HOMOTOPY MINIMAL PERIODS FOR HYPERBOLIC MAPS ON INFRA-NILMANIFOLDS

Abstract

In this paper, we show that for every nonnilpotent hyperbolic map $f$ on an infra-nilmanifold, the set $\operatorname{HPer}(f)$ is cofinite in $\mathbb{N}$. This is a generalization of a similar result for expanding maps in Lee and Zhao (J. Math. Soc. Japan 59(1) (2007), 179–184). Moreover, we prove that for every nilpotent map $f$ on an infra-nilmanifold, $\operatorname{HPer}(f)=\{1\}$.

1 Infra-nilmanifolds

Let $f:X\rightarrow X$ be a map on a topological space $X$ . We say that $x\in X$ is a periodic point of $f$ if $f^{n}(x)=x$ for some positive integer $n$ . If this is the case, we say that this positive integer $n$ is the pure period of $x$ if $f^{l}(x)\neq x$ for all $l<n$ . In this paper, we study these periodic points when $X$ is an infra-nilmanifold and we show that for a large class of maps $f$ on such manifolds, there exists a positive integer $m$ such that any map $g$ homotopic to $f$ admits points of pure period $k$ for any $k\in [m,+\infty )$ . In the first section, we recall the necessary background on the class of infra-nilmanifolds and their maps. In the next section, we give a more detailed description of the theory of fixed and periodic points. The third and last section is devoted to the proof of our main result.

Every infra-nilmanifold is modeled on a connected and simply connected nilpotent Lie group. Given such a Lie group $G$ , we consider its group of affine transformations $\operatorname{Aff}(G)=G\mathbb{o}\operatorname{Aut}(G)$ , which admits a natural left action on the Lie group $G$ :

$$\begin{eqnarray}\forall (g,\unicode[STIX]{x1D6FC})\in \operatorname{Aff}(G),\quad \forall h\in G:\;\;^{(g,\unicode[STIX]{x1D6FC})}h=g\unicode[STIX]{x1D6FC}(h).\end{eqnarray}$$

Note that when $G$ is abelian, $G$ is isomorphic to $\mathbb{R}^{n}$ for some $n$ and $\operatorname{Aff}(G)$ is the usual affine group $\operatorname{Aff}(\mathbb{R}^{n})$ with its usual action on the affine space $\mathbb{R}^{n}$ . Let $p:\operatorname{Aff}(G)=G\mathbb{o}\operatorname{Aut}(G)\rightarrow \operatorname{Aut}(G)$ denote the natural projection onto the second factor.

Definition 1.1. A subgroup $\unicode[STIX]{x1D6E4}\subseteq \operatorname{Aff}(G)$ is called almost-crystallographic if and only if $p(\unicode[STIX]{x1D6E4})$ is finite and $\unicode[STIX]{x1D6E4}\cap G$ is a uniform and discrete subgroup of $G$ . The finite group $F=p(\unicode[STIX]{x1D6E4})$ is called the holonomy group of $\unicode[STIX]{x1D6E4}$ .

The action of $\unicode[STIX]{x1D6E4}$ on $G$ is properly discontinuous and cocompact and when $\unicode[STIX]{x1D6E4}$ is torsion-free, this action becomes a free action, from which we can conclude that the resulting quotient space $\unicode[STIX]{x1D6E4}\backslash G$ is a compact manifold with fundamental group $\unicode[STIX]{x1D6E4}$ .

Definition 1.2. A torsion-free almost-crystallographic group $\unicode[STIX]{x1D6E4}\subseteq \operatorname{Aff}(G)$ is called an almost-Bieberbach group, and the corresponding manifold $\unicode[STIX]{x1D6E4}\backslash G$ is called an infra-nilmanifold (modeled on $G$ ).

When the holonomy group is trivial, $\unicode[STIX]{x1D6E4}$ will be a lattice in $G$ and the corresponding manifold $\unicode[STIX]{x1D6E4}\backslash G$ is a nilmanifold. When $G$ is abelian, $\unicode[STIX]{x1D6E4}$ will be called a Bieberbach group and $\unicode[STIX]{x1D6E4}\backslash G$ a compact flat manifold. When $G$ is abelian and the holonomy group of $\unicode[STIX]{x1D6E4}$ is trivial, then $\unicode[STIX]{x1D6E4}$ is just a lattice in some $\mathbb{R}^{n}$ and $\unicode[STIX]{x1D6E4}\backslash G$ is a torus.

Now, define the semigroup $\operatorname{aff}(G)=G\mathbb{o}\operatorname{Endo}(G)$ , where $\operatorname{Endo}(G)$ is the set of continuous endomorphisms of $G$ . Note that $\operatorname{aff}(G)$ acts on $G$ in a similar way as $\operatorname{Aff}(G)$ , that is, any element $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ of $\operatorname{aff}(G)$ can be seen as a self-map of $G$ :

$$\begin{eqnarray}(\unicode[STIX]{x1D6FF},\mathfrak{D}):\;G\rightarrow G:\;h\mapsto \unicode[STIX]{x1D6FF}\mathfrak{D}(h)\end{eqnarray}$$

and we refer to $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ as an affine map of $G$ . One of the nice features of infra-nilmanifolds is that any map on a infra-nilmanifold is homotopic to a map which is induced by an affine map of $G$ . One can prove this by using the following result by Lee.

Theorem 1.3. (Lee [18])

Let $G$ be a connected and simply connected nilpotent Lie group and suppose that $\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D6E4}^{\prime }\subseteq \operatorname{Aff}(G)$ are two almost-crystallographic groups modeled on $G$ . Then for any homomorphism $\unicode[STIX]{x1D711}:\unicode[STIX]{x1D6E4}\rightarrow \unicode[STIX]{x1D6E4}^{\prime }$ there exists an element $(\unicode[STIX]{x1D6FF},\mathfrak{D})\in \operatorname{aff}(G)$ such that

$$\begin{eqnarray}\forall \unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}:\;\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D6FF},\mathfrak{D})=(\unicode[STIX]{x1D6FF},\mathfrak{D})\unicode[STIX]{x1D6FE}.\end{eqnarray}$$

Note that we can consider the equality $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D6FF},\mathfrak{D})=(\unicode[STIX]{x1D6FF},\mathfrak{D})\unicode[STIX]{x1D6FE}$ in $\operatorname{aff}(G)$ , since $\operatorname{Aff}(G)$ is contained in $\operatorname{aff}(G)$ . With this equality in mind, it is easy to see that the affine map $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ induces a well-defined map

$$\begin{eqnarray}\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}:\unicode[STIX]{x1D6E4}\backslash G\rightarrow \unicode[STIX]{x1D6E4}^{\prime }\backslash G:\;\unicode[STIX]{x1D6E4}h\rightarrow \unicode[STIX]{x1D6E4}^{\prime }\unicode[STIX]{x1D6FF}\mathfrak{D}(h),\end{eqnarray}$$

which exactly induces the morphism $\unicode[STIX]{x1D711}$ on the level of the fundamental groups.

