Let $(E,{\cal A},\mu,T)$ be a dynamical system and let $\Phi$ be a function defined on $E$ with values in $\mathbb{R}^2$. We give a criterion, the central limit theorem along subsequences of positive density, for the recurrence of the corresponding ‘stationary walk’ defined as the cocycle $\big(\sum^{n-1}_{j=0}\Phi(T^jx)\big)_{n\geq1}$.

This criterion is satisfied by functions which are homologous to a martingale difference (a property which holds for regular functions in many systems). It can also be applied to the periodic Lorentz gas in the plane and shows recurrence for this model.

Given a free action of a group $G$ on a directed graph $E$ we show that the crossed product of $C^* (E)$, the universal $C^*$-algebra of $E$, by the induced action is strongly Morita equivalent to $C^* (E/G)$. Since every connected graph $E$ may be expressed as the quotient of a tree $T$ by an action of a free group $G$ we may use our results to show that $C^* (E)$ is strongly Morita equivalent to the crossed product $C_0 ( \partial T ) \times G$, where $\partial T$ is a certain zero-dimensional space canonically associated to the tree.

We show that if the geodesic flow of a compact analytic Riemannian manifold $M$ of non-positive curvature is either $C^{k}$-topologically stable or satisfies the $\epsilon$-$C^{k}$-shadowing property for some $k > 0$ then the universal covering of $M$ is a Gromov hyperbolic space. The same holds for compact surfaces without conjugate points.

In this paper we prove that any ergodic measurably distal system can be realized as a minimal topologically distal system with an invariant Borel measure of full support. The proof depends upon a theorem stating that every measurable function from a measurable system with its base space being a compact metric space to a connected compact group is cohomologous to a continuous function.

Consider a magnetic field on a closed Riemannian manifold of negative curvature. A geometric study provides dynamical properties of the associated flow stronger than expected for general perturbations of the geodesic flow. Under natural assumptions, a magnetic flow on a closed surface cannot be ${\mathcal C}^1$-conjugate to a geodesic flow.

We show that the horocycle foliations on a compact $C^{\infty}$ (or even $C^{\omega}$) surface of non-positive curvature can fail to be Lipschitz, even if the curvature vanishes only along a single closed geodesic. We calculate the Hölder exponents of these foliations at vectors tangent to geodesics along which the curvature vanishes.

We give sharp regularity results for the invariant subbundles of hyperbolic dynamical systems in terms of contraction and expansion rates and prove optimality in a strong sense: we construct open dense sets of codimension one systems where this regularity is not exceeded. Furthermore, we exhibit open dense sets of symplectic, geodesic, and codimension one systems where the analogous regularity results of [PSW] are optimal. As our main result we produce open sets of symplectic Anosov diffeomorphisms and flows with low transverse Hölder regularity of the invariant foliations almost everywhere. Prevalence of low regularity of conjugacies is a corollary. We also establish a new connection between the transverse regularity of foliations and their tangent subbundles.

For every irrational number $\alpha$ satisfying the property $\lim_{n\to\infty}|\!\sin \pi\alpha n|^{-1/n}=1$ and for every number $\beta>1$, it is shown that the difference equation $$ \xi_{n+1}+\xi_{n-1} +2\beta\cos(2\pi\alpha n+\th)\xi_n=0, \quad n\in\mathbb{Z} $$ has a non-trivial solution $\{\xi_n\}$ satisfying $\mathop{\overline{\lim}}_{|n|\to\infty}|\xi_n|^{1/|n|}\le|\beta|^{-1}$ if and only if $\theta=2\pi\alpha n+2\pi k\pm{\pi/2}$ for some $n,k\in\mathbb{Z}$.

We present numerous examples of ways that a Bernoulli shift can behave relative to a family of factors. This shows the similarities between the properties which collections of ergodic transformations can have and the behavior of a Bernoulli shift relative to a collection of its factors. For example, we construct a family of factors of a Bernoulli shift which have the same entropy, and any extension of one of these factors has more entropy, yet no two of these factors sit the same. This is the relative analog of Ornstein and Shields uncountable collection of nonisomorphic $K$ transformations of the same entropy. We are able to construct relative analogs of almost all the zero entropy counter-examples constructed by Rudolph (1979), as well as the $K$ counterexamples constructed by Hoffman (1997). This paper provides a solution to a problem posed by Ornstein (1975).

We prove that the transfer operator for a general class of rational maps converges exponentially fast in the supremum norm and in Hölder norms, for small enough Hölder exponents, to its principal eigendirection.

The modified Jacobi–Perron algorithm was shown by Ito etal in 1993 to be strongly convergent (in the sense of Brentjes 1981) almost everywhere with exponential rate. Their proof relies on very complicated computations. We will show that the original article by Podsypanin in 1977 on the modified Jacobi–Perron algorithm almost contains a proof of this convergence with exponential rate. The only ingredients missing in that article are some ergodic-theoretical facts about the transformation generating the approximations. This leads to a very simple proof of the before mentioned exponential convergence in the modified Jacobi–Perron algorithm.

It is well known that the spectrum of dynamical systems arising from generalized Morse sequences (M. Keane. Generalized Morse sequences. Z. Wahr. Verw. Geb.10 (1968), 335–353.) is simple as soon as it is non-discrete. We give, in this paper, a necessary and sufficient condition for the existence of such a transformation with non-purely singular spectrum. From this, it follows that this problem is equivalent to an open problem of the existence of ‘flat’ polynomials on the Torus group. We show that this latter question can be given an affirmative answer on some other group, and this allows us to construct a countable abelian group action with simple spectrum whose spectral type is the sum of a discrete measure and of the Haar measure on the dual group.

