}(\operatorname {\mathrm {Int}}(J))$ . Since $|X_{2k+1}| = p/q$ , we have $R^{i}(d_{2j+1}) = r_{2j+1}$ and so then for $x>d_{2j+1}$ in J, we have $i\not \in M_{2j+1}(x)$ and for $x<d_{2j+1}$ , $i\in M_{2j+1}(x)$ , completing the proof of (b).

If $d_{2j-1} = d_{2j+1}$ every $x\in \Sigma \setminus \{d_{2j-1}\}$ has the same number of indices in $M_{2j}$ , so assume that $d_{2j-1} < d_{2j+1}$ , with the other inequality being similar. If i is such that $R^i(J)\subset X_{2j}$ then $i\in M_{2j}(x)$ for all $x\in J$ . If i is such that $R^i(J)\cap X_{2j} = \emptyset $ then $i\not \in M_{2j}(x)$ for all $x\in J$ . If i is such that $\ell _{2j}\in R^i(J)$ then $i\in M_{2j}(x)$ if and only if $x> d_{2j-1}$ in J. If i is such that $\ell _{2j+1} = r_{2j}\in R^i(J)$ then $i\in M_{2j}(x)$ if and only if $x < d_{2j+1}$ in J, finishing the proof.

Corollary 12.3. If for some m, the non-empty connected components of $\Sigma \setminus \bigcup _{j=0}^{k-1} \{d_{2j+1}\}$ are $K_1, \ldots , K_m$ , then $B_k(p/q, \nu )$ consists of exactly m distinct periodic orbits $P_1, \ldots , P_m$ with $\zeta (x)\in P_j$ if and only if $x\in o(K_j, R)$ . Further, $\unicode{x3bb} _k(p/q,\nu ) = \sum N |K_j| \delta _j$ with $\Upsilon _j$ the unique invariant probability measure supported in $P_j$ and $N = qk/\gcd (p,k)$ .

Proof. As noted above, J is a fundamental domain for the action of R on $S_k$ and so it suffices to study $\zeta (x)$ for $x\in J$ .

Combining Lemma 12.2(a) and (b), we have that for $x\in \Sigma $ , $\zeta (x) = \zeta (x')$ if and only if x and $x'$ are in the same component $K_j$ . Further, using Lemma 12.2(c), $\zeta (x)$ and $\zeta (x')$ can be on the same $\sigma $ -orbit if and only if they are in the same component $K_j$ , proving the first sentence of the corollary. The second sentence follows from the definition of $\unicode{x3bb} _k$ , the fact that $S_k = \bigcup _{i=1}^N R^i(J)$ , and that R preserves Lebesgue measure.

12.3 The pure lattice and the structure of $\operatorname {\mathrm {HM}}_{k p/q}$

We now describe the pure affine lattice in more detail with an eye towards counting the number of $p/q$ -periodic kfsm sets. For this a new method of specifying the address system in $S_k$ will be useful. We fix a k and an $\omega = p/q$ and sometimes suppress dependence on them

Recall that a pair $(p/q, \vec {\nu })$ specifies an address system $\{X_j(p/q, \vec {\nu })\}$ with each $X_j(p/q, \vec {\nu }) = [\ell _j, r_j]$ . For each $i = 1, \ldots , k-1$ let $\xi _i$ be the signed displacement of the address system from its totally symmetric position given by $(\omega , \vec {\nu }_s(\omega ))$ . Thus

(12.1) $$ \begin{align} \xi_i( \vec{\nu}) = (\nu_1 + \cdots \nu_i) - i(1-\omega). \end{align} $$

Since in the HM construction $X_{2k}$ is fixed for all $\vec {\nu }$ , the vector $\vec {\xi }(\omega )$ is $(k-1)$ -dimensional and so $\vec {\xi }:{\mathcal {D}}_{k,p/q} \rightarrow \vec {\xi }({\mathcal {D}}_{k,p/q})$ is an affine map from the simplex $\sum \nu _i = k (1-p/q)$ to a subset of ${\mathbb R}^{k-1}$ . Note that $\vec {\xi }(\vec {\nu }_s) = \vec {0}$ .

Lemma 12.4. Given k and $ p/q$ , there exists a $\vec {\eta }$ with $\|\vec {\eta }\|_\infty \leq \gcd (p,k)/(2q)$ so that $\vec {\nu }\in {\mathcal {D}}_{k,p/q}$ is a pure parameter for $p/q$ if and only if $\vec {\xi }(p/q, \vec {\nu }) \in \vec {\eta } + (\gcd (p,k)/q) {\mathbb Z}^{k-1}$ in $\vec {\xi }({\mathcal {D}}_{k,p/q})$ .