On the other hand, if we choose an arbitrary map $f:\unicode[STIX]{x1D6E4}\backslash G\rightarrow \unicode[STIX]{x1D6E4}^{\prime }\backslash G$ between two infra-nilmanifolds and choose a lifting $\tilde{f}:G\rightarrow G$ of $f$ , then there exists a morphism $\tilde{f}_{\ast }:\unicode[STIX]{x1D6E4}\rightarrow \unicode[STIX]{x1D6E4}^{\prime }$ such that $\tilde{f}_{\ast }(\unicode[STIX]{x1D6FE})\circ \tilde{f}=\tilde{f}\circ \unicode[STIX]{x1D6FE}$ , for all $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$ . By Theorem 1.3, an affine map $(\unicode[STIX]{x1D6FF},\mathfrak{D})\in \operatorname{aff}(G)$ exists which also satisfies $\tilde{f}_{\ast }(\unicode[STIX]{x1D6FE})\circ (\unicode[STIX]{x1D6FF},\mathfrak{D})=(\unicode[STIX]{x1D6FF},\mathfrak{D})\circ \unicode[STIX]{x1D6FE}$ for all $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$ . Therefore, the induced map $\overline{(\unicode[STIX]{x1D6FF},D)}$ and $f$ are homotopic. We call $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ an affine homotopy lift of $f$ .

We end this introduction about infra-nilmanifolds with the definition of a hyperbolic map on an infra-nilmanifold. We denote by $\mathfrak{D}_{\ast }$ the Lie algebra endomorphism induced by $\mathfrak{D}$ on the Lie algebra $\mathfrak{g}$ associated to $G$ .

Definition 1.4. Let $M$ be an infra-nilmanifold and $f:M\rightarrow M$ be a continuous map, with $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ as an affine homotopy lift. We say that $f$ is a hyperbolic map if $\mathfrak{D}_{\ast }$ has no eigenvalues of modulus  $1$ .

Remark 1.5. The map $\mathfrak{D}$ , and hence also $\mathfrak{D}_{\ast }$ depends on the choice of the lift $\tilde{f}$ . Once the lift $\tilde{f}$ is fixed, and hence the morphism $\tilde{f}_{\ast }$ is fixed, the $\mathfrak{D}$ – part of the map $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ in Theorem 1.3 is also fixed (although the $\unicode[STIX]{x1D6FF}$ – part is not unique in general). It follows that $f$ determines $\mathfrak{D}$ only up to an inner automorphism of $G$ . But as inner automorphisms have no effect on the eigenvalues of $\mathfrak{D}_{\ast }$ (in the case of a nilpotent Lie group $G$ ) the notion of a hyperbolic map is well defined.

Two important classes of maps on infra-nilmanifolds which are hyperbolic are the expanding maps and the Anosov diffeomorphisms.

Remark 1.6. Due to [4, Lemma 4.5], it is known that every nowhere expanding map on an infra-nilmanifold only has eigenvalues $0$ or eigenvalues of modulus $1$ . This means that every hyperbolic map for which $\mathfrak{D}_{\ast }$ is not nilpotent has an eigenvalue of modulus strictly bigger than $1$ .

2 Nielsen theory, dynamical zeta functions and $\operatorname{HPer}(f)$

Let $f:X\rightarrow X$ be a self-map of a compact polyhedron $X$ . There are different ways to assign integers to this map $f$ that give information about the fixed points of $f$ . One of these integers is the Lefschetz number $L(f)$ which is defined as

$$\begin{eqnarray}L(f)=\mathop{\sum }_{i=0}^{\operatorname{dim}\;X}(-1)^{i}\operatorname{Tr}(f_{\ast ,i}:H_{i}(X,\mathbb{R})\rightarrow H_{i}(X,\mathbb{R})).\end{eqnarray}$$

In our situation, the space $X=\unicode[STIX]{x1D6E4}\backslash G$ will be a infra-nilmanifold, which is an aspherical space, and hence the (co)homology of the space $X=\unicode[STIX]{x1D6E4}\backslash G$ equals the (co)homology of the group $\unicode[STIX]{x1D6E4}$ . It follows that in this case we have (see also [13, p. 36])

$$\begin{eqnarray}\displaystyle L(f) & = & \displaystyle \mathop{\sum }_{i=0}^{\operatorname{dim}\;X}(-1)^{i}\operatorname{Tr}(f_{\ast ,i}:H_{i}(\unicode[STIX]{x1D6E4},\mathbb{R})\rightarrow H_{i}(\unicode[STIX]{x1D6E4},\mathbb{R}))\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{i=0}^{\operatorname{dim}\;X}(-1)^{i}\operatorname{Tr}(f_{i}^{\ast }:H^{i}(\unicode[STIX]{x1D6E4},\mathbb{R})\rightarrow H^{i}(\unicode[STIX]{x1D6E4},\mathbb{R})).\nonumber\end{eqnarray}$$

The Lefschetz fixed point theorem states that if $L(f)\neq 0$ , then $f$ has at least one fixed point. Because the Lefschetz number is only defined in terms of (co)homology groups, it remains invariant under a homotopy and hence, if $L(f)\neq 0$ , the Lefschetz fixed point theorem guarantees that any map homotopic to $f$ also has at least one fixed point.

Another integer giving information on the fixed points of $f$ is the Nielsen number $N(f)$ . It is a homotopy-invariant lower bound for the number of fixed points of $f$ . To define $N(f)$ , fix a reference lifting $\tilde{f}$ of $f$ with respect to a universal cover $(\tilde{X},p)$ of $X$ and denote the group of covering transformations by ${\mathcal{D}}$ . For $\unicode[STIX]{x1D6FC}\in {\mathcal{D}}$ , the sets $p(\operatorname{Fix}(\unicode[STIX]{x1D6FC}\circ \tilde{f}))$ form a partition of the fixed point set $\operatorname{Fix}(f)$ . These sets are called fixed point classes. By using the fixed point index, we can assign an integer to each fixed point class in such a way that if a nonzero integer is assigned, the fixed point class cannot completely vanish under a homotopy. Such a nonvanishing fixed point class will be called essential and $N(f)$ is defined as the number of essential fixed point classes of $f$ .

By definition, it is clear that $N(f)$ will indeed be a homotopy-invariant lower bound for the number of fixed points of $f$ . Hence, in general, $N(f)$ will give more information about the fixed points of $f$ than $L(f)$ . The downside, however, is that Nielsen numbers are often much harder to compute than Lefschetz numbers, because the fixed point index can be a tedious thing to work with. Luckily, on infra-nilmanifolds there exists an algebraic formula to compute $N(f)$ , which makes them a convenient class of manifolds to study Nielsen theory on. More information on both $L(f)$ and $N(f)$ can be found in for example, [3, 14, 15].

