We prove that the return time statistics of a dynamical system do not change if one passes to an induced (i.e. first return) map. We apply this to show exponential return time statistics in (i) smooth interval maps with nowhere-dense critical orbits and (ii) certain interval maps with neutral fixed points. The method also applies to (iii) certain quadratic maps of the complex plane.

A classical approach to the study of a dynamical system is to code it using a partition. This leads to the question of the quality of this coding and in particular to the problem of finding checkable conditions ensuring that the coding is essentially one-to-one. We prove that, with respect to an invariant measure with only positive Lyapunov exponents, it is basically enough that the map be one-to-one on each piece of the partition. This is a multi-dimensional generalization of a well-known fact in one-dimensional dynamics. We apply this result to the study of the measure of maximal entropy of endomorphisms of complex projective spaces.

Nous étudions la croissance des variétés instables des automorphismes (i.e. des difféomorphismes holomorphes) des surfaces complexes. Ces variétés sont paramétrées holomorphiquement par la droite complexe \mathbb{C} et la croissance est à prendre au sens de Nevanlinna. La formule obtenue exprime le taux de croissance en fonction des exposants de Lyapunov et de l'entropie topologique. La technique employée est également valable pour les difféomorphismes Anosov de \mathbb{T}^2.

We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling space to be weakly mixing, and for the dynamics of two such spaces to be topologically conjugate.

We study the convergence of pointwise ergodic means for random subsequences, in a universal framework, together with ergodic means which are modulated by random weights. The methods used in this work mainly involve Gaussian tools, transference principles and new results on oscillation functions.

Suppose that S is a transitive locally compact Markov shift with Gurevic entropy less than log p, where p is an integer. If the inequality is strict, then S admits a graph presentation with degree p. Additionally, we realize any pre-assigned values for the Salama entropies. This result is best possible, since there are examples of Gurevic entropy log p which do not admit a graph presentation with degree p. We also show that every non-compact locally compact Markov shift has presentations without synchronizing blocks.

A rational map f is called geometrically finite if every critical point contained in its Julia set is eventually periodic. If a perturbation of f into another geometrically finite rational map is horocyclic and preserves the critical orbit relations with respect to the Julia set of f, then we can construct a semiconjugacy or a topological conjugacy between their dynamics on the Julia sets.

An action of $\mathbb{Z}^k$ is associated to a higher rank graph $\Lambda$ satisfying a mild assumption. This generalizes the construction of a topological Markov shift arising from a non-negative integer matrix. We show that the stable Ruelle algebra of $\Lambda$ is strongly Morita equivalent to $C^*(\Lambda)$. Hence, if $\Lambda$ satisfies the aperiodicity condition, the stable Ruelle algebra is simple, stable and purely infinite.

Two measure-preserving dynamical systems are weakly disjoint if some pointwise convergence property is satisfied by ergodic averages on their direct product. Disjointness implies weak disjointness. We start studying this new concept, both by stating some general properties and by giving various examples.

For each positive integer n we use the concept of ‘admissible arrays on n symbols’ to define a set of positive integers $Q(n)$ which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In earlier joint work with M. Scheutzow, it was shown that the set $Q(n)$ is intimately connected to the set of periods of periodic points of classes of non-expansive nonlinear maps defined on the positive cone in $\mathbb{R}^n$. In this paper we continue the characterization of $Q(n)$ and present precise asymptotic estimates for the largest element of $Q(n)$. For example, if $\gamma(n)$ denotes the largest element of $Q(n)$, then we show that $\lim_{n \to \infty} (n\log n)^{-1/2}\log \gamma(n) = 1$. We also discuss why understanding further details about the fine structure of $Q(n)$ involves some delicate number theoretical issues.

In this paper, we study the behaviour of the entropy function of higher-dimensional shifts of finite type. We construct a topologically mixing $\mathbb{Z}^2$ shift of finite type whose ergodic invariant measures are connected in the $\overline{d}$ topology and whose entropy function has a strictly local maximum. We also construct a topologically mixing $\mathbb{Z}^2$ shift of finite type X with the property that there is a uniform gap between the topological entropy of X and the topological entropy of any subshift of X with stronger mixing properties. Our examples illustrate the necessity of strong topological mixing hypotheses in existing higher-dimensional representation and embedding theorems.

On montre que la condition combinatoire de fortes coïncidences est suffisante pour qu'un système substitutif de type Pisot soit isomorphe en mesure à un échange de morceaux dans un compact autosimilaire de mesure non nulle dans le produit d'un espace euclidien et d'extensions finies de corps p-adiques. En particulier, tout système substitutif de type Pisot est une extension finie de son facteur équicontinu maximal ; en règle générale, ce dernier contient une translation p-adique si et seulement si la matrice d'incidence de la substitution est nilpotente modulo p.

Let $r \geq 1$; the Følner sequences $\{F^{n,1}\}_{n=1}^{\infty}, \{F^{n,2}\}_{n=1}^{\infty}, \dotsc, \{F^{n,r}\}_{n=1}^{\infty}$ satisfy the bounded intersection property if there is a constant p such that, for any $n\in\mathbb{N}$ and $1\leq i\leq r$, each $F^{n,i}$ can intersect no more than p disjoint translates of $F^{n,1}, F^{n,2},\dotsc, F^{n,r}$. They have comparable magnitudes if $0<\varliminf_{n}|F^{n,i}|/|F^{n,j}|<\infty$ for $1\leq i,j\leq r$. Suppose that G is a countable Abelian group with an element of infinite order, and let $\mathcal{X}$ be a mixing finite (or $\beta$-local, with $\beta>1/2$) rank action of G on a probability space. Suppose that the Følner sequences of $\mathcal{X}$ satisfy the bounded intersection property, and have comparable magnitudes. Then $\mathcal{X}$ is mixing of all orders. We follow Ryzhikov's joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of k copies of $\mathcal{X}$ is necessarily product measure.