Proof. Theorem 12.3 implies that $B_k(p/q, \vec {\nu })$ is a single periodic orbit if and only if no $d_{2j-1}$ is in the interior of J. This happens if and only if all $\ell _{2j-1}$ are contained in $o(0, R_{p/q})$ . Now $o(0, R_{p/q})$ divides $S_k$ evenly into subintervals of length $\gcd (p,k)/q$ . For each $j = 1, \ldots , k-1$ let $m_j$ be such that $R^{m_j}_{p/q}(0)$ is the point on $o(0, R_{p/q})$ that is closest to $\ell _{2j-1}$ and define $\eta _j = \ell _{2j-1} - R^{m_j}_{p/q}(0)$ . Thus $\|\vec {\eta }\|_\infty \leq \gcd (p,k)/(2q)$ and $\vec {\nu }$ is pure if and only if $\phi (\vec {\nu }) \in \vec {\eta } + (\gcd (p,k)/q) {\mathbb Z}^{k-1}$ .

Definition 12.5. The set $L = \vec {\eta } + (\gcd (p,k)/q) {\mathbb Z}^{k-1}\cap \vec {\xi }({\mathcal {D}}_{k,p/q})$ is called the $p/q$ -pure affine lattice as is its pre-image $\vec {\xi }^{-1}(L)\subset {\mathbb R}^{n-1}$ .

12.4 Sub-resonance and the size of clusters

Definition 12.6. When $\omega =p/q$ , the pair $(p/q, \nu )$ is called sub-resonant if for some $qk/\gcd (p,k)>n>1$ and $j\not =j'$ , $R_\omega ^n(\ell _j) = \ell _{j'}$ .

It follows from Theorem 12.3 that the number of sub-resonances in $(p/q, \vec {\nu }) \in {\mathcal {D}}_{k, p/q}$ controls the number of distinct periodic orbits in a cluster $B_k(p/q, \vec {\nu })$ . No sub-resonance corresponds to k distinct periodic orbits and when all the $\ell _j$ are on a single $R_{p/q}$ orbit, $B_k(p/q, \vec {\nu })$ is a single periodic orbit. In the latter case, $(p/q, \vec {\nu })$ is a pure parameter.

The set of sub-resonance parameters is a finite collection of codimension-one affine subspaces in ${\mathcal {D}}_{k, p/q}$ . Thus the set of parameters with no sub-resonance case is an open, dense and full measure subset of $ {\mathcal {D}}_{k, p/q}$ . It follows then that in $\operatorname {\mathrm {HM}}_{k, p/q}$ the typical parameter corresponds to a cluster of k periodic orbits. It also follows that, restricted to $\operatorname {\mathrm {HM}}_{k, p/q}$ , the assignment $\vec {\nu }\mapsto B_k(p/q, \vec {\nu })$ is constant, and thus is continuous on connected components of the no sub-resonance parameters and is discontinuous at the sub-resonance parameters.

12.5 Estimating the number of $p/q$ -kfsm sets

For a given $g\in {\mathcal {G}}$ the number of points from the pure $p/q$ -lattice $\operatorname {\mathrm {Pure}}_{k, p/q}$ contained in $\operatorname {\mathrm {HM}}_{k,p/q}(g)$ tells us how many distinct periodic orbits there are in ${\hat {{\mathcal {B}}}}_k(g)$ . So by Lemma 10.7 it tells us how many distinct periodic $p/q$ -kfsm sets g has. We get an estimate for this number using the continuity properties of $B_k$ from Theorem 9.5 and the relationship of kfsm sets in $S_k$ to those in $S^1$ . The next result proves Theorem 1.2(b).

Theorem 12.7. If $\alpha \in \operatorname {\mathrm {Int}}(\rho (g))$ , $\alpha \not \in {\mathbb Q}$ , $k>0$ , and $p_n/q_n$ is a sequence of rationals in lowest terms with $p_n/q_n \rightarrow \alpha $ , then there exists a $C>0$ so that for sufficiently large n the number of distinct periodic $p_n/q_n$ -kfsm sets in $\Lambda _1(g)$ is greater than or equal to $C q_n^{k-1}$ .

Proof. By Theorem 9.5(b) there is an $\epsilon _1> 0$ so that $N_{\epsilon _1}(\vec {\nu }_s(\alpha ))\subset \operatorname {\mathrm {HM}}_k(g)$ , where recall that $\vec {\nu }_s(\alpha )$ is the Sturmian $\vec {\nu }$ for $\alpha $ on the diagonal of ${\mathcal {D}}_k$ . Since $\vec {\xi }$ is a homeomorphism there is an $\epsilon $ -ball H in the max norm with $\epsilon>0$ about $\vec {0}$ in $\vec {\xi }({\mathcal {D}}_k)$ with $\vec {\xi }^{-1}(H) \subset \operatorname {\mathrm {HM}}_k(g)$ . Thus if $|p_n/q_n - \alpha | < \epsilon $ there is a $\epsilon $ -ball in the max norm, that is, a $(k-1)$ -dimensional hypercube $H_1$ , about $(p_n/q_n, \vec {0})$ in $\vec {\xi }({\mathcal {D}}_{k,p_n/q_n})$ with $\vec {\xi }^{-1}(H_1) \subset \operatorname {\mathrm {HM}}_{k,p_n/q_n}(g)$ .