By using the Lefschetz and Nielsen numbers of iterates of $f$ as coefficients, it is possible to define the so-called dynamical zeta functions. The Lefschetz zeta function was introduced by Smale in [21]:

$$\begin{eqnarray}L_{f}(z)=\exp \left(\mathop{\sum }_{k=1}^{+\infty }\frac{L(f^{k})}{k}z^{k}\right).\end{eqnarray}$$

In his paper, Smale also proved that the Lefschetz zeta function is always rational for self-maps on compact polyhedra.

The proof is actually quite straightforward. Let the $\unicode[STIX]{x1D706}_{ij}$ ’s denote the eigenvalues of $f_{\ast }^{i}:H^{i}(X,\mathbb{R})\rightarrow H^{i}(X,\mathbb{R})$ , with $j\in \{1,\ldots ,\operatorname{dim}(H^{i}(X,\mathbb{R}))\}$ . Because the trace of a matrix is the sum of the eigenvalues, we find

$$\begin{eqnarray}L_{f}(z)=\exp \left(\mathop{\sum }_{k=1}^{+\infty }\left(\mathop{\sum }_{i=0}^{\operatorname{dim}\;X}(-1)^{i}\mathop{\sum }_{j=1}^{\operatorname{dim}\;H^{i}(X)}\unicode[STIX]{x1D706}_{ij}^{k}\right)\frac{z^{k}}{k}\right).\end{eqnarray}$$

By reordering the terms and by using the fact that

$$\begin{eqnarray}\mathop{\sum }_{k=1}^{+\infty }\frac{a^{k}z^{k}}{k}=-\text{log}(1-az)\quad \text{for}\;|z|<|a|^{-1},\end{eqnarray}$$

it is easy to derive that

(1) $$\begin{eqnarray}L_{f}(z)=\mathop{\prod }_{i=0}^{\operatorname{dim}\;X}\mathop{\prod }_{j=1}^{\operatorname{dim}\;H^{i}(X)}(1-\unicode[STIX]{x1D706}_{ij}z)^{(-1)^{i+1}}.\end{eqnarray}$$

Remark 2.1. Suppose that $\unicode[STIX]{x1D6EC}$ is a lattice of a connected and simply connected nilpotent Lie group $G$ and $f:\unicode[STIX]{x1D6EC}\backslash G\rightarrow \unicode[STIX]{x1D6EC}\backslash G$ is a self-map of the nilmanifold $\unicode[STIX]{x1D6EC}\backslash G$ with affine homotopy lift $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ . Let $\mathfrak{D}_{\ast }$ be the induced linear map on the Lie algebra $\mathfrak{g}$ of $G$ as before. The main result of [19] states that there are natural isomorphisms

$$\begin{eqnarray}H^{i}(\unicode[STIX]{x1D6EC},\mathbb{R})\cong H^{i}(\unicode[STIX]{x1D6EC}\backslash G,\mathbb{R})\cong H^{i}(\mathfrak{g},\mathbb{R}).\end{eqnarray}$$

The naturality of these automorphisms implies that there is a commutative diagram

Here $\mathfrak{D}_{\ast }^{i}$ is the map induced by $\mathfrak{D}_{\ast }$ on the $i$ th cohomology space of $\mathfrak{g}$ . Recall, that the cohomology of $\mathfrak{g}$ is defined as the cohomology of a cochain complex, where the $i$ th term is Hom $(\bigwedge ^{i}\mathfrak{g},\mathbb{R})=(\bigwedge ^{i}\mathfrak{g})^{\ast }$ , the dual space of $\bigwedge ^{i}\mathfrak{g}$ . So, $\mathfrak{D}_{\ast }^{i}$ is induced by the dual map of $\bigwedge ^{i}\mathfrak{D}_{\ast }$ . Since this dual map and $\bigwedge ^{i}\mathfrak{D}_{\ast }$ have the same eigenvalues, it follows that the set of eigenvalues of $\mathfrak{D}_{\ast }^{i}$ , hence also the set of eigenvalues $\unicode[STIX]{x1D706}_{i,j}$ of $f_{i}^{\ast }$ in expression (1), is a subset of the set of eigenvalues of $\bigwedge ^{i}\mathfrak{D}_{\ast }:\bigwedge ^{i}\mathfrak{g}\rightarrow \bigwedge ^{i}\mathfrak{g}$ . (This fact is also reflected in the formula obtained in [7, Theorem 23].)

The Nielsen zeta function was introduced by Fel’shtyn in [10, 20] and is defined in a similar way as the Lefschetz zeta function:

$$\begin{eqnarray}N_{f}(z)=\exp \left(\mathop{\sum }_{k=1}^{+\infty }\frac{N(f^{k})}{k}z^{k}\right).\end{eqnarray}$$

It is known that this zeta function does not always have to be a rational function. A counterexample for this can be found in [7], for example, in Remark 7.

For self-maps on infra-nilmanifolds, however, the Nielsen zeta function will always be rational. To prove this, one can exploit the fact that $N(f)$ and $L(f)$ are very closely related. In [5], we defined a subgroup $\unicode[STIX]{x1D6E4}_{+}$ of $\unicode[STIX]{x1D6E4}$ , which equals $\unicode[STIX]{x1D6E4}$ or is of index $2$ in $\unicode[STIX]{x1D6E4}$ . The precise definition is not of major significance for the rest of this paper. However, it allowed us to write $N_{f}(z)$ as a function of $L_{f}(z)$ if $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6E4}_{+}$ , and as a combination of $L_{f}(z)$ and $L_{f_{+}}(z)$ if $[\unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6E4}_{+}]=2$ . Here, $f_{+}:\unicode[STIX]{x1D6E4}_{+}\backslash G\rightarrow \unicode[STIX]{x1D6E4}_{+}\backslash G$ is a lift of $f$ to the $2$ -folded covering space $\unicode[STIX]{x1D6E4}_{+}\backslash G$ of $\unicode[STIX]{x1D6E4}\backslash G$ . The following theorem, together with the fact that Lefschetz zeta functions are always rational, therefore proves the rationality of Nielsen zeta functions for infra-nilmanifolds.

Theorem 2.2. [5, Theorem 4.6]

Let $M=\unicode[STIX]{x1D6E4}\backslash G$ be an infra-nilmanifold and let $f:M\rightarrow M$ be a self-map with affine homotopy lift $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ . Let $p$ denote the number of positive real eigenvalues of $\mathfrak{D}_{\ast }$ which are strictly greater than $1$ and let $n$ denote the number of negative real eigenvalues of $\mathfrak{D}_{\ast }$ which are strictly less than $-1$ . Then we have the following table of equations:

Moreover, this theorem also tells us that we can write $N_{f}(z)$ in a similar form as in equation (1), since every Lefschetz zeta function is of this form. More information about dynamical zeta functions can be found in [7].

