Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.

The structures of the enveloping semigroups of certain elementary finite- and infinite-dimensional distal dynamical systems are given, answering open problems posed in 1982 by Namioka [Ellis groups and compact right topological groups. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26). American Mathematical Society, Providence, RI, 1984, 295–300]. The universal minimal system with (topological) quasi-discrete spectrum is obtained from the infinite-dimensional case. It is proved that, on the one hand, a minimal system is a factor of this universal system if and only if its enveloping semigroup has quasi-discrete spectrum and that, on the other hand, such a factor need not have quasi-discrete spectrum in itself. This leads to a natural generalization of the property of having quasi-discrete spectrum, which is named the ${\mathcal{W}}$ -property.

We study the different rates of escape of points under iteration by holomorphic self-maps of $\mathbb{C}^{\ast }=\mathbb{C}\setminus \{0\}$ for which both zero and infinity are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either zero, infinity or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points, each of which has $J(f)$ as its boundary.

We construct an area preserving homeomorphism ƒ: T2→T2 that is isotopic to the identity and fixed point free, but has the property that every essential simple closed curve C satisfies ƒ(C)∩C ≠ Ø. This answers a question of Guillou.

We prove that the restriction of the graph of a subfactor, ΓN,M, to an infinite subset of vertices with finite boundary has the same norm as ΓN,W. In particular, if N φ M is extremal with [M : N] > 4 and ΓN,M has an A∞, tail then ΓN, M = A∞.

Let $f:M\rightarrow M$ be a $C^{1}$ self-map of a smooth Riemannian manifold $M$ and $\unicode[STIX]{x1D707}$ be an $f$-invariant ergodic Borel probability measure with a compact support $\unicode[STIX]{x1D6EC}$. We prove that if $f$ is Hölder mild on the intersection of the singularity set and $\unicode[STIX]{x1D6EC}$, then the pointwise dimension of $\unicode[STIX]{x1D707}$ can be controlled by the Lyapunov exponents of $\unicode[STIX]{x1D707}$ with respect to $f$ and the entropy of $f$. Moreover, we establish the distinction of the Hausdorff dimension of the critical points sets of maps between the $C^{1,\unicode[STIX]{x1D6FC}}$ continuity and Hölder mildness conditions. Consequently, this shows that the Hölder mildness condition is much weaker than the $C^{1,\unicode[STIX]{x1D6FC}}$ continuity condition. As applications of our result, if we study the recurrence rate of $f$ instead of the pointwise dimension of $\unicode[STIX]{x1D707}$, then we deduce that the analogous relation exists between recurrence rate, entropy and Lyapunov exponents.

This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

In this paper, we consider random dynamical systems (abbreviated as RDSs) generated by compositions of one-sided stationary random endomorphisms of class C^2 of a compact manifold. We will first introduce the notions of entropy and Lyapunov exponents for such RDSs, and then prove that the entropy formula of Pesin type holds if the sample measures of an invariant measure are absolutely continuous with respect to the Lebesgue measure on the manifold. Our result covers those obtained by Ledrappier and Young [11], and Liu [12] for i.i.d. random diffeomorphisms or (non-invertible) endomorphisms, and that obtained by [12] for two-sided stationary random endomorphisms. If the phase spaces are compact and finite-dimensional manifolds without boundary, this result may be considered as the almost-final form of Pesin entropy formula for RDSs with absolutely continuous invariant or sample measures.

We consider inner functions on the unit disk which have a finite number of singularities on the unit circle. The restriction of such functions to the circle are maps onto the circle. We give sufficient conditions that these restrictions are exact endomorphisms whose natural extensions are Bernoulli and that the entropy is given by Rohlin's formula, We also give the entropy in closed form if ƒ' is in the Nevalinna class N. An example is considered. In the last section we show that if two restrictions are metrically isomorphic, they are diffeomorphic.

We consider the functional in a periodic setting. We discuss whether the minimizers or the stable solutions satisfy some symmetry or monotonicity properties, with special emphasis on the autonomous case when F is x-independent. In particular, we give an answer to a question posed by Victor Bangert when F is autonomous in dimension n≤3 and in any dimension for non-zero rotation vectors.

We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure-preserving $\mathbb {Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a nullsequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third, and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on ${\mathbb Z}^{d}$-systems.

We examine multiple ergodic averages of commuting transformations with polynomial iterates in which the polynomials may be pairwise dependent. In particular, we show that such averages are controlled by the Gowers–Host–Kra seminorms whenever the system satisfies some mild ergodicity assumptions. Combining this result with the general criteria for joint ergodicity established in our earlier work, we determine a necessary and sufficient condition under which such averages are jointly ergodic, in the sense that they converge in the mean to the product of integrals, or weakly jointly ergodic, in that they converge to the product of conditional expectations. As a corollary, we deduce a special case of a conjecture by Donoso, Koutsogiannis, and Sun in a stronger form.