We next estimate the number of pure resonance $\vec {\nu }$ in $H_1$ . By Lemma 12.4, the pure $\vec {\nu }$ form an affine lattice with linear separation $\gcd (p_n, k)/q_n$ . Thus for $p_n/q_n$ close enough to $\alpha $ , the number of lattice points in $H_1$ is larger than

$$ \begin{align*} \bigg( \frac{\epsilon q_n}{ \gcd(p_n, k)}\bigg)^{k-1} \geq \bigg(\frac{\epsilon q_n}{ k}\bigg)^{k-1} \end{align*} $$

since $\gcd (p_n, k) \leq k$ . Thus since $\vec {\xi }$ is a homeomorphism the same estimate holds for the number of pure lattice points in $\vec {\xi }^{-1}(H_1)\subset \operatorname {\mathrm {HM}}_{k, p_n/q_n}(g)$ . By Theorem 9.5 this tells us how many distinct periodic $p_n/q_n$ are in ${\hat {{\mathcal {B}}}}_k(g)$ and thus in ${\mathcal {B}}_k(g)$ by Theorem 7.6.

To project this estimate to kfsm sets in $S^1$ , recall from Theorem 10.10 that $\theta _k:\overline {\operatorname {\mathrm {HM}}}_{k, p_n/q_n}(g)\rightarrow {\mathcal {C}}_{k, p_n/q_n}(g)$ is injective on the pure lattice. The projection $\operatorname {\mathrm {Pure}}(k, p_n/q_n) \rightarrow \overline {\operatorname {\mathrm {Pure}}}(k, p_n/q_n)$ is at most k to $1$ and so the number of distinct $p_n/q_n$ periodic orbits in ${\mathcal {C}}(g)$ is greater than or equal to

$$ \begin{align*} \frac{1}{k}\bigg(\frac{\epsilon}{k}\bigg)^{k-1} q_n^{k-1}.\\[-34pt] \end{align*} $$

Remark 12.8. Using Lemma 12.1 for a pure $(p/q, \vec {\nu })$ , if there is a symmetry of the form $\tau ^j(\vec {\nu }) = \vec {\nu }$ for some $0 < j < k$ , then the period of the $C_k(p/q, \vec {\nu })$ counted in the theorem is $jq/\gcd (j, p)$ . In the typical case of no such symmetry the period is $kq/\gcd (k, p)$ . So, for example, when p and k are relatively prime, the counted periodic orbit has rotation type $(pk, qk)$ , and when k divides p, the rotation type is $(p,q)$ . By making judicious choices of the sequence $p_n/q_n\rightarrow \alpha $ , one can control the rotation types of the counted periodic orbits.

13 Parameterization via the interpolated family of maps

We return now to the heuristic description of kfsm sets in the Introduction using a family of interpolated semi-monotone maps, and prove results and connections to the HM parameterization. Since we are mainly developing a heuristic, some details are left to the reader. In many ways this perspective is better for studying kfsm sets, while the HM construction is better for measures. Initially the parameterization depends on the map $\tilde {g}\in {\mathcal {G}}$ but using the model map we will get a uniform parameterization.

13.1 The family of k-fold interpolated maps for $g\in {\mathcal {G}}$

Fix $g\in {\mathcal {G}}$ with preferred lift $\tilde {g}$ . For $y\in [g(x_{\min } + n), g(x_{\max }+ n)]$ there is a unique $x \in [x_{\min }+n, x_{\max }+n]$ with $\tilde {g}(x) = y$ . Denote this x by $b_n(y)$ (b for branch). Let $L_g = \tilde {g}(\min (\Lambda _\infty (g) \cap I_0))$ and $U_g = \tilde {g}(\max (\Lambda _\infty (g) \cap I_{-1}))$ with the $I_i$ as defined in §5.1. Note that, from the definition of the class ${\mathcal {G}}$ , $0 \leq L_g < U_g \leq 1$ and, by equivariance, $L_g + j = \tilde {g}(\min (\Lambda _\infty (g) \cap I_{2j}))$ and $U_g + j= \tilde {g}(\max (\Lambda _\infty (g) \cap I_{2j-1}))$ .

Definition 13.1. For $\vec {c}\in {\mathbb R}^k$ define $\tilde {c}\in {\mathbb R}^k$ via $\tilde {c}_j = c_j + j-1$ for $j = 1, \ldots , k$ .

Fix $k>0$ . For $\vec {c}\in [L_g, U_g]^{k}$ and for $j = 1, 2, \ldots , k$ define $\tilde {H}_{k \vec {c}}(x)$ on $[b_{-1}(0), b_{-1}(0) + k ]$ as

$$ \begin{align*} \tilde{H}_{k\vec{c}}(x) = \begin{cases} \tilde{c}_j & \text{when} \ x\in[b_{j-2}(c_j), b_{j-1}(c_j)], \\ \tilde{g}(x) & \text{otherwise}, \end{cases} \end{align*} $$

and extend to $\tilde {H}_{k\vec {c}}:{\mathbb R}\rightarrow {\mathbb R}$ so that $\tilde {H}_{k\vec {c}}(x+k) = \tilde {H}_{k\vec {c}}(x) + k$ . See Figure 1. Next define $H_{k\vec {c}}:S_k\rightarrow S_k$ as the descent of $H_{k,\vec {c}}$ to $S_k$ .

