A framework for understanding the geometry of continuous actions of $\mathbb Z^d$ was developed by Boyle and Lind using the notion of expansive behaviour along lower-dimensional subspaces. For algebraic $\mathbb Z^d$-actions of entropy rank one, the expansive subdynamics are readily described in terms of Lyapunov exponents. Here we show that periodic point counts for elements of an entropy rank-one action determine the expansive subdynamics. Moreover, the finer structure of the non-expansive set is visible in the topological and smooth structure of a set of functions associated to the periodic point data.

Let f be a transcendental meromorphic function with finitely many poles such that the finite singularities of f-1 lie in a bounded set. We show that the Julia set of f has Hausdorff dimension strictly greater than one and packing dimension equal to two. The proof for Hausdorff dimension simplifies the earlier argument given for transcendental entire functions.

Suppose $M$ is a compact surface and $\Lambda\subset M$ is a non-trivial mixing hyperbolic attractor for some $f\in\mathrm{Diff}(M)$. We show that if $\Lambda$ is a hyperbolic set for some $g\in\mathrm{Diff}(M)$, then $\Lambda$ is a non-trivial mixing hyperbolic attractor or repeller for $g$.

Consider a tiling $\mathcal T$ of the two-dimensional Euclidean space made with copies up to translation of a finite number of polygons meeting each other full edge to full edge. In this paper, we prove that, associated with $\mathcal T$, there exists a tiling of a (compact) translation surface made with copies up to translation of some of the polygons used to construct $\mathcal T$. Furthermore, when $\mathcal T$ is repetitive, there exists a tiling of a translation surface, made with copies up to translation of arbitrarily large polygons chosen in a finite collection of patches of $\mathcal T$; each of these patches contain copies of all the polygons used to construct $\mathcal T$.

In this paper, we consider tilings of the hyperbolic 2-space ${\mathbb H}^2$, built with a finite number of polygonal tiles, up to affine transformation. To such a tiling $T$, we associate a space of tilings: the continuous hull $\Omega(T)$ on which the affine group acts. This space $\Omega(T)$ inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First, we prove that the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally, we give examples with an arbitrary number of ergodic invariant probability measures.

Let $h:D\rightarrow {\mathbb R}^2$ be a homeomorphism of a disk into the plane. Let $C\subset {\rm int}\,D$ be a Cantor set such that $(C,h|_C)$ is an adding machine. We give a condition under which there are points outside of C which have their entire orbits contained in D.

Let S be a compact surface without boundary, with Euler characteristic $\chi(S)>0$, and let $K \subset S$ be a compact set such that each component of K is cellular. The aim of this paper is to prove that, if $f:S \setminus K \rightarrow S \setminus K$ is a homeomorphism which has a certain instability property in the proximity of K, then the set of periodic orbits of f is non-empty. This result give us some corollaries:

1. there is no minimal homeomorphism $f:S^2 \setminus K \rightarrow S^2 \setminus K$ for K an arbitrary compact set;

2. if $f:S^2 \rightarrow S^2$ is an area-preserving homeomorphism, then $K=\text{cl(Per($f$))}$ is not an isolated invariant set.

We introduce the notions of a rational point, a rational sub-nilmanifold and a rational homomorphism of compact nilmanifolds and describe elementary properties thereof. We also show that the closure of the orbit of a point of a nilmanifold $X$ under a polynomial action of the group $\mathbb{Z}^{d}$ by translations is, up to a shift, a union of rational sub-nilmanifolds of $X$.

