We analyze in detail the notions of induction and restriction for cellular automata. As a by-product we extend a few classical and recent theorems on cellular automata to uncountable groups.

We prove that if $G$ is a countable, discrete group having infinite normal subgroups with the relative property (T) of Kazhdan–Margulis, then the Bernoulli shift action of $G$ on $\prod_{g \in G} (X_0, \mu_0)_g$, for $(X_{0},\mu_{0})$ an arbitrary non-trivial probability space, has first cohomology group isomorphic to the character group of $G$.

Equilibrium states of rational maps for Hölder continuous potentials are not $\phi$-mixing, mainly due to the presence of critical points. Here we prove that for disks the normalized return times of arbitrary orders are, in the limit, Poisson distributed as the radius of the disks go to zero. The return times are normalized by the measure of the disks. We also show that rational maps are weakly Bernoulli with respect to the partition given by Denker and Urbanski.

For continuous maps ƒ of the circle to itself, we show: (A) the set of nonwandering points of ƒ coincides with that of ƒn for every odd n; (B) ƒ has a horseshoe if and only if it has a non-wandering homoclinic point; (C) if the set of periodic points is closed and non-empty, then every non-wandering point is periodic.

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is,

$$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$

$\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$

$\unicode[STIX]{x1D719}$

$G$

$H$

$\unicode[STIX]{x1D719}H=H$

$\ker (\unicode[STIX]{x1D719})\leq H$

$s(\unicode[STIX]{x1D719})$

$\log s(\unicode[STIX]{x1D719})$

$\unicode[STIX]{x1D719}$

$\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$

Dans cet article nous allons commencer par démontrer un résultat perturbatif, nous en déduirons différentes manières de créer des connexions à l'aide de perturbations de difféomorphismes petites en topologie C1. Ensuite, nous nous intéresserons plus spécifiquement à la place des variétés stables des points périodiques hyperboliques ainsi qu'à la fermeture d'orbites récurrentes en un sens faible; puis, nous ferons une étude plus fine des difféomorphismes de classe C1 ‘génériques’, montrant en particulier l'existence d'attracteurs, au sens de Liapounov, transitif en un certain sens faible ; enfin, nous expliquerons ce qui se passe pour les difféomorphismes qui préservent une forme volume finie.

For a rational function with Herman rings, we define a tree and a piece-wise linear map on it, which reflect the configuration of the Herman rings. Their properties are investigated and some examples are given. Moreover, it is possible to define a similar tree associated with (super) attractive basins or Siegel disks.

Let $\mu$ be a probability measure on a locally compact group $G$, and suppose $G$ acts measurably on a probability measure space $(X,m)$, preserving the measure $m$. We study ergodic theoretic properties of the action along $\mu$-i.i.d. random walks on $G$. It is shown that under a (necessary) spectral assumption on the $\mu$-averaging operator on $L^2(X,m)$, almost surely the mean and the pointwise (Kakutani's) random ergodic theorems have roughly $n^{-1/2}$ rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonal $G$-action on $(X\times X,m\times m)$, an almost surely exponential rate of mixing along random walks is obtained.

The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of $m$-integration as a $G$-invariant mean on $L^\infty(X,m)$. These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable $G$. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups' homogeneous spaces, for which our results can be applied.

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

Almost-sure convergence of (l/k) log Wk(x, y) to entropy for weak Bernoulli processes is proved, where Wk (x, y) is the waiting time until an initial segment of length k of a sample path x is seen in an independently chosen sample path y. Analogous almost-sure results are obtained in the approximate match case for very weak Bernoulli processes. The weak Bernoulli proof uses recent results obtained by the authors about the estimation of joint distributions, while the very weak Bernoulli result utilizes a new characterization of such processes in terms of a blowing-up property.

This article is devoted to studying which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things, we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand, we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.

In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $\unicode[STIX]{x1D719}$ is a Hölder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb{R}$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.

For a map $T:[0,1]\rightarrow [0,1]$ with an invariant measure $\unicode[STIX]{x1D707}$, we study, for a $\unicode[STIX]{x1D707}$-typical $x$, the set of points $y$ such that the inequality $|T^{n}x-y|<r_{n}$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D719}}$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

Let G be a connected finite-dimensional Lie group and M a compact surface. We investigate whether, for a given G and M, every continuous action of G on M must have a fixed (stationary) point. It is shown that when G is nilpotent and M has non-zero Euler characteristic that every action of G on M must have a fixed point. On the other hand, it is shown that the non-abelian 2-dimensional Lie group (affine group of the line) acts without fixed points on every compact surface. These results make it possible to complete this investigation for Lie groups of dimension at most 3.

We prove the meromorphicity of the zeta function on shifts of finite type for Hölder continuous functions assuming that the essential spectrum of the associated Ruelle operator is contained in the open unit disc. This result allows to extend the region of meromorphicity of the zeta function for Axiom A flows by a strip whose width is determined by the contraction rate of the flow.

The crossed product of an AF-algebra by an automorphism, a power of which is approximately inner, is shown to be embeddable into an AF-algebra. The proof uses almost inductive limit automorphisms, i.e. automorphisms possessing a sequence of almost invariant finite-dimensional C*-subalgebras converging to the given AF-algebra.

By employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses.

We show that the n-center problem in \mathbb{R}3 has positive topological entropy for n\ge 3. The proof is based on global regularization of singularities and the results of Gromov and Paternain on the topological entropy of geodesic flows. The n-center problem in S3 is also studied.

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.

We give a partial answer to a question posed by C. Pugh and M. Shub in [PS2]. A partially hyperbolic diffeomorphism $f\in \mathit{Diff}^{2}(M)$ is accessible if every two points of M can be connected by a path which is piecewise in a stable or an unstable leaf. We prove that accessibility is open in $\mathit{Diff}^{2}(M)$ among the partially hyperbolic diffeomorphisms with one-dimensional center bundle.