Example: The model map. For the model map $f_m$ , $x_{\min } = 0, x_{\max } = 1/2, L = 0, U = 1/2$ and $b_j(y) = (y+ 2j)/3$ .

Given a compact $Z\subset \Lambda _k(g)$ , for $j = 0, \ldots , k$ let

$$ \begin{align*} \ell_j'(Z) = \tilde{g}_k( \max\{Z\cap I_{2j-1}\}) - (j-1)\quad \text{and} \quad r_j'(Z) = \tilde{g}_k( \min\{Z\cap I_{2j}\}) - (j-1). \end{align*} $$

If for some j we have $\ell _j' < L_g$ let $\ell _j = L_g$ , otherwise let $\ell _j = \ell _j'$ . Similarly, if for some j we have $r_j'> R_g$ let $r_j = R_g$ , otherwise let $r_j = r_j'$ . Not that these rs and $\ell $ s are unrelated to those in §9.1.

Theorem 13.2. Assume $Z\subset \Lambda _k(g)$ is compact and invariant. The following assertions are equivalent.

1. (a) Z is a kfsm set.

2. (b) For $j = 1, \ldots , k$ , $\ell _j(Z) \leq r_j(Z)$ .

3. (c) $Z\subset P(H_{k\vec {c}})$ for

   (13.1) $$ \begin{align} \vec{c}\in \prod_{j=1}^k [ \ell_j(Z), r_j(Z)], \end{align} $$
   thus $(\tilde {g}_k)_{\vert Z} = (H_{k\vec {c}})_{\vert Z}$ .

Proof. If for some j, $\ell _j(Z)> r_j(Z)$ then g restricted to Z does not preserve the cyclic order, and so (a) implies (b). Assertion (c) implies (a) since invariant sets in non-decreasing maps are always kfsm. Finally, (b) says that $Z\subset P(H_{k\vec {c}})$ for $\vec {c}$ in the given range.

Definition 13.3. For $Z\in {\mathcal {B}}_k(g)$ , let

$$ \begin{align*}\operatorname{\mathrm{Box}}_g(Z) = \prod_{j=1}^k [ \ell_j(Z), r_j(Z)], \end{align*} $$

and so $\operatorname {\mathrm {Box}}_g(Z)\subset [L_g, U_g]^{k}$ .

Remark 13.4

1. (a) Nothing in the theorem requires Z to be recurrent. If it is then $Z\in {\mathcal {B}}_k(g)$ and so by Theorem 8.5, $\iota _k(Z) = B_k(\omega , \vec {\nu })$ where $\omega = \rho _k(Z)$ and $\vec {\nu }\in {\mathcal {D}}_{k\omega }$ .

2. (b) When Z is a periodic orbit or cluster, $\operatorname {\mathrm {Box}}_g(Z)$ is k-dimensional. When Z is a periodic orbit cluster its box is equal to the intersections of the boxes of its constituent single periodic orbits.

3. (c) When Z is a semi-Denjoy minimal set contained in $P(H_{k\vec {c}})$ , recall that a tight flat spot of Z is one for which both endpoints of a flat spot of $H_{k\vec {c}}$ are in Z. The dimension of $\operatorname {\mathrm {Box}}_g(Z)$ is the same as the number of loose flat spots in Z. Since, by Lemma 3.5(b), Z cannot have k tight flat spots, the dimension of $\operatorname {\mathrm {Box}}_g(Z)$ is between $0$ and $k-1$ .

4. (d) Note that in contrast, in the HM parameterization, each single periodic orbit or semi-Denjoy minimal set corresponds to just one point.

5. (e) When $\alpha \not \in {\mathbb Q}$ , if $Z = \iota _k^{-1}(B_k(\alpha , \vec {\nu }))$ from the HM construction then the number of loose flat spots of Z is the same as the number of resonances of $(\alpha , \vec {\nu })$ , that is, $j\not = j'$ with $R_\alpha ^N(\ell _j) = \ell _{j'}$ for some $n>1$ , which is then the same as the dimension of $\operatorname {\mathrm {Box}}_g(Z)$ .

13.2 Non-recurrence and kfsm sets that hit the negative-slope region

Throughout this paper we have assumed that the kfsm sets were recurrent and avoided the negative-slope region. In this section we use the interpolated maps to motivate and explain these assumptions.

Assume now that Z is a kfsm set for some $g\in {\mathcal {G}}$ and Z contains points in the negative-slope region of g. It still follows that Z is an invariant set of some $H_{k\vec {c}}$ . Let $Z'$ be the maximal recurrent set in $P(H_{k\vec {c}})$ . A gap of $Z'$ is a component of the complement of $Z'$ that contains a flat spot of $H_{k\vec {c}}$ . In formulas, a gap is an interval $ (\max \{Z'\cap I_{2j-1}\}, \min \{Z'\cap I_{2j}\}) $ for some j. Since $g_k$ acting on Z is semi-monotone, Z can contain at most one point $p_j$ in the negative-slope region within each gap.

