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.

On a surface of negative curvature, we study the distribution of closed geodesics which are not far from minimizing length in their homology classes. These geodesics do not have the statistical behaviour obtained by Margulis for ‘generic’ geodesics and later extended by Lalley to geodesics satisfying certain homological constraints. We cannot use the usual techniques of ‘thermodynamic formalism for Anosov flows’, but we study the problem using ideas from Aubry–Mather theory, as well as some facts of low-dimensional topology.

A \mathbb{G}-solenoid is a laminated space whose leaves are copies of a single Lie group \mathbb{G} and whose transversals are totally disconnected sets. It inherits a \mathbb{G}-action and can be considered as a dynamical system. Free \mathbb{Z}^d-actions on the Cantor set as well as a large class of tiling spaces possess such a structure of \mathbb{G}-solenoids. For a large class of Lie groups, we show that a \mathbb{G}-solenoid can be seen as a projective limit of branched manifolds modeled on \mathbb{G}. This allows us to give a topological description of the transverse invariant measures associated with a \mathbb{G}-solenoid in terms of a positive cone in the projective limit of the dim(\mathbb{G})-homology groups of these branched manifolds. In particular, we exhibit a simple criterion implying unique ergodicity. Particular attention is paid to the case when the Lie group \mathbb{G} is the group of affine orientation-preserving isometries of the Euclidean space or its subgroup of translations.

In this paper the Poincaré–Hopf inequalities are shown to be necessary and sufficient conditions for an abstract Lyapunov graph L to be continued to an abstract Lyapunov graph of Morse type. The Lyapunov graphs considered may represent smooth flows on closed orientable n-manifolds, n \geq 2. The continuation which is presented by means of a constructive algorithm, is shown to be unique in dimensions two and three. In all other dimensions, the exact number of possible continuations of L are presented.

We show that if G is a connected semisimple Lie group with finite center, and if G admits a locally faithful, non-tame action by isometries of a connected Lorentz manifold, then $\mathfrak{g}$ has an ideal which is Lie algebra isomorphic to $\mathfrak{sl}_2(\mathbb{R})$. We also analyze the collection of connected Lie groups G admitting a free action by isometries of a connected Lorentz manifold such that the action is properly ergodic with respect to the Lorentz volume form.

We study the structure of two-fold self-joinings for a typical transformation of the standard non-atomic Borel space $(X,\mu)$. We prove that the typical automorphism is not simple and semisimple. This is an answer to questions posed by del Junco, Rudolph, Lemanczyk and Mentzen. In order to answer these questions we proved that for a typical transformation there exists an ergodic self-joining concentrated on two fibers, i.e. there exists the simplest non-trivial ergodic self-joining.

In this paper we will consider a bifurcation that occurs in the topology of the Julia set of meromorphic maps as they become entire. For some meromorphic maps the Julia set will be a Cantor set. We will investigate how the Julia set changes as these meromorphic maps approach the entire function whose Julia set is a Cantor bouquet. Other meromorphic maps have a Julia set that is a Jordan curve. Again we will study how this curve changes as these meromorphic maps approach the entire function whose Julia set is a Cantor bouquet.

Given a unitary representation u of the locally compact group G on the Hilbert space H, we investigate the notion of infinite tensor power of u. Then we apply the results to study covariance of the associated canonical action of G on the generalized Cuntz algebra $\mathcal{O}_H$ in the GNS representation of (the gauge-invariant extension of) some pure quasi-free state. We examine in detail the case of a measure-preserving action of G on a measure space X. In this case, covariance is (almost) equivalent to the existence of a G-invariant state on $L^\infty(X)$.

In this paper we prove that on an Anosov manifold the space of symmetric m-tensor fields of vanishing energy is finite dimensional modulo the space of potential tensor fields for an arbitrary m and coincides with the latter for m=0 and m=1. For m=2 this question relates to the spectral rigidity problem.

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.