The problem under consideration is: when is a Markov endomorphism (one-sided shift) $T=T_P$ with transition matrix $P$, isomorphic to a Bernoulli endomorphism $\tilde{T}_\rho$ with an appropriate stationary vector $\rho=\{\rho_i\}_{i\in I}$? An obvious necessary condition is that there exists an independent complement $\delta$ of the measurable partition $T^{-1}\varepsilon$ with $\distr \delta = \rho$. In this case the cofiltration (decreasing sequence of measurable partitions) $\xi(T)=\{T^{-n}\varepsilon\}^{\infty}_{n=1}$ generated by $T$ is finitely isomorphic to the standard Bernoulli cofiltration $\xi(\tilde{T}_{\rho})=\{\tilde{T}^{-n}_{\rho}\varepsilon\}^{\infty}_{n=1}$ and $T$ is called finitely $\rho$-Bernoulli.

We show that every ergodic Markov endomorphism $T_P$, which is finitely Bernoulli, can be represented as a skew product over $\tilde{T}_{\rho}$ with $d$-point fibres $(d\in \mathbb{N})$. We compute the minimal $d=d(T_P)$ in these skew-product representations by means of the transition matrix, and obtain necessary and sufficient conditions under which $d(T_P)=1$, i.e. $T_P$ is isomorphic to $\tilde{T}_{\rho}$. The cofiltration $\xi(T)$ of any finitely Bernoulli ergodic Markov endomorphism $T=T_P$ is represented as a $d$-point extension of the standard cofiltration $\xi(\tilde{T}_{\rho})$, and we show that the minimal $d=d_{\xi}(T)$ in these extensions is equal to $d(T)$. In particular, $d(T)=1\Longleftrightarrow d_{\xi}(T)=1$, that is, a Markov endomorphism $T$ is isomorphic to $\tilde{T}_{\rho}$ iff $\xi(T)$ is isomorphic to the Bernoulli cofiltration $\xi(\tilde{T}_{\rho})$.

Many applications of nonlinear dynamics involve forced systems. We consider the case where for a fixed input the driven system is contracting; this is for instance the situation in certain classes of filters, and in the study of synchronization. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the driving state space to the driven state space; this is a special case of the well known concept of an inertial manifold for more general systems. If the driving state space is a manifold and the contraction is sufficiently strong this invariant set is a normally hyperbolic manifold, and hence smooth. The aim of this paper is to extend this result in two directions: firstly, where we only have uniform contraction for a compact invariant set of input states, and secondly where the contraction rates are non-uniform (and hence defined by Liapunov exponents and analogous quantities). In both cases the invariant graph is only defined over closed subsets of the input space, and hence we need to define an appropriate notion of smoothness for such functions. This is done in terms of the Whitney extension theorem: a function is considered Whitney smooth if it satisfies the conditions of this theorem and hence can be extended to a smooth function of the whole input space.

This paper is concerned with the dynamics of transcendental entire functions. Let $f(z)$ be a transcendental entire function. We shall study the boundedness of the components of the Fatou set $F(f)$ under some restrictions on the growth of the function. This relates to a problem due to Baker in 1981.

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a variational principle for the metric entropies. The pressure of $\phi =0$ is the Gurevic entropy. This pressure may be finite even if the topological entropy is infinite. Let $L_\phi$ denote the Ruelle operator for $\phi$. We offer a definition of positive recurrence for $\phi$ and show that it is a necessary and sufficient condition for a Ruelle–Perron–Frobenius theorem to hold: there exist a $\sigma$-finite measure $\nu $, a continuous function $h>0$ and $\lambda >0$ such that $L_\phi ^{*}\nu =\lambda \nu$, $L_\phi h=\lambda h $ and $\lambda ^{-n}L_\phi ^nf\rightarrow h\int f\,d\nu$ for suitable functions $f$. We show that under certain conditions this convergence is uniform and exponential. We prove a decomposition theorem for positive recurrent functions and construct conformal measures and equilibrium measures. We give complete characterization of the situation when the equilibrium measure is a Gibbs measure. We end by giving examples where positive recurrence can be verified. These include functions of the form $$ \phi =\log f\left( \cfrac{1}{x_0+ \cfrac{1}{x_1+\dotsb }}\right), $$ where $f$ is a suitable function on a suitable shift $X$.

We prove a collection of results inspired by Krengel's theorem on the existence of partitions with infinitely many independent iterates in any weakly mixing measure-preserving dynamical system. Our approach avoids Krengel's use of two-fold mixing thereby obtaining stronger results, including characterizations of mild and strong mixing, as well as weak mixing. We also obtain results for non-weakly mixing systems and for more general group actions.

The billiard in the exterior of a finite disjoint union $K$ of strictly convex bodies in ${\mathbb R}^d$ with smooth boundaries is considered. The existence of global constants $0 < \delta < 1$ and $C > 0$ is established such that if two billiard trajectories have $n$ successive reflections from the same convex components of $K$, then the distance between their $j$th reflection points is less than $C(\delta^j + \delta^{n-j})$ for a sequence of integers $j$ with uniform density in $\{1,2,\dots,n\}$. Consequently, the billiard ball map (although not continuous in general) is expansive. As applications, an asymptotic of the number of prime closed billiard trajectories is proved which generalizes a result of Morita [Mor], and it is shown that the topological entropy of the billiard flow does not exceed $\log (s-1)/a$, where $s$ is the number of convex components of $K$ and $a$ is the minimal distance between different convex components of $K$.

We present an original approach which allows us to investigate the statistical properties of a non-uniformly hyperbolic map on the interval. Based on a stochastic approximation of the deterministic map, this method essentially gives the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral fixed point.

We answer a question proposed by Barge and Franks in 1993 by showing that a bounded connected set in the plane which is closed or open and meets an infinite number of its iterates under an orientation-preserving fixed point free homeomorphism must meet all of them.