There are two cases. In the first, which may happen for both rational and irrational rotation numbers, for all $j'$ there is some n so that $f^i(p_{j'})\not \in \{p_j\}$ for all $i\geq n$ . This implies that $H_{k\vec {c}}^i(p_j)\in P(H_{k\vec {c}})$ for all $i\geq n$ , and so, by Lemma 3.5(c), there is an $n'$ so that $H_{k\vec {c}}^i(p_j)\in Z'$ for all $i\geq n'$ . Thus in this case negative-slope orbits add no additional recurrent dynamics. In Figure 4 the disks give part of a periodic kfsm set and the squares show additional homoclinic points to this kfsm set in the negative- and positive-slope region.

Figure 4 A semi-monotone set with homoclinic points.

The second case holds for just rational rotation numbers and occurs when some $p_j$ is a periodic point; this does add new recurrent kfsm sets. Since the periodic points on the endpoint of a gap always return, there is, in fact, a periodic point in the negative-slope region of each gap of $Z'$ . By adjusting $\vec {c}$ we can assume that all these gap periodic orbits are also superstable periodic orbits of $H_{k\vec {c}}$ . Since the periodic points in $Z'$ are all unstable the periodic points of $Z'$ must alternate with these gap periodic points. In particular, the number of gap periodic orbits equals the number of periodic orbits in $Z'$ . Thus the addition of the negative-slope periodic orbits just adds a factor of two to the basic estimates of §12.5.

We again use Figure 4, but this time to discuss the holes in the space of recurrent kfsm sets. Let $H_{c_0}$ be an interpolated map whose flat spot contains the homoclinic points indicated by squares. Assume we are in the $k=1$ case and so each interpolated map $H_c$ contains exactly one recurrent semi-monotone set $Z_c$ in its positive-slope region. As $c_n$ increases to $c_0$ , the sets $Z_{c_n}$ converge in the Hausdorff topology not just to $Z_{c_0}$ , but to that set union the boxed point shown in the positive slope region. A similar phenomenon happens as $c_n$ decreases to $c_0$ . This phenomenon also clearly happens for loose gaps of semi-Denjoy minimal sets and for all k. This is the geometric explanation of the holes in ${\mathcal {B}}_k(g)$ and the discontinuity of $B_k$ discussed in §9.1.

13.3 The rotation number diagram

Fix $g\in {\mathcal {G}}$ . Since $H_{k\vec {c}}:S_k\rightarrow S_k$ we define $R_k(\vec {c}) = k \rho (D_k\circ H_{k\vec {c}} \circ D_k^{-1})$ . Thus if $Z\subset P( H_{k\vec {c}})$ is compact invariant then $\rho _k(Z) = R_k(\vec {c})$ . We treat $R_k$ as a function $R_k:[L_g, U_g]^{k}\rightarrow {\mathbb R}$ .

Let ${\mathbb R}_+^k = \{\vec {u}\in {\mathbb R}^k \colon \ \text {all} \ u_i> 0\}$ . The open projective positive cone in ${\mathbb R}^k$ is $Q_k = \{\vec {u}\in {\mathbb R}^k_+\colon \|\vec {u}\|_2 = 1\}$ . For a given k and $\omega $ , define $\varphi _{-,\omega },\varphi _{+,\omega } :Q_k\rightarrow {\mathbb R}^+$ as

$$ \begin{align*} \varphi_{-,\omega}(\vec{u})&= \min\{t\in{\mathbb R}^+\colon R_k(t\vec{u}+\vec{L}) = \omega\},\\ \varphi_{+,\omega}(\vec{u})&= \max\{t\in{\mathbb R}^+\colon R_k(t\vec{u}+\vec{L}) = \omega\}, \end{align*} $$

where $\vec {L} = (L_g, L_g, \ldots , L_g)$ . So $\varphi _{-,\omega }$ and $\varphi _{+,\omega }$ give the top and bottom edges of the level set $R_k^{-1}(\omega )$ when viewed from the origin.

Theorem 13.5. Assume $g\in {\mathcal {G}}$ and construct $R_k$ as above.

1. (a) $R_k$ is continuous function and is non-decreasing in t along any line $\vec {c} = t \vec {u} + \vec {v}$ with all $v_i \geq 0$ .

2. (b) For all $\omega $ the functions $\varphi _{-,\omega }$ and $\varphi _{+,\omega }$ are continuous.

3. (c) For rational $\omega $ , $\varphi _{-,p/q}< \varphi _{+,p/q}$ , while for $\alpha \not \in {\mathbb Q}$ , $\varphi _{-,\alpha }=\varphi _{+,\alpha }$ . Thus each level set $R_k^{-1}(p/q)$ is homeomorphic to a $(k-1)$ -dimensional open disk product a non-trivial closed interval, while each $R_k^{-1}(\alpha )$ is homeomorphic to a $(k-1)$ -dimensional open disk.

4. (d) $\rho (g) = [\rho (H_{L_g}), \rho (H_{U_g})] = \rho (\Lambda _1(g), g)$ .