Closely related to fixed point theory, is periodic point theory. We call $x\in X$ a periodic point of $f$ if there exists a positive integer $n$ , such that $f^{n}(x)=x$ . Of course, when $f^{n}(x)=x$ , this does not automatically imply that the actual period of $x$ is $n$ . For example, it is immediately clear that every fixed point is also a periodic point of period $n$ , for all $n>0$ . In order to exclude these points, we define the set of periodic points of pure period $n$ :

$$\begin{eqnarray}P_{n}(f)=\{x\in X\;\Vert \;f^{n}(x)=x\text{ and }f^{k}(x)\neq x,\forall k|n\}.\end{eqnarray}$$

The set of homotopy minimal periods of $f$ is then defined as the following subset of the positive integers:

$$\begin{eqnarray}\operatorname{HPer}(f)=\mathop{\bigcap }_{f\simeq g}\{n|P_{n}(g)\neq \emptyset \}.\end{eqnarray}$$

This set has been studied extensively, for example, in [1] for maps on the torus, in [12] for maps on nilmanifolds and in [9, 17] for maps on infra-nilmanifolds.

Just as Nielsen fixed point theory divides $\operatorname{Fix}(f)$ into different fixed point classes, Nielsen periodic point theory divides $\operatorname{Fix}(f^{n})$ into different fixed point classes, for all $n>0$ and looks for relations between fixed point classes on different levels. This idea is covered by the following definition.

Definition 2.3. Let $f:X\rightarrow X$ be a self-map. If $\mathbb{F}_{k}$ is a fixed point class of $f^{k}$ , then $\mathbb{F}_{k}$ will be contained in a fixed point class $\mathbb{F}_{kn}$ of $(f^{k})^{n}$ , for all $n$ . We say that $\mathbb{F}_{k}$ boosts to $\mathbb{F}_{kn}$ . On the other hand, we say that $\mathbb{F}_{kn}$ reduces to $\mathbb{F}_{k}$ .

An important definition that gives some structure to the boosting and reducing relations is the following.

Definition 2.4. A self-map $f:X\rightarrow X$ will be called essentially reducible if, for all $n,k$ , essential fixed point classes of $f^{kn}$ can only reduce to essential fixed point classes of $f^{k}$ . A space $X$ is called essentially reducible if every self-map $f:X\rightarrow X$ is essentially reducible.

It can be shown that the fixed point classes for maps on infra-nilmanifolds always have this nice structure for their boosting and reducing relations.

Theorem 2.5. [17]

Infra-nilmanifolds are essentially reducible.

One of the consequences of having this property, is the following.

Theorem 2.6. [1]

Suppose that $f$ is essentially reducible and suppose that

$$\begin{eqnarray}N(f^{k})>\mathop{\sum }_{p\text{ prime},\,p|k}N(f^{k/p}),\end{eqnarray}$$

then $k\in \operatorname{HPer}(f)$ .

The idea of this theorem is actually quite easy to grasp. Because maps on infra-nilmanifolds are essentially reducible, every reducible essential fixed point class on level $k$ will reduce to an essential fixed point class on level $\frac{k}{p}$ , with $p$ a prime divisor of $k$ . Therefore, the condition

$$\begin{eqnarray}N(f^{k})>\mathop{\sum }_{p\text{ prime},\,p|k}N(f^{k/p})\end{eqnarray}$$

actually tells us that there is definitely one irreducible essential fixed point class on level $k$ , which means that there is at least one periodic point of pure period $k$ .

For this paper, this is all we need to know about Nielsen periodic point theory. More information about Nielsen periodic point theory in general can be found in [11, 13] or [14].

3 $\operatorname{HPer}(f)$ for hyperbolic maps on infra-nilmanifolds

3.1 The nonnilpotent case

We begin with the following definition, which tells us something about the asymptotic behavior of the sequence $\left\{N(f^{k})\right\}_{k=1}^{\infty }$ .

Definition 3.1. The asymptotic Nielsen number of $f$ is defined as

$$\begin{eqnarray}N^{\infty }(f)=\max \left\{1,\limsup _{k\rightarrow \infty }N(f^{k})^{1/k}\right\}.\end{eqnarray}$$

By $\operatorname{sp}(A)$ we mean the spectral radius of the matrix or the operator $A$ . It equals the largest modulus of an eigenvalue of $A$ .

Theorem 3.2. [8, Theorem 4.3]

For a continuous map $f$ on an infra-nilmanifold, with affine homotopy lift $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ , such that $\mathfrak{D}_{\ast }$ has no eigenvalue $1$ , we have

$$\begin{eqnarray}N^{\infty }(f)=\operatorname{sp}\Bigl(\bigwedge \mathfrak{D}_{\ast }\Bigr).\end{eqnarray}$$

If $\{\unicode[STIX]{x1D708}_{i}\}_{i\in I}$ is the set of eigenvalues of $\mathfrak{D}_{\ast }$ , we know that

$$\begin{eqnarray}\operatorname{sp}\Bigl(\bigwedge \mathfrak{D}_{\ast }\Bigr)=\left\{\begin{array}{@{}ll@{}}\mathop{\prod }_{|\unicode[STIX]{x1D708}_{i}|>1}|\unicode[STIX]{x1D708}_{i}|\quad & \text{if }\operatorname{sp}(\mathfrak{D}_{\ast })>1,\\ 1\quad & \text{if }\operatorname{sp}(\mathfrak{D}_{\ast })\leqslant 1.\end{array}\right.\end{eqnarray}$$

Therefore, we have the following corollary of Theorem 3.2.

Corollary 3.3. Let $f$ be a hyperbolic, continuous map on an infra-nilmanifold. Let $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ be an affine homotopy lift of $f$ and let $\{\unicode[STIX]{x1D708}_{i}\}_{i\in I}$ be the set of eigenvalues of $\mathfrak{D}_{\ast }$ . If $\mathfrak{D}_{\ast }$ is not nilpotent, then

$$\begin{eqnarray}N^{\infty }(f)=\mathop{\prod }_{|\unicode[STIX]{x1D708}_{i}|>1}|\unicode[STIX]{x1D708}_{i}|.\end{eqnarray}$$

Proof. When $\mathfrak{D}_{\ast }$ is not nilpotent, we know by Remark 1.6 that $\operatorname{sp}(\mathfrak{D}_{\ast })>1$ . Because $f$ is hyperbolic, $1$ is certainly not an eigenvalue of $\mathfrak{D}_{\ast }$ and therefore, we can use the result of Theorem 3.2.◻

Because of Theorem 2.2, we know that $N_{f}(z)$ can be written as the quotient of Lefschetz zeta functions. Since every Lefschetz zeta function on a compact polyhedron is of the form

$$\begin{eqnarray}L_{f}(z)=\mathop{\prod }_{i=1}^{m}(1-\unicode[STIX]{x1D707}_{i}z)^{\unicode[STIX]{x1D6FE}_{i}},\end{eqnarray}$$

with $\unicode[STIX]{x1D707}_{i}\in \mathbb{C}$ and $\unicode[STIX]{x1D6FE}_{i}\in \{1,-1\}$ , the same will hold for $N_{f}(z)$ . Also, it is easy to check that

$$\begin{eqnarray}N_{f}(z)=\mathop{\prod }_{i=1}^{n}(1-\unicode[STIX]{x1D706}_{i}z)^{-\unicode[STIX]{x1D700}_{i}}\Rightarrow N(f^{k})=\mathop{\sum }_{i=1}^{n}\unicode[STIX]{x1D700}_{i}\unicode[STIX]{x1D706}_{i}^{k},\end{eqnarray}$$

for all $k\in \mathbb{N}$ .

