We consider open sets of transformations in a manifold M, exhibiting non-uniformly expanding behavior in some forward invariant domain U\subset M. Assuming that each transformation has a unique Sinai–Ruelle–Bowen (SRB) measure in U, and some general uniformity conditions, we prove that the SRB measure varies continuously with the dynamics in the L1-norm. We also describe a concrete application.

We provide new non-trivial quantitative information on the behavior of Poincaré recurrence. In particular we establish the almost everywhere coincidence of the recurrence rate and of the pointwise dimension for a large class of repellers, including repellers without finite Markov partitions.

Using this information, we are able to show that for locally maximal hyperbolic sets the recurrence rate possesses a certain local product structure, which closely imitates the product structure provided by the families of local stable and unstable manifolds, as well as the almost product structure of hyperbolic measures.

For a closed lamination on the unit circle invariant under z\mapsto z^d we prove an inequality relating the number of points in the ‘gaps’ with infinite pairwise disjoint orbits to the degree; in particular, this gives estimates on the cardinality of any such ‘gap’ as well as on the number of distinct grand orbits of such ‘gaps’. As a tool, we introduce and study a dynamically defined growing tree in the quotient space. We also use our techniques to obtain for laminations an analog of Sullivan's no wandering domain theorem. Then we apply these results to Julia sets of polynomials.

In this paper, which is a sequel to an earlier paper, we consider special examples, methods, and invariants for the isomorphism problem for ordered dimension groups built on square primitive integer matrices A. We use the algorithm of our first paper to derive techniques and computable invariants for deciding isomorphism for subclasses of the matrices A.

We prove the existence of a Sinai–Ruelle–Bowen (SRB) measure and the exponential decay of correlations for smooth observables for mixing Anosov C^{1+\alpha} diffeormorphisms on a d-dimensional (d \geq 2) Riemannian manifold. The novelty lies in the very simple method of proof. We construct explicitly a coupling between two initial densities so that under the action of the diffeomorphism, both components get closer and closer. The speed of this convergence can be explicitly estimated and is directly related to the speed of decay of the correlations. The existence of the SRB measure and its properties readily follow.

Soit (\Omega,{\cal B},m) un espace de probabilité. À une action T stationnaire de \mathbb{Z}^d est associée une notion algébrique de cocycle de degré \geq 1, le degré 1 correspondant au sens habituel. Celle-ci a été étudiée entre autres par Westman (1971), Feldman et Moore (1977). Une autre notion de cocycle, présentant des analogies avec le calcul différentiel, à été étudiée par A. et S. Katok (1995). Nous présentons un mécanisme de génération associant, à tout cocycle de ce dernier type, un cocycle algébrique de même degré. De plus, toutes les classes de cohomologie algébrique peuvent être engendrées ainsi. Ce résultat a pour conséquence que tout cocycle algébrique de degré \geq d+1 est un cobord. Il permet aussi de montrer que dans la classe des cocycles intégrables, la cohomologie algébrique de degré \leq d n'est pas triviale, alors qu'elle l'est dans la classe des cocycles mesurables dès le degré 2. Enfin, la méthode de génération présentée permet d'obtenir des cocycles algébriques vérifiant les théorèmes ergodiques correspondant à leur degré.

Soit E\subset \mathbb{C} un compact de capacité logarithmique positive. Supposons que pour tout polynôme P\not=\text{id} on a P^{-1}(E)\not=E. Alors pour tout polynômes non constants f et g vérifiant f^{-1}(E)=g^{-1}(E), on a f=g, c'est-à-dire que E est un ensemble d'unicité pour les polynômes. La preuve de ce résultat utilise la théorie de systèmes dynamiques holomorphes en une et plusieurs variables.