Proof. Part (a) follows directly from Lemma 3.1(b) and (c). For (b) assume to the contrary that $\varphi _-$ is not continuous. Then there is a sequence $\vec {u}_n\rightarrow \vec {u}_0$ with $\varphi _-(\vec {u}_n)\not \rightarrow \varphi _-(\vec {u}_0)$ . Passing to a subsequence if necessary, there is some $t_0$ with $\varphi _-(\vec {u}_n) \vec {u}_n + \vec {L} \rightarrow t_0 \vec {u}_0 + \vec {L}$ . By the continuity of $R_k$ , $R_k( t_0 \vec {u}_0 + \vec {L}) = \omega $ , and by the non-convergence assumption, there is some $t' < t_0$ with $R_k( t' \vec {u}_0 + \vec {L}) = \omega $ . Thus again by the continuity of $R_k$ for n large enough there is some $t^{\prime \prime }_n < \varphi _-(\vec {u}_n)$ with $R_k( t^{\prime \prime }_n \vec {u}_n + \vec {L}) = \omega $ , a contradiction. Therefore, $\varphi _-$ is continuous; the continuity of $\varphi _+$ is similar.

For (c), pick any $t_0$ and $\vec {u}_0$ with $R_k(t_0\vec {u}_0 + \vec {L}) = p/q$ and let $\vec {c} = t_0\vec {u}_0 + \vec {L}$ . Then by Lemma 3.5, $H_{\vec {c}}$ has a periodic orbit $Z\subset P(H_{\vec {c}})$ . Since Z is a finite set there is a non-trivial interval I so that $t\in I$ implies $R_k(t\vec {u}_0 + \vec {L}) = p/q$ and so $\varphi _{-,p/q}< \varphi _{+,p/q}$ .

To complete (c), assume to the contrary that for some $\vec {u}_0$ , $\varphi _{-,\alpha }(\vec {u}_0) < \varphi _{+,\alpha }(\vec {u}_0)$ . Thus by the continuity of $R_k$ there is an open ball $N\subset R_k^{-1}(\alpha )$ . Pick $\vec {c}\in N$ and let Z be the semi-Denjoy minimal set in $P(H_{\vec {c}})$ guaranteed by Lemma 3.5 which has at least one tight gap, say the gap associated with $c_1$ , the first coordinate of $\vec {c}$ . Let y be the x-coordinate of the right-hand endpoint of this gap and so $\tilde {g}_k(y) = c_1$ . Since Z is minimal under $H_{\vec {c}}$ there are points $z\in Z$ with $z> y$ and arbitrarily close to y which have a $n>0$ with $y < H_{\vec {c}}^n(z) < z$ . Now let $c_1' = \tilde {g}_k(z)$ and $\vec {c}' = (c_1', c_2, \ldots , c_k)$ and we have that $H^n_{\vec {c}'}(z) < z$ , which says that the nth iterate of the first coordinate flat spot of $H_{\vec {c}'}$ is in that flat spot. Thus $H_{\vec {c}'}$ has a periodic orbit and so $R_k(\vec {c}') \not = \alpha $ for some $\vec {c}'$ arbitrarily close to $\vec {c}$ , a contradiction.

For (d), assume $k=1$ and $\rho (g) = [\rho _1, \rho _2]$ . Let $H_T$ be the semi-monotone map constructed from g to have a single flat spot of height $\tilde {g}(x_{\max })$ and $H_B$ similarly constructed to have a single flat spot of height $\tilde {g}(x_{\min })$ . Since $H_T \geq \tilde {g}$ , we have $\rho (H_T) \geq \rho _1$ . Now by Lemma 3.5, there is a compact invariant $Z\subset P(H_T)$ and so $g_{\vert Z} = (H_T)_{\vert Z}$ and so $\rho (H_T) = \rho (Z, g) \in \rho (g)$ and so $\rho (H_T) = \rho _1$ . Similarly, $\rho (H_B) = \rho _2$ . Note that by definition of $H_U$ , the compact invariant $Z\subset P(H_T)$ also satisfies $Z\subset P(H_U)$ and so $\rho (H_T) = \rho (H_U)$ . Similarly, $\rho (H_B) = \rho (H_L)$ . Thus $\rho (g) = [\rho (H_L), \rho (H_U)]$ . Finally, consider the entire family $H_c$ for $c\in [L, U]$ . Since $\rho (H_c)$ is continuous in c, for each $\omega \in [\rho (H_L), \rho (H_U)]$ there is a c with $\rho (H_c) = \omega $ . Further, for each c there is a compact invariant $Z_c\subset P(H_c)$ and $Z_c\subset \Lambda _1(g)$ , and thus $\omega \in \rho (\Lambda _1(g))\subset \rho (g)$

Remark 13.6

1. (a) Note that $H_T(x) \leq x+ 1$ and $H_B(x)\geq x$ , and thus $g\in {\mathcal {G}}$ implies $ \rho (g) \subset [0,1]$ . Further, it follows from (d) that the image of each $R_k$ is $\rho (g)$ .

2. (b) Part (b) deals only with the part of the level sets of $R_k$ in the open set $(L_g, U_g)^k$ . The extension to all of $[L_g, U_g]^{k}$ is technical and not very illuminating, so we leave it to the interested reader.