In Remark 2.1 we already mentioned the fact that for nilmanifolds the $\unicode[STIX]{x1D707}_{i}$ ’s appearing in the expression for $L_{f}(z)$ are eigenvalues of $\bigwedge \mathfrak{D}_{\ast }$ . We now claim that the same holds for maps on infra-nilmanifolds. Consider an infra-nilmanifold $\unicode[STIX]{x1D6E4}\backslash G$ and a self-map $f$ of $\unicode[STIX]{x1D6E4}\backslash G$ with affine homotopy lift $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ . Without loss of generality, we may assume that $f=\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}$ . We now fix a fully characteristic subgroup $\unicode[STIX]{x1D6EC}$ of finite index in $\unicode[STIX]{x1D6E4}$ that is contained in $G$ (e.g., see [16]). Hence for the induced morphism $f_{\ast }:\unicode[STIX]{x1D6E4}\rightarrow \unicode[STIX]{x1D6E4}$ we have that $f_{\ast }(\unicode[STIX]{x1D6EC})\subseteq \unicode[STIX]{x1D6EC}$ . It follows that $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ also induces a map $\hat{f}$ on the nilmanifold $\unicode[STIX]{x1D6EC}\backslash G$ and that $\hat{f}_{\ast }=f_{\ast |\unicode[STIX]{x1D6EC}}$ . By [2, Theorem III 10.4] we know that the restriction map induces an isomorphism $\text{res}:H^{i}(\unicode[STIX]{x1D6E4},\mathbb{Q})\rightarrow H^{i}(\unicode[STIX]{x1D6EC},\mathbb{Q})^{\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6EC}}$ . As the restriction map is natural, we obtain the following commutative diagram:

It follows that each of the eigenvalues of $f_{\ast }^{i}$ is also an eigenvalue of $\hat{f}_{\ast }^{i}$ . Since the latter ones are all eigenvalues of $\bigwedge ^{i}\mathfrak{D}_{\ast }$ , by Remark 2.1, it follows that all eigenvalues of $f_{\ast }^{i}$ are also eigenvalues of $\bigwedge ^{i}\mathfrak{D}_{\ast }$ . This means that the $\unicode[STIX]{x1D707}_{i}$ ’s appearing in the expression for $L_{f}(z)$ are eigenvalues of $\bigwedge \mathfrak{D}_{\ast }$ and of course, because $f_{+}$ has the same affine homotopy lift as $f$ , the same applies to $L_{f_{+}}(z)$ .

By Theorem 2.2, we know that $N_{f}(z)$ can be written as a combination of $L_{f}(z)$ and possibly $L_{f_{+}}(z)$ , or as a combination of $L_{f}(-z)$ and possibly $L_{f_{+}}(-z)$ . In the first case, by the previous discussion we see that all $\unicode[STIX]{x1D706}_{i}$ ’s in the expression for $N_{f}(z)$ are eigenvalues of $\bigwedge \mathfrak{D}_{\ast }$ . In the latter case, all $\unicode[STIX]{x1D706}_{i}$ ’s are the opposite of eigenvalues of $\bigwedge \mathfrak{D}_{\ast }$ . This means that we can write

$$\begin{eqnarray}N(f^{k})=\mathop{\sum }_{i=1}^{n}\unicode[STIX]{x1D700}_{i}\unicode[STIX]{x1D706}_{i}^{k},\end{eqnarray}$$

such that all $\unicode[STIX]{x1D706}_{i}$ ’s or all $-\unicode[STIX]{x1D706}_{i}$ ’s are eigenvalues of $\bigwedge \mathfrak{D}_{\ast }$ .

Lemma 3.4. If $f$ is a nonnilpotent hyperbolic map on an infra-nilmanifold, with $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ as affine homotopy lift, it is possible to write

$$\begin{eqnarray}N(f^{k})=\mathop{\sum }_{i=1}^{m}a_{i}\unicode[STIX]{x1D706}_{i}^{k},\end{eqnarray}$$

with $a_{i}\in \mathbb{Z}$ , $a_{1}\geqslant 1$ and such that

$$\begin{eqnarray}|\unicode[STIX]{x1D706}_{1}|=\unicode[STIX]{x1D706}_{1}=\operatorname{sp}\Bigl(\bigwedge \mathfrak{D}_{\ast }\Bigr)>|\unicode[STIX]{x1D706}_{2}|\geqslant \cdots \geqslant |\unicode[STIX]{x1D706}_{m}|.\end{eqnarray}$$

Proof. By previous arguments, we know that it is possible to write

$$\begin{eqnarray}N(f^{k})=\mathop{\sum }_{i=1}^{n}\unicode[STIX]{x1D700}_{i}\unicode[STIX]{x1D706}_{i}^{k},\end{eqnarray}$$

where all $\unicode[STIX]{x1D706}_{i}$ ’s or all $-\unicode[STIX]{x1D706}_{i}$ ’s are eigenvalues of $\bigwedge \mathfrak{D}_{\ast }$ . By grouping the $\unicode[STIX]{x1D706}$ ’s that appear more than once and by changing the order, we obtain the desired form

$$\begin{eqnarray}N(f^{k})=\mathop{\sum }_{i=1}^{m}a_{i}\unicode[STIX]{x1D706}_{i}^{k},\end{eqnarray}$$

with $a_{i}\in \mathbb{Z}$ and $|\unicode[STIX]{x1D706}_{1}|\geqslant |\unicode[STIX]{x1D706}_{2}|\geqslant \ldots \geqslant |\unicode[STIX]{x1D706}_{m}|.$ There is a unique eigenvalue of $\bigwedge \mathfrak{D}_{\ast }$ of maximal modulus, namely the product

$$\begin{eqnarray}\mathop{\prod }_{|\unicode[STIX]{x1D706}_{i}|\geqslant 1}\unicode[STIX]{x1D706}_{i}=\unicode[STIX]{x1D707}_{1}.\end{eqnarray}$$

Note that the product is real, because for every $\unicode[STIX]{x1D706}\not \in \mathbb{R}$ , we know that if $|\unicode[STIX]{x1D706}|>1$ , then $|\overline{\unicode[STIX]{x1D706}}|>1$ and both are eigenvalues of $\bigwedge \mathfrak{D}_{\ast }$ , because $\mathfrak{D}_{\ast }$ is a real matrix. It is unique because $f$ is hyperbolic and $\mathfrak{D}_{\ast }$ has no eigenvalues of modulus  $1$ .