Let ${\mathbb{M}}={\mathbb{Z}}^D$ be a $D$-dimensional lattice, and let $({\mathcal{A}},+)$ be an abelian group. ${\mathcal{A}}^{\mathbb{M}}$ is then a compact abelian group under componentwise addition. A continuous function $\Phi:{\mathcal{A}}^{\mathbb{M}}\longrightarrow{\mathcal{A}}^{\mathbb{M}}$ is called a linear cellular automaton if there is a finite subset ${\mathbb{F}}\subset{\mathbb{M}}$ and non-zero coefficients $\varphi_{\textsf{f}}\in{\mathcal{Z}}$ so that, for any ${\bf{a}}\in{\mathcal{A}}^{\mathbb{M}}\, \Phi({\bf{a}}) = \sum_{{\textsf{f}}\in{\mathbb{F}}}\varphi_{\textsf{f}}\cdot\sigma^{{\textsf{f}}}({\bf{a}})$. Suppose that $\mu$ is a probability measure on ${\mathcal{A}}^{\mathbb{M}}$ whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on $\Phi$ and $\mu$) under which $\Phi$asymptotically randomizes$\mu$, meaning that $\mathrm{wk}^*-\lim_{{\mathbb{J}}\ni j\rightarrow\infty} \Phi^j\mu = \eta$, where $\eta$ is the Haar measure on ${\mathcal{A}}^{\mathbb{M}}$, and ${\mathbb{J}}\subset{\mathbb{N}}$ has Cesàro density one. In the case when $\Phi=1+\sigma$ and ${\mathcal{A}}=({{\mathbb{Z}}_{/p}})^s$ ($p$ prime), we provide a condition on $\mu$ that is both necessary and sufficient. We then use this to construct zero-entropy measures which are randomized by $1+\sigma$.

We give a non-commutative generalization of classical symbolic coding in the presence of a synchronizing word. This is done by a scattering theoretical approach. Classically, the existence of a synchronizing word turns out to be equivalent to asymptotic completeness of the corresponding Markov process. A criterion for asymptotic completeness in general is provided by the regularity of an associated extended transition operator. Commutative and non-commutative examples are analysed.

In this paper we revisit once again, see Shub and Sullivan (Ergod. Th. & Dynam. Sys.5 (1985), 285–289), a family of expanding circle endomorphisms. We consider a family $\{B_\theta\}$ of Blaschke products acting on the unit circle, $\mathbb{T}$, in the complex plane obtained by composing a given Blaschke product $B$ with the rotations about zero given by multiplication by $\theta \in \mathbb{T}$. While the initial map $B$ may have a fixed sink on $\mathbb{T}$, there is always an open set of $\theta$ for which $B_\theta$ is an expanding map. We prove a lower bound for the average measure theoretic entropy of this family of maps in terms of $\int \ln|B'(z)|\,{\it dz}$.

We study the spectral properties of the Ruelle–Perron–Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the ${\mathcal C}^\infty$ case, the essential spectral radius is arbitrarily small, which yields a description of the correlations with arbitrary precision. Moreover, we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the Sinai–Ruelle–Bowen measure, the variance for the central limit theorem, the rates of decay for smooth observable, etc.).

We study the algebraic properties of automorphism groups of two-sided, transitive, countable-state Markov shifts together with the dynamics of those groups on the shift space itself as well as on periodic orbits and the 1-point-compactification of the shift space. We present a complete solution to the cardinality question of the automorphism group for locally compact and non-locally compact, countable-state Markov shifts, shed some light on its huge subgroup structure and prove the analogue of Ryan's theorem about the center of the automorphism group in the non-compact setting. Moreover, we characterize the 1-point-compactification of locally compact, countable-state Markov shifts, whose automorphism groups are countable and show that these compact dynamical systems are conjugate to synchronized systems on doubly transitive points. Finally, we prove the existence of a class of locally compact, countable-state Markov shifts whose automorphism groups split into a direct sum of two groups, one being the infinite cyclic group generated by the shift map, the other being a countably infinite, centerless group, which contains all automorphisms that act on the orbit-complement of certain finite sets of symbols like the identity.

A dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta{\mathbb{N}}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, Glasner (On tame enveloping semigroups. Colloq. Math.105 (2006), 283–395) calls the system tame. In this paper, we study the tame system under an Abelian group action. We introduce the notion of scrambled pairs for a dynamical system under an Abelian group action and show that a tame system has no scrambled pair. At the same time, we give some sufficient conditions such that a pair is a scrambled one. Moreover, using these sufficient conditions we prove that a minimal tame system under an Abelian group action is almost automorphic and uniquely ergodic. This gives a positive answer to Problem 2.5 in the paper cited above. Finally, we prove that for ${\mathbb Z}$-actions a tame system is $M$-null, that is, the metric sequence entropy is zero for any invariant measure and any increasing sequence of natural numbers. We also give examples to show that tameness is strictly weaker than nullness for ${\mathbb Z}$-actions.