Let $(X,\mathcal{B},\mu, T)$ be an ergodic dynamical system on the finite measure space $(X,\mathcal{B},\mu )$. Let $\mathcal{K}$ denote the Kronecker factor of T, i.e. the closed linear span in L2 of the eigenfunctions for T. We say that $(X,\mathcal{B},\mu ,T)$ is a Wiener–Wintner (WW) dynamical system of power type $\alpha$ in L1 if there exists in $\mathcal{K}^{\bot}$ a dense set of functions f for which the following holds: there exists a finite positive constant Cf such that \[\bigg\| \sup_{\varepsilon} \bigg|\frac{1}{N} \sum_{n=1}^Nf\circ T^n e^{2\pi in\varepsilon}\bigg| \bigg\|_1\leq \frac{C_f}{N^{\alpha}}\] for all positive integers N. Examples of ergodic dynamical systems with this WW property include K automorphisms as well as some skew products over irrational rotations. For WW dynamical systems a simpler proof of the almost everywhere double recurrence property, random weights with a break of duality can be obtained. They also provide naturally almost everywhere continuous random Fourier series related to the spectral measure of the transformation.

Let r, s be natural numbers, r, s \geq 2, such that {\rm log}\,r/{\rm log}\,s is irrational. Let \mu_j = \sum_{k=0}^{s-1} p^{(j)}_{k}\delta_{k/s^j}, where p^{(j)}_k \in [0,1), \sum_{k=0}^{s-1} p^{(j)}_k =1, j = 1, 2, \dotsc and {\delta}_{x} denotes the probability atom at x. We prove that if there exists k_0\in \{0,\dotsc , s-2\}, such that \inf \{ p^{(j)}_{k_0}, p^{(j)}_{k_0+1} : j\in \mathbb N \} > 0 and \mu is the infinite convolution measure \mu = *^{\infty}_{j=1} {\mu_{j} }, where the convergence is in the weak* sense, then \mu-almost all numbers are normal to base r. This generalizes some earlier results of Feldman, Smorodinsky and Pushkin.

In an explicit family of partially hyperbolic diffeomorphisms of the torus T3, Shub and Wilkinson recently succeeded in perturbing the Lyapunov exponents of the center direction. We present here a local version of their argument, allowing one to perturb the center Lyapunov exponents of any partially hyperbolic system, in any dimension and with arbitrary dimension of the center bundle.

A combinatorial model for a property of continuous self-maps of a compact interval is a self-map \pi of a finite ordered set such that every continuous \pi-weakly monotone self-map of a compact interval has that property. We identify the minimal combinatorial models for the property ‘the set of periods is a given set’. Here the word minimal refers to the number of points in the domain of the model. We also identify the minimal permutation models and, in appropriate cases, the minimal combinatorial models for properties involving ‘horseshoes’.

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.

Nous donnons des invariants complets pour l'équivalence topologique de champs de vecteurs, en dimension 3, au voisinage d'une connexion (par des variétés invariantes de dimension 1) entre des selles possédant des valeurs propres complexes.

The exact order of mixing for zero-dimensional algebraic dynamical systems is not entirely understood. Here non-Archimedean norms in function fields of positive characteristic are used to exhibit an asymptotic shape in non-mixing sequences for algebraic \mathbb{Z}^2-actions. This gives a relationship between the order of mixing and the convex hull of the defining polynomial. Using these methods, we show that an algebraic dynamical system for which any shape of cardinality three is mixing, is mixing of order three, and for any k\geq 1 exhibit examples that are k-fold mixing but not (k+1)-fold mixing.

A partially hyperbolic diffeomorphism is dynamically coherent if its center, center-stable and center-unstable invariant distributions are integrable, i.e. tangent to foliations. Dynamical coherence is a key assumption in the theory of stable ergodicity. The main result: a partially hyperbolic diffeomorphism f\colon M\to M is dynamically coherent if the strong stable and unstable foliations are quasi-isometric in the universal cover \widetilde{M}, i.e. for any two points in the same leaf, the distance between them in \widetilde{M} is bounded from below by a linear function of the distance along the leaf.

We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps using their Markov diagrams.

Hölder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion).

Using an equivariant version of the Ruelle transfer operator studied by Parry and Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.