In this paper we consider diffeomorphisms preserving hyperbolic Sinaĭ–Ruelle–Bowen (SRB) probability measures ${\it\mu}$ having intersections for almost every pair of the stable and unstable manifolds. In this context, when the dimension of the unstable manifold is constant almost everywhere, we show the ergodicity of ${\it\mu}$. As an application we obtain another proof of the ergodicity of a hyperbolic SRB measure for transitive surface diffeomorphisms, which is shown by Rodriguez Hertz, Rodriguez Hertz, Tahzibi and Ures [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].

Anderson and Putnam have recently shown that the space of all tilings of a substitution tiling scheme is a special case of the present author's ‘expanding attractor’. This resurrects an earlier conjecture of the present author in this special case: The tiling space of a substitution tiling space is a fiber bundle over the torus, with Cantor set fiber. If true, this could be important in the classification of these spaces, just as in the one-dimensional case. These spaces were known (in these two disciplines) to be locally the product of a disk and a Cantor set. Farrell and Jones gave an example showing our more inclusive conjecture is false. Here we show that the Penrose, Ammann and ‘stepped plane’ tiling spaces are in fact such bundles. Among dynamicists, the stepped plane tiling space (in special cases) is known as the ‘DA’ attractor of Smale. The formalism of substitution systems provides a Markov partition, and this is certainly part of their intuitive appeal. However, in dynamics such partitions are rare in dimensions greater than 2 and do not occur for the DA attractor. We consider a weaker concept, that omits this requirement. Finally, we construct an example like the Farrell–Jones example, in sufficient detail that it can be visualized as a smooth tiling space, as opposed to the more rigid geometric tiling space.

To obtain connections between grammars of languages and subshifts, Krieger introduced the idea of associating to a subshift X a certain increasing sequence of shifts of finite type X_{n} sitting inside X. Their union is the set of presynchronizing points. A system is called presynchronized if there is a sequence of irreducible components C_{n} \subset X_{n}, n \in \mathbb{N}, increasing with n and with dense union (a characterization of coded systems shows that presynchronized systems are coded). We investigate the uniqueness (up to eventual equality) of such sequences. For that an equivalence relation on the periodic presynchronizing points captures all the possible choices of increasing sequences of irreducible components. Dense equivalence classes correspond to dense unions. There is a one-sided and a two-sided theory. In the one-sided setting we show the uniqueness of a dense equivalence class and identify the right presynchronized systems as the well-known half-synchronized systems. The two-sided theory turns out to be more complicated (and interesting). There can be more than one dense class. Many examples are discussed in detail. There is a natural ordering on the set of (dense) equivalence classes. For each finite-order structure we find a representative in the countable class of systems which are intersections of Dyck shifts with shifts of finite type. Finally we discuss the relation to other known classes of subshifts, which is more subtle than in the one-sided setting. Almost all systems considered in this work will be coded.

It is shown that for certain classes of S-unimodal maps with aperiodic kneading sequences, the topological conjugacies are also quasisymmetric. This includes some infinitely renormalizable polynomials of unbounded type.

We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions as Poincaré maps of measured foliations, a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet, $A$ . The set of return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$ .

A graph surface P is a two-dimensional polyhedron having the simplest kind of non-trivial singularities which result from gluing surfaces with compact boundaries along boundary components. We study the behavior of the volume entropy h(g) of hyperbolic metrics g on a closed graph surface P depending on the lengths of singular geodesics $Q\subset P$. We show that always h(g) > 1 and $h(g)\to\infty$ as $L_g(Q)\to\infty$ for at least one singular geodesic Q.

Let $(X_k)_{k\geq 0}$ be a stationary and ergodic process with joint distribution $\mu $ , where the random variables $X_k$ take values in a finite set $\mathcal {A}$ . Let $R_n$ be the first time this process repeats its first n symbols of output. It is well known that $({1}/{n})\log R_n$ converges almost surely to the entropy of the process. Refined properties of $R_n$ (large deviations, multifractality, etc) are encoded in the return-time $L^q$ -spectrum defined as

provided the limit exists. We consider the case where $(X_k)_{k\geq 0}$ is distributed according to the equilibrium state of a potential with summable variation, and we prove that

where $P((1-q)\varphi )$ is the topological pressure of $(1-q)\varphi $ , the supremum is taken over all shift-invariant measures, and $q_\varphi ^*$ is the unique solution of $P((1-q)\varphi ) =\sup _\eta \int \varphi \,d\eta $ . Unexpectedly, this spectrum does not coincide with the $L^q$ -spectrum of $\mu _\varphi $ , which is $P((1-q)\varphi )$ , and it does not coincide with the waiting-time $L^q$ -spectrum in general. In fact, the return-time $L^q$ -spectrum coincides with the waiting-time $L^q$ -spectrum if and only if the equilibrium state of $\varphi $ is the measure of maximal entropy. As a by-product, we also improve the large deviation asymptotics of $({1}/{n})\log R_n$ .