In this paper, we study the display of weak mixing by reparameterized linear flows on the torus \mathbb{T}^d, d\geq 2. We show that if the vector of the translation flow is Liouvillian (i.e. well approximated by rationals), then for a residual set of time change functions in the C^\infty topology, the reparameterized flow is weak mixing. If this is not the case, i.e. if the vector of the linear flow is Diophantine, it follows from a result of Kolmogorov on the two torus, and its generalization to any dimension by Herman, that any C^\infty reparameterization of the flow is C^\infty conjugate to a linear flow. More generally, in any given class of differentiability C^r for the time change function \phi, we give the optimal arithmetical condition on the vector of the translation flow that guarantees the existence of a residual set in the C^r topology of weak mixing reparameterizations. In the real analytic case, the optimal arithmetical condition for the generic display of weak mixing under time change is also given.

As a consequence of our results on reparameterizations of Liouvillian linear flows, we obtain that an aperiodic smooth flow on the two-dimensional torus is in general weak mixing. We also deduce the existence on the torus of analytic diffeomorphisms that are rank one and weak mixing.

We consider intermittency in one-parameter families of unimodal maps, induced by saddle node and boundary crisis bifurcations. In these bifurcations either a periodic orbit or a periodic interval disappears to give rise to chaotic bursts. We prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the attractor. For this, we extend Pianigiani's results on conditionally invariant measures for the logistic family to more general families.

We prove the following result in commutative harmonic analysis.

THEOREM 1. Suppose\muis a probability measure on the dual\hat Gof a countable Abelian groupG, Eis a symmetric subset ofGand\sum_{g \in E-E, g \neq 0} |\widehat{\mu} (g)| < \delta < 1/3.Then there is anf \in C^+(\hat G)such that\|f-1\| < \tfrac{3}{2}\delta, (\widehat{f\mu})(g)=0for allg\in E\backslash \{0\}and(\widehat{f\mu})(0)=1.

This has the following corollary which leads to simple proofs and improvements of many of the results in a recent paper of Bergelson et al.

THEOREM 2.Ifg \mapsto U_gis a unitary action of a discrete Abelian groupGon a Hilbert space{\cal H}and\{g_i\} \subset Gis a sequence such that both\{g_i\}and\{2g_i\}are mixing for this action then for allx\in{\cal H}and\epsilon>0there are anx'\buildrel\epsilon\over\sim xand an IP-set\Gammagenerated by a subsequence of\{g_i\}such that the vectors\{U_gx':g\in\Gamma\}are pairwise orthogonal.

We also show that even in the case of a \mathbb{Z}-action (i.e. a single unitary operator U) the hypothesis \{g_i\} mixing alone is not sufficient for the conclusion of Theorem 2. Finally we give conditions on G and on the spectrum of the action under which that assumption is sufficient.

We investigate uniform ergodic type theorems for almost additive and subadditive functions on a subshift over a finite alphabet. We show that every uniquely ergodic subshift admits a uniform ergodic theorem for Banach-space-valued almost additive functions. We then give a necessary and sufficient condition on a minimal subshift to allow for a uniform subadditive ergodic theorem. This provides, in particular, a sufficient condition for unique ergodicity.

The question addressed in this paper is the worst-case growth rate, for ergodic processes, in the number of conditional measures on n-steps in the future, given the past, that are a fixed distance apart. It is shown that if error is measured using the variational (i.e. distributional) distance then doubly exponential growth is possible, while if error is measured using the \bar{d}-metric then more than exponential growth is possible. The question of whether doubly exponential growth is possible in the \bar{d}case is left open.

The connected configuration space of a so-called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The dynamical system describes the uniform motion of a point particle in this configuration space with specular reflections at the boundaries of the removed cylinders. It is proven here that under a certain geometric condition a cylindric billiard flow is completely hyperbolic. As a consequence, every hard ball system is completely hyperbolic.

Given a twist map on the torus, we prove that the non-existence of a rotational invariant circle implies that the shear-rotation interval is non-trivial, i.e. the existence of orbits with non-zero mean acceleration. Under the same assumption, we prove the existence of infinitely many periodic orbits and positive entropy invariant measures with non-zero shear-rotation number. The method is variational, but relies on topological understanding of the shear-rotation interval of torus homeomorphisms homotopic to a Dehn twist.