3. (c) Let $\tau $ act on $\vec {c}$ as the left cyclic shift. It easily follows that $R_k(\tau (\vec {c})) = R_k(\vec {c})$ .

4. (d) When $k=1$ there is a one-dimensional family $H_c$ for $c\in [L_g,U_g]$ . The rotation number $R_1(c)$ is non-decreasing in c and assumes each irrational value at a point and each rational value on an interval by (a) and (c). For each c there is a unique recurrent $Z_c\subset P(H_c)$ and $\iota _1(Z_c)$ is the Sturmian minimal set with the given rotation number. This, along with the geometry of the family $H_c$ , gives the proof of Lemma 10.15.

13.4 Comparing $g\in {\mathcal {G}}$ to the model map

In this section we use the interpolation parameter $\vec {c}$ to parameterize all the $Z\in {\mathcal {B}}_k(g)$ for a general $g\in {\mathcal {G}}$ . Notice that for the model map, ${\hat {\Lambda }}_k(f_m)$ is all of $\Omega _k$ . Thus ${\hat {{\mathcal {B}}}}_k(g) \subset {\hat {{\mathcal {B}}}}_k(f_m)$ and we can pass back to ${\mathcal {B}}(g)$ using the inverse of the itinerary map. Thus we can use a subset of the interpolation parameters of the model map to parameterize ${\mathcal {B}}(g)$ using the symbolic representation of a kfsm set as the link. This subset turns out to be a square of the form $[L', U']^k$ . In this section we often add an additional f or g subscript to indicate which map $f_m$ or g is involved.

Since $\iota _{k,g}(\Lambda _k(g)) = {\hat {\Lambda }}_k(g) \subset \Omega _k = {\hat {\Lambda }}_k(f)$ we may define $\psi ':\Lambda _k(g) \rightarrow \Lambda _k(f)$ by $\psi ' = \iota _{kf} \circ \iota _{kg}^{-1}$ . By Theorem 6.1, $\psi '$ is an orientation-preserving homeomorphism onto its image as well as a conjugacy. It thus induces a map $\overline {\psi }:{\mathcal {B}}_k(g) \rightarrow {\mathcal {B}}_k(f)$ .

Recall that the parameters for the model map are $[L_{f}, U_{f}]^{k} = [0,1/2]^{k}$ . For a map $\phi : [a,b] \rightarrow [a,b]$ extend it to the Cartesian product as $\phi ^{(k)} = (\phi , \phi , \ldots , \phi )$ .

Theorem 13.7. Given $g\in {\mathcal {G}}$ and $k>0$ , construct the interpolation parameters $[L_g,U_g]$ . There exist an interval $[L', U'] \subset [0, 1/2]$ and an orientation-preserving homeomorphism $\phi :[U_g, L_g] \rightarrow [L', U']$ so that for all $Z\in {\mathcal {B}}_k(g)$ , $ \phi ^{(k)}(\operatorname {\mathrm {Box}}_g(Z))) = \operatorname {\mathrm {Box}}_f(\overline {\psi }(Z))$ , and for all $\omega \in \rho (g)$ , $\phi ^{(k)}\rho _{k,g}^{-1}(\omega ) = \rho _{k,f}^{-1}(\omega )$ .

Proof. Construct $\psi '$ as above. Its properties imply that

(13.2) $$ \begin{align} \psi'(\ell_j(Z)) = \ell_j(\overline{\psi}(Z)) \quad \text{and} \quad \psi'(r_j(Z)) = r_j(\overline{\psi}(Z)) \end{align} $$

for all $Z\subset {\mathcal {B}}_k(g)$ and $j = 1, \ldots , k$ . Let $L' = \psi '(L_g)$ and $U' = \psi '(U_g)$ . Then $\psi '$ restricts to $\psi :\Lambda _k(g)\cap [L_g, U_g]^{k} \rightarrow \Lambda _k(f)\cap [L', U']^{k}$ . Since $\psi T_k = T_k \psi $ and $\Lambda _k(g)\cap [L_g, U_g]^{k}$ is compact we can extend $\psi $ equivariantly to a homeomorphism $\Psi :[L_g, U_g]^{k} \rightarrow [L', U']^{k}$ which, using (13.2), satisfies $\Psi \circ \operatorname {\mathrm {Box}}_g = \operatorname {\mathrm {Box}}_f\circ \overline {\psi }$ . Finally, since $\Psi \circ \tau = \tau \circ \Psi $ (recall $\tau $ is the left cyclic shift) there is a $\phi :[U_g, L_g] \rightarrow [L', U']$ with $\Psi = \phi ^{(k)}$ .

This result implies that the $\rho _k$ -diagram for g looks like a k-dimensional cube cut from inside the $\rho _k$ -diagram of the model map and perhaps rescaled.

Figure 5 The rotation number diagram for the model map with $k=2$ , reparameterized for clarity.

13.5 The case $k=2$ : numerics

Figure 5 shows the $k=2$ rotation number diagram for the model map $f_m$ . Each connected union of rectangles is the level set of some rational. Rationals with denominator less than $6$ are shown. Only the center rectangle is labeled for each rational. Each rectangle in the figure corresponds to a different $2$ -fold semi-monotone periodic orbit. The intersections of these rectangles correspond to $H_{\vec {c}}$ which have a cluster of two periodic orbits.