Let ${\mathbb{M}}={\mathbb{N}}^D$ be the positive orthant of a $D$-dimensional lattice and let $({\mathcal{G}},+)$ be a finite abelian group. Let ${\mathfrak{G}} \subseteq{\mathcal{G}}^{\mathbb{M}}$ be a subgroup shift, and let $\mu$ be a Markov random field whose support is ${\mathfrak{G}}$. Let $\Phi: {\mathfrak{G}} \longrightarrow {\mathfrak{G}}$ be a linear cellular automaton. Under broad conditions on ${\mathcal{G}}$, we show that the Cesaro average $N^{-1} \sum_{n=0}^{N-1} \Phi^n (\mu)$ converges to a measure of maximal entropy for the shift action on ${\mathfrak{G}}$.

For a given factor map $\pi:X\longrightarrow Y$ between two topological dynamical systems and a Borel cover ${\mathcal U}$, two notions of measure-theoretical conditional entropy $h_\mu^+(T,{\mathcal U}\mid Y)$ and $h_\mu^-(T,{\mathcal U}\mid Y)$ for an invariant Borel probability measure $\mu$ are introduced. It is shown that $h_\mu^+(T,{\mathcal U}\mid Y)=h_\mu^-(T,{\mathcal U}\mid Y)$. Moreover, $\max_{\mu}h_\mu^+(T,{\mathcal U}\mid Y)=h_{{\rm top}}(T,{\mathcal U}\mid Y)$ when $\mathcal U$ is an open cover. The relative variational principle is a consequence of the results.

For the family of exponential maps $E_{\kappa}(z)=\exp(z)+\kappa$, we prove an analog of Böttcher's theorem by showing that any two exponential maps $E_{\kappa_1}$ and $E_{\kappa_2}$ are conjugate on suitable subsets of their escaping sets, and this conjugacy is quasiconformal. Furthermore, we prove that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's ‘pinched disk model’ for the Julia sets of these maps. On the other hand, we show that two exponential maps are generally not conjugate on their sets of escaping points. We also answer several questions about escaping endpoints of dynamic rays. In particular, we give a necessary and sufficient condition for the ray to be continuously differentiable in such a point, and show that escaping points can escape arbitrarily slowly. Furthermore, we show that the principle of topological renormalization is false for attracting exponential maps.

It is a safe conjecture that most (not necessarily periodic) two-dimensional Lorentz gases with finite horizon are recurrent. Here we formalize this conjecture by means of a stochastic ensemble of Lorentz gases, in which independent identically distributed random scatterers are placed in each cell of a co-compact lattice in the plane. We prove that the typical Lorentz gas, in the sense of Baire, is recurrent, and give results in the direction of showing that recurrence is an almost sure property (including a zero--one law that holds in every dimension). A few toy models illustrate the extent of these results.

We propose to study the renormalization operator acting on critical $\mathcal{C}^r$ circle mappings. (More precisely, the operator acts on critical commuting pairs.) Assuming that there is a Banach manifold of critical analytic commuting pairs on which the renormalization operator acts hyperbolically (with non-trivial hyperbolic attractor), we prove that, for r > 2, the operator remains hyperbolic with the same expanding subspaces when acting on $\mathcal{C}^r$ commuting pairs. By this we mean that the tangent renormalization operator admits a hyperbolic splitting with the same unstable subbundle.

We consider a class of expanding maps of the 2-torus of the form $f(x,y)=(a(x,y), b(y))$ that are C$^2$-perturbations of linear ones. In that class we consider invariant sets $\Lambda$ possessing a simple Markov structure, and show there exist ergodic invariant measures supported on $\Lambda$ with Hausdorff dimension arbitrarily close to the Hausdorff dimension of $\Lambda$. For f and $\Lambda$ as above we also show that the Birkhoff exceptional set has full Hausdorff dimension.