Because of Theorem 3.2, we know that $N^{\infty }(f)=\operatorname{sp}(\bigwedge \mathfrak{D}_{\ast })=|\unicode[STIX]{x1D707}_{1}|$ . Suppose now that $\unicode[STIX]{x1D707}_{1}$ or $-\unicode[STIX]{x1D707}_{1}$ does not appear as one of the $\unicode[STIX]{x1D706}$ ’s in the expression of $N(f^{k})$ . Then, it should still hold that

$$\begin{eqnarray}1=\limsup _{k\rightarrow \infty }\left(\frac{\mathop{\sum }_{i=1}^{m}a_{i}\unicode[STIX]{x1D706}_{i}^{k}}{\unicode[STIX]{x1D707}_{1}^{k}}\right)^{1/k}.\end{eqnarray}$$

Let $a_{\max }=\max \{|a_{i}|\}$ , then it is easy to derive that for all $k$ :

$$\begin{eqnarray}\frac{\mathop{\sum }_{i=1}^{m}a_{i}\unicode[STIX]{x1D706}_{i}^{k}}{\unicode[STIX]{x1D707}_{1}^{k}}\leqslant \mathop{\sum }_{i=1}^{m}|a_{i}|\left|\frac{\unicode[STIX]{x1D706}_{i}}{\unicode[STIX]{x1D707}_{1}}\right|^{k}\leqslant ma_{ max}\left|\frac{\unicode[STIX]{x1D706}_{1}}{\unicode[STIX]{x1D707}_{1}}\right|^{k}.\end{eqnarray}$$

So, we would have that

$$\begin{eqnarray}1\leqslant \limsup _{k\rightarrow \infty }\left(ma_{\max }\left|\frac{\unicode[STIX]{x1D706}_{1}}{\unicode[STIX]{x1D707}_{1}}\right|^{k}\right)^{1/k}=\left|\frac{\unicode[STIX]{x1D706}_{1}}{\unicode[STIX]{x1D707}_{1}}\right|<1,\end{eqnarray}$$

where the last inequality follows from the fact that $\unicode[STIX]{x1D707}_{1}$ is the unique eigenvalue of maximal modulus. Moreover, an easy argument shows that $a_{1}<0$ or $\unicode[STIX]{x1D706}_{1}<0$ cannot occur in the expression of $N(f^{k})$ , because otherwise $N(f^{k})$ would be negative for sufficiently large $k$ . As we have already proved that $a_{1}=0$ is impossible, we know that $a_{1}\geqslant 1$ and that $\operatorname{sp}(\bigwedge \mathfrak{D}_{\ast })$ will appear as one of the $\unicode[STIX]{x1D706}$ ’s in the expression for $N(f^{k})$ .◻

Remark 3.5. The fact that $\operatorname{sp}(\bigwedge \mathfrak{D}_{\ast })$ has to appear in the expression for $N(f^{k})$ was proved in a more general setting in [9].

Lemma 3.6. If $f$ is a hyperbolic map on an infra-nilmanifold, then $N(f^{k})\neq 0$ for all $k>0$ .

Proof. Let $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ be an affine homotopy lift of $f$ and let $F$ be the holonomy group of the infra–nilmanifold. By [16], we know that

$$\begin{eqnarray}N(f^{k})=\frac{1}{\#F}\mathop{\sum }_{\mathfrak{A}\in F}|\text{det}(I-\mathfrak{A}_{\ast }\mathfrak{D}_{\ast }^{k})|.\end{eqnarray}$$

Because all the terms make a nonnegative contribution to this sum, we know that

$$\begin{eqnarray}N(f^{k})\geqslant \frac{1}{\#F}|\text{det}(I-\mathfrak{D}_{\ast }^{k})|=\frac{1}{\#F}\mathop{\prod }_{i=1}^{n}|1-\unicode[STIX]{x1D707}_{i}^{k}|>0,\end{eqnarray}$$

where the $\unicode[STIX]{x1D707}_{i}$ are all the eigenvalues of $\mathfrak{D}_{\ast }$ . The last inequality follows from the fact that $f$ is hyperbolic and so there are no eigenvalues of modulus  $1$ .◻

From now on, we consider $f$ to be a hyperbolic map on an infra-nilmanifold and $N(f^{k})$ to be of the form

$$\begin{eqnarray}N(f^{k})=\mathop{\sum }_{i=1}^{m}a_{i}\unicode[STIX]{x1D706}_{i}^{k},\end{eqnarray}$$

with $a_{i}\in \mathbb{Z}$ , $a_{1}\geqslant 1$ and such that

$$\begin{eqnarray}|\unicode[STIX]{x1D706}_{1}|=\unicode[STIX]{x1D706}_{1}=\operatorname{sp}\Bigl(\bigwedge \mathfrak{D}_{\ast }\Bigr)>|\unicode[STIX]{x1D706}_{2}|\geqslant \ldots \geqslant |\unicode[STIX]{x1D706}_{m}|.\end{eqnarray}$$

For the sake of clarity, we keep using this notation in the rest of this paragraph.

Lemma 3.7. For all $\unicode[STIX]{x1D707}$ such that $\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D707}>1$ , there exists $k_{0}\in \mathbb{N}$ , such that for all $k\geqslant k_{0}$ and for all $n\in \mathbb{N}$ , we have the following inequality:

$$\begin{eqnarray}N(f^{k+n})>\unicode[STIX]{x1D707}^{n}N(f^{k}).\end{eqnarray}$$

Proof. Let $1>\unicode[STIX]{x1D700}>0$ , such that

$$\begin{eqnarray}\frac{\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D707}}\geqslant \unicode[STIX]{x1D700}>0.\end{eqnarray}$$

Note that this implies that

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{1}\frac{1-\unicode[STIX]{x1D700}}{1+\unicode[STIX]{x1D700}}\geqslant \unicode[STIX]{x1D707}.\end{eqnarray}$$

Now, choose $k_{0}\in \mathbb{N}$ such that, for all $i\in \{2,\ldots ,m\}$ ,

$$\begin{eqnarray}\left|\frac{a_{i}}{a_{1}}\right|\left|\frac{\unicode[STIX]{x1D706}_{i}}{\unicode[STIX]{x1D706}_{1}}\right|^{k_{0}}<\frac{\unicode[STIX]{x1D700}}{m}.\end{eqnarray}$$

Because of Lemma 3.4, we know that $|\unicode[STIX]{x1D706}_{1}|>|\unicode[STIX]{x1D706}_{i}|$ , for all these $i$ ’s, so the inequality will hold for $k_{0}$ sufficiently large.