The computation of this diagram used a discrete version of the HM construction. The construction depends on integers $p, q, \mu $ with $0 < p/q < 1$ , p and q relatively prime, and $0 \leq \mu \leq 2(q-p)$ . The discrete circle is the finite cyclic group ${\mathbb Z}/2q{\mathbb Z} = {\mathbb Z}_{2q}$ and it is acted on by $R_p: n \mapsto n+p$ . The address intervals are $X_0' = [1, \mu ], X_1' = [\mu + 1, \mu + p], X_2' = [\mu + p + 1, 2q - p]$ , and $X_3' = [2q - p + 1, 2(q-p)]$ . Let $B'(p, q, \mu )$ be the itinerary of the point $1$ under $R_p$ .

Using Theorem 8.5(b), when p is odd, $R_p$ has a single period $2q$ orbit in ${\mathbb Z}_{2q}$ . Expanding the points in ${\mathbb Z}_{2q}$ to intervals in the circle as in the proof of Theorem 8.5(c), we see that by varying $\mu $ the construction generates all the symbolic $p/q$ -periodic $2$ -fold semi-monotone sets in $\Omega _2$ .

Now when p is even, $R_p$ has a pair of period q orbits. When $\mu $ is odd, these generate different periodic orbits $B'(p, q, \mu )$ . However, $\mu $ even corresponds to a pure parameter and so varying $\mu $ through the even $\mu $ generates all the symbolic $p/q$ -periodic $2$ -fold semi-monotone sets in $\Omega _2$ .

The next step is to use $B'(p, q, \mu )$ to compute its symbolic box as in Corollary 14.1 below. Finally, we take the inverse of the itinerary map for the model map to get a box in the $\vec {c}$ parameter. Because the map $f_m$ has uniform slope of three in its positive-slope region the formula for this inverse is ${\underline s}\in \Sigma _2^+$ ,

(13.3) $$ \begin{align} \iota_1^{-1}({\underline s}) = \sum_{j=0}^\infty \frac{s_j}{3^{j+1}}. \end{align} $$

14 Symbolic kfsm sets and the map $z\mapsto z^n$

Using the model map, the characterization of ‘physical’ kfsm sets in Theorem 13.2 can be transformed into a characterization of symbolic kfsm sets. For compact ${\hat {Z}}\subset \Omega _k$ for $j = 1, \ldots , k$ , define

$$ \begin{align*} {\hat{\ell}}_j(Z) = \sigma_k( \max\{{\hat{Z}}\cap [2j -1]\})\quad \text{and} \quad {\hat{r}}_j(Z) = \sigma_k( \min\{{\hat{Z}}\cap [2j]\}). \end{align*} $$

Since $\iota _k$ is order-preserving and onto for the model map we have the following corollary.

Corollary 14.1. Assume ${\hat {Z}}\subset \Omega _k$ is compact and shift invariant. The following assertions are equivalent.

1. (1) ${\hat {Z}}$ is kfsm.

2. (2) For $j = 1, \ldots , k$ , ${\hat {\ell }}_j({\hat {Z}}) \leq {\hat {r}}_j({\hat {Z}})$ with indices reduced mod $2k$ .

If Z is recurrent we know that each $Z\in {\mathcal {B}}_k(f)$ has $\iota _k({\hat {Z}}) = B_k(\omega , \vec {\nu })$ for some allowable $(\omega , \vec {\nu })$ which yields an indirect connection between the interpolated semi-monotone maps and HM parameterization.

There is a well-known connection between the dynamics of $d_n:z\mapsto z^n$ and the full shift on n symbols. This yields a connection of the symbolic kfsm sets as described by this corollary to invariant sets of the circle on which the action of $d_n$ is semi-monotone, sometimes called circular orbits.

In Figure 6 we show the conditions forced by Corollary 14.1 as flat spots in the graph of $d_n$ for $k=3$ and $n=6$ . There are two classes of flat spots. Those in class A are forced by the condition that ${\hat {Z}}\subset \Omega _k$ and thus satisfies (5.1). These are the intervals of width $1/6$ , $[1/18, 2/9], [7/18, 5/9]$ and $[13/18, 8/9]$ . These conditions are satisfied by all symbolic kfsm sets in the corollary. The other three flat spots in class B are determined by the conditions in part (b) of the corollary and vary with the symbolic kfsm set. Note that adding all the flat spots yields a degree-one semi-monotone circle map as expected. See figures in [21, 34].

This figure also illustrates a clear difference between the kfsm sets for bimodal circle maps and circular orbits for $d_n$ . Specifically, the kfsm sets correspond to a specific subclass of circular orbits for $d_{2k}$ . On the other hand, there is clearly a tight relationship between the theories which needs to be investigated. Perhaps the degree reduction process described in [8, 34] would be a good place to start.

Figure 6 The semi-monotone map corresponding to a symbolic $3$ -fold semi-monotone set interpolated into $z\mapsto z^6$ .