Now, consider the fraction

$$\begin{eqnarray}\frac{N(f^{k+n})}{N(f^{k})}=\frac{a_{1}\unicode[STIX]{x1D706}_{1}^{k+n}+\mathop{\sum }_{i=2}^{m}a_{i}\unicode[STIX]{x1D706}_{i}^{k+n}}{a_{1}\unicode[STIX]{x1D706}_{1}^{k}+\mathop{\sum }_{i=2}^{m}a_{i}\unicode[STIX]{x1D706}_{i}^{k}}=\frac{\unicode[STIX]{x1D706}_{1}^{n}+\mathop{\sum }_{i=2}^{m}\frac{a_{i}}{a_{1}}\frac{\unicode[STIX]{x1D706}_{i}}{\unicode[STIX]{x1D706}_{1}}^{k}\unicode[STIX]{x1D706}_{i}^{n}}{1+\displaystyle \mathop{\sum }_{i=2}^{m}\frac{a_{i}}{a_{1}}\frac{\unicode[STIX]{x1D706}_{i}}{\unicode[STIX]{x1D706}_{1}}^{k}}.\end{eqnarray}$$

Note that $N(f^{k})\neq 0$ , according to Lemma 3.6, so the fraction is well defined. It is now easy to see that this equality implies the following inequalities:

$$\begin{eqnarray}\frac{N(f^{k+n})}{N(f^{k})}\geqslant \frac{\unicode[STIX]{x1D706}_{1}^{n}-\left|\mathop{\sum }_{i=2}^{m}\frac{a_{i}}{a_{1}}\frac{\unicode[STIX]{x1D706}_{i}}{\unicode[STIX]{x1D706}_{1}}^{k}\right|\unicode[STIX]{x1D706}_{1}^{n}}{1+\left|\mathop{\sum }_{i=2}^{m}\frac{a_{i}}{a_{1}}\frac{\unicode[STIX]{x1D706}_{i}}{\unicode[STIX]{x1D706}_{1}}^{k}\right|}>\unicode[STIX]{x1D706}_{1}^{n}\frac{1-\unicode[STIX]{x1D700}}{1+\unicode[STIX]{x1D700}}\geqslant \unicode[STIX]{x1D706}_{1}^{n}\left(\frac{1-\unicode[STIX]{x1D700}}{1+\unicode[STIX]{x1D700}}\right)^{n}\geqslant \unicode[STIX]{x1D707}^{n}.\end{eqnarray}$$

◻

Corollary 3.8. There exists $\unicode[STIX]{x1D708}$ , such that $\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D708}>1$ and an $l_{0}\in \mathbb{N}$ , such that for all $l\geqslant l_{0}$ and for all $k<l$ :

$$\begin{eqnarray}N(f^{l})>\unicode[STIX]{x1D708}^{l-k}N(f^{k}).\end{eqnarray}$$

Proof. Fix $\unicode[STIX]{x1D707}$ as in Lemma 3.7 and let $k_{0}$ be the resulting integer from this lemma. Note that Lemma 3.7 actually tells us that the sequence $\{N(f^{k})\}_{k=1}^{\infty }$ will be strictly increasing from a certain point onwards. Because all Nielsen numbers are integers, this means that there will exist $l_{0}\geqslant k_{0}$ , such that $N(f^{l_{0}})>N(f^{l})$ , for all $l<l_{0}$ , so also for all $l<k_{0}$ .

Now, let us define the following number

$$\begin{eqnarray}\unicode[STIX]{x1D70F}=\min \left\{\left(\frac{N(f^{l_{0}})}{N(f^{l})}\right)^{1/(l_{0}-l)}\Vert l<l_{0}\right\}.\end{eqnarray}$$

It is clear that $\unicode[STIX]{x1D70F}>1$ . Let $\unicode[STIX]{x1D708}=\min \left\{\unicode[STIX]{x1D707},(1+\unicode[STIX]{x1D70F})/2\right\}$ . Clearly, $\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D708}>1$ and, for all $k<l_{0}$ , we have the following inequalities:

$$\begin{eqnarray}\frac{N(f^{l_{0}})}{N(f^{k})}\geqslant \unicode[STIX]{x1D70F}^{l_{0}-k}>\unicode[STIX]{x1D708}^{l_{0}-k}.\end{eqnarray}$$

Because of Lemma 3.7 and the fact that $\unicode[STIX]{x1D707}\geqslant \unicode[STIX]{x1D708}$ , we know this inequality also applies to all $l\geqslant l_{0}$ .◻

Theorem 3.9. If $f$ is a hyperbolic map on an infra-nilmanifold, with affine homotopy lift $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ , such that $\mathfrak{D}_{\ast }$ is not nilpotent, then there exists an integer $m_{0}$ , such that

$$\begin{eqnarray}[m_{0},+\infty )\subset \operatorname{HPer}(f).\end{eqnarray}$$

Proof. Choose $\unicode[STIX]{x1D708}$ and $l_{0}$ as in Corollary 3.8. Since

$$\begin{eqnarray}\lim _{k\rightarrow \infty }\frac{\unicode[STIX]{x1D708}^{2^{k-1}}}{k}=+\infty ,\end{eqnarray}$$

we know there exists a $k_{0}$ , such that $\unicode[STIX]{x1D708}^{2^{k-1}}>k$ for all $k\geqslant k_{0}$ . Define $m_{0}=\max \{2^{k_{0}},2l_{0}+1\}$ .

Now, suppose that $m\geqslant m_{0}$ and $m$ is even. Let $K$ denote the number of different prime divisors of $m$ . As $m\geqslant 2l_{0}+1$ , we know that $m/2>l_{0}$ and hence the result of Corollary 3.8 applies. Therefore, we have the following inequalities

$$\begin{eqnarray}\mathop{\sum }_{p\text{ prime},p|m}N(f^{m/p})\leqslant K\cdot N(f^{m/2})<\frac{K}{\unicode[STIX]{x1D708}^{m/2}}\cdot N(f^{m}).\end{eqnarray}$$

By Theorem 2.6, it now suffices to show that

$$\begin{eqnarray}\frac{K}{\unicode[STIX]{x1D708}^{m/2}}\leqslant 1.\end{eqnarray}$$

Because $K$ denotes the number of different prime divisors of $m$ , we certainly know that $m>2^{K}$ . By the definition of $m_{0}$ , we also know that $m\geqslant 2^{k_{0}}$ . If $K\geqslant k_{0}$ , then

$$\begin{eqnarray}\unicode[STIX]{x1D708}^{m/2}>\unicode[STIX]{x1D708}^{2^{K-1}}>K,\end{eqnarray}$$

which is sufficient. If $k_{0}>K$ , we have that

$$\begin{eqnarray}\unicode[STIX]{x1D708}^{m/2}\geqslant \unicode[STIX]{x1D708}^{2^{k_{0}-1}}>k_{0}>K.\end{eqnarray}$$

So, when $m\geqslant m_{0}$ is even, $m\in \operatorname{HPer}(f)$ .

When $m\geqslant m_{0}$ is odd, a similar argument holds. Let $K$ again be the number of different prime divisors of $m$ and note that $m\geqslant 2l_{0}+1$ implies that $(m-1)/2\geqslant l_{0}$ . Again, by using Corollary 3.8, we obtain the following inequalities:

$$\begin{eqnarray}\mathop{\sum }_{p\text{prime},p|m}N(f^{m/p})\leqslant K\cdot N(f^{(m-1)/2})<\frac{K}{\unicode[STIX]{x1D708}^{(m+1)/2}}\cdot N(f^{m}).\end{eqnarray}$$

Again, $m>2^{K}$ and by definition $m\geqslant 2^{k_{0}}$ . When $K\geqslant k_{0}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D708}^{(m+1)/2}>\unicode[STIX]{x1D708}^{(2^{K}+1)/2}>\unicode[STIX]{x1D708}^{2^{K-1}}>K.\end{eqnarray}$$

When $k_{0}>K$ , the same reasoning gives us

$$\begin{eqnarray}\unicode[STIX]{x1D708}^{(m+1)/2}\geqslant \unicode[STIX]{x1D708}^{(2^{k_{0}}+1)/2}>\unicode[STIX]{x1D708}^{2^{k_{0}-1}}>k_{0}>K.\end{eqnarray}$$

This concludes the proof of this theorem. ◻

Remark 3.10. Having obtained Lemma 3.4, it is also possible to prove our main theorem in an alternative way, by following the approach of [8, Section 6].

Remark 3.11. Note that our proof also applies to every essentially irreducible map $f$ (on any manifold) for which there exists $\unicode[STIX]{x1D707}>1$ and $k_{0}\in \mathbb{N}$ , such that for all $k\geqslant k_{0}$ and for all $n\in \mathbb{N}$ , we have that

$$\begin{eqnarray}N(f^{k+n})>\unicode[STIX]{x1D707}^{n}N(f^{k}).\end{eqnarray}$$

This condition is therefore sufficient for $\operatorname{HPer}(f)$ to be cofinite in $\mathbb{N}$ .

3.2 The nilpotent case

For the sake of completeness, in this section we also treat the case where $\mathfrak{D}_{\ast }$ is nilpotent.

The following two theorems can be found in [6].

Theorem 3.12. Let $\unicode[STIX]{x1D6E4}\subseteq \operatorname{Aff}(G)$ be an almost-Bieberbach group with holonomy group $F\subseteq \operatorname{Aut}(G)$ . Let $M=\unicode[STIX]{x1D6E4}\backslash G$ be the associated infra-nilmanifold. If $f:M\rightarrow M$ is a map with affine homotopy lift $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ , then

$$\begin{eqnarray}R(f)=\infty \text{if and only if }\exists \mathfrak{A}\in F\text{ such that }\det (I-\mathfrak{A}_{\ast }\mathfrak{D}_{\ast })=0.\end{eqnarray}$$

Theorem 3.13. Let $f$ be a map on an infra-nilmanifold such that $R(f)<\infty$ , then

$$\begin{eqnarray}N(f)=R(f).\end{eqnarray}$$

Proposition 3.14. When $f$ is a hyperbolic map on an infra-nilmanifold with affine homotopy lift $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ such that $\mathfrak{D}_{\ast }$ is nilpotent then, for all $k$ ,

$$\begin{eqnarray}N(f^{k})=R(f^{k})=1.\end{eqnarray}$$

Proof. By combining Theorems 3.12 and 3.13 we know that every fixed point class of $f^{k}$ is essential if and only if for all $\mathfrak{A}\in F$ (where $F$ is the holonomy group of our infra-nilmanifold), it is true that

$$\begin{eqnarray}\det (I-\mathfrak{A}_{\ast }\mathfrak{D}_{\ast }^{k})\neq 0.\end{eqnarray}$$

By [4, Lemma 3.1], we know that there exists $\mathfrak{B}\in F$ , and an integer $l$ , such that

$$\begin{eqnarray}(\mathfrak{B}_{\ast }\mathfrak{D}_{\ast }^{k})^{l}=\mathfrak{D}_{\ast }^{lk}\qquad \text{and}\qquad \det (I-\mathfrak{A}_{\ast }\mathfrak{D}_{\ast }^{k})=\det (I-\mathfrak{B}_{\ast }\mathfrak{D}_{\ast }^{k}).\end{eqnarray}$$

Note that $\det (I-\mathfrak{B}_{\ast }\mathfrak{D}_{\ast }^{k})=0$ implies that $\mathfrak{B}_{\ast }\mathfrak{D}_{\ast }^{k}$ has an eigenvalue $1$ , but this would mean that $\mathfrak{D}_{\ast }^{lk}$ has an eigenvalue $1$ , which is in contradiction with the hyperbolicity of our map. Therefore, $R(f^{k})=N(f^{k})$ .

Note that $\mathfrak{D}_{\ast }$ only has eigenvalue $0$ . The fact that there exists $\mathfrak{B}\in F$ and an integer $l$ such that

$$\begin{eqnarray}(\mathfrak{B}_{\ast }\mathfrak{D}_{\ast }^{k})^{l}=\mathfrak{D}_{\ast }^{lk}\qquad \text{and}\qquad \det (I-\mathfrak{A}_{\ast }\mathfrak{D}_{\ast }^{k})=\det (I-\mathfrak{B}_{\ast }\mathfrak{D}_{\ast }^{k}),\end{eqnarray}$$

implies that $\mathfrak{B}_{\ast }\mathfrak{D}_{\ast }^{k}$ only has eigenvalue $0$ . As a consequence

$$\begin{eqnarray}\det (I-\mathfrak{A}_{\ast }\mathfrak{D}_{\ast }^{k})=\det (I-\mathfrak{B}_{\ast }\mathfrak{D}_{\ast }^{k})=1,\end{eqnarray}$$

for all $\mathfrak{A}\in F$ . By applying the main formula from [16], an easy computation shows that $N(f^{k})=1$ .◻

In [8], we find the following proposition.

Proposition 3.15. If $\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}:M\rightarrow M$ is a continuous map on an infra-nilmanifold, induced by an affine map, then every nonempty fixed point class is path-connected and

1. (1) Every essential fixed point class of $\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}$ consists of exactly one point.

2. (2) Every nonessential fixed point class of $\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}$ is empty or consists of infinitely many points.

Theorem 3.16. If $f$ is a hyperbolic map on an infra-nilmanifold with affine homotopy lift $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ such that $\mathfrak{D}_{\ast }$ is nilpotent, then

$$\begin{eqnarray}\operatorname{HPer}(f)=\{1\}.\end{eqnarray}$$

Proof. Let $\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}$ be the induced map of $(\unicode[STIX]{x1D6FF},\mathfrak{D})$ on the infra-nilmanifold. It suffices to show that $\operatorname{Per}(\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})})=\{1\}$ , because $N(f)=1$ immediately implies that $1\in \operatorname{HPer}(f)$ .

By Propositions 3.15 and 3.14, we know that $\operatorname{Fix}(\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}^{k})$ consists of precisely one point, for all $k>0$ . Because, for all $k>0$ , it holds that

$$\begin{eqnarray}\operatorname{Fix}(\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})})\subset \operatorname{Fix}(\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}^{k}),\end{eqnarray}$$

we know that $\operatorname{Fix}(\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}^{k})=\operatorname{Fix}(\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})})$ , for all $k>0$ . From this, it follows that $\overline{(\unicode[STIX]{x1D6FF},\mathfrak{D})}$ only has periodic points of pure period  $1$ .◻