We show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called ‘Topological Collet–Eckmann’. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each Hölder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.

We survey a collection of results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many interesting and non-trivial developments, only the case of conformal dynamics is completely understood. The study of the dimension of invariant sets of non-conformal maps has unveiled several new phenomena, but it still lacks today a satisfactory general approach. Indeed, we have a complete understanding of only a few classes of invariant sets of non-conformal maps satisfying certain simplifying assumptions. For example, the assumptions may ensure that there is a clear separation between different Lyapunov directions or that number-theoretical properties do not influence the dimension.

For n≥1, let H be the (2n+1)-dimensional real Heisenberg group, and let Λ be a lattice in H. Let Γ be the group of automorphisms of the corresponding nilmanifold Λ∖H and U the associated unitary representation of Γ on L2 (Λ∖H) . Denote by T the maximal torus factor associated to Λ∖H. Using Weil’s representation (also known as the metaplectic representation), we show that a dense set of matrix coefficients of the restriction of U to the orthogonal complement of L2 (T) in L2 (Λ∖H) belong to ℓ4n+2+ε (Γ) for every ε>0 . We give the following application to random walks on Λ∖H defined by a probability measure μ on Aut (Λ∖H) . Denoting by Γ(μ) the subgroup of Aut (Λ∖H) generated by the support of μ and by U0 and V0 the restrictions of U to, respectively, the subspaces of L2 (Λ∖H) and L2 (T) with zero mean, we prove the following inequality: where λ is the left regular representation of Γ(μ) on ℓ2 (Γ(μ)) . In particular, the action of Γ(μ) on Λ∖H has a spectral gap if and only if the corresponding action of Γ(μ) on T has a spectral gap.

This paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of:

• • either a robust local phenomenon;

• • or a global structure forbidding this local phenomenon.

We consider Anosov actions of ℝk, k≥2, on a closed connected orientable manifold M, of codimension one, i.e. such that the unstable foliation associated to some element of ℝk has dimension one. We prove that if the ambient manifold has dimension greater than k+2, then the action is topologically transitive. This generalizes a result of Verjovsky for codimension-one Anosov flows.

In this paper we study linearly repetitive Delone sets and prove, following the work of Bellissard, Benedetti and Gambaudo, that the hull of a linearly repetitive Delone set admits a properly nested sequence of box decompositions (tower system) with strictly positive and uniformly bounded (in size and norm) transition matrices. This generalizes a result of Durand for linearly recurrent symbolic systems. Furthermore, we apply this result to give a new proof of a classic estimation of Lagarias and Pleasants on the rate of convergence of patch frequencies.

Let n>m≥1 be two integers. At first we obtain general results for the multifractal analysis of the orthogonal projections on m-dimensional linear subspaces of singular measures μ on ℝn satisfying the multifractal formalism. The results hold for γn,m-almost every such subspace, where γn,m is the uniform measure on the Grassmannian manifold Gn,m. Let μ be such a measure and suppose that its upper Hausdorff dimension is less than or equal to m. Let I stand for the interval over which the singularity spectrum of μ is increasing. We prove that there exists a non-trivial subinterval of I such that for every , for γn,m-almost every m-dimensional subspace V, the multifractal formalism holds at α for μV, the orthogonal projection of μ on V. Moreover, in some cases the result is optimal in the sense that the interval is maximal in I. Also, we determine the Lq-spectrum τμV(q) on the minimal interval J necessary to recover the singularity spectrum of μV over as the Legendre transform of τμV. The interval J and the function τμV(q) do not depend on V, and τμV(q) can differ from τμ on a non-trivial interval. For Gibbs measures and some of their discrete counterparts, we show the stronger uniform result: for γn,m-almost every m-dimensional subspace V, the multifractal formalism holds for μV over the whole interval . As an application, we obtain a part of the singularity spectrum of some self-similar measures on attractors of iterated function systems which do not satisfy the weak separation condition.

The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or a topological Markov chain. As the main summands, these principles contain the integrals over invariant measures and the Kolmogorov–Sinai entropy. In the paper we derive the variational principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov–Sinai entropy but a new invariant of entropy type—the t-entropy.

In this paper, we investigate the boundary of the escaping set I(f) for quasiregular mappings on ℝn, both in the uniformly quasiregular case and in the polynomial type case. The aim is to show that ∂I(f) is the Julia set J(f) when the latter is defined, and shares properties with the Julia set when J(f) is not defined.

We consider an ergodic process on finitely many states, with positive entropy. Our main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set B is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function 1−e−t. That is, the occurrences of B along the time axis can appear either with gap sizes of nearly the exponential distribution (as in an independent and identically distributed process), or they attract each other and create ‘series’. We recall that in [T. Downarowicz, Y. Lacroix and D. Leandri. Spontaneous clustering in theoretical and some empirical stochastic processes. ESAIM Probab. Stat. to appear] it is proved that in a typical (in the sense of category) ergodic process (of any entropy), all cylinders B of selected lengths (such lengths have upper density 1 in ℕ) reveal strong attracting. Combining this with the result of this paper, we obtain, globally in ergodic processes of positive entropy and for long cylinder sets, the prevalence of attracting and deficiency of repelling. This phenomenon resembles what in real life is known as the law of series; the common-sense observation that a rare event, having occurred, has a mysterious tendency to untimely repetitions.

We prove the conjecture of F. Rodriguez Hertz and J. Rodriguez Hertz [Expansive attractors on surfaces. Ergod. Th. & Dynam. Sys.26(1) (2006), 291–302; MR 2201950(2006j:37049)] that every non-trivial transitive expansive attractor of a homeomorphism of a compact surface is a derived from pseudo-Anosov attractor.

In this article, we consider solutions that start close to some linearly stable invariant tori in an analytic Hamiltonian system, and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The proof combines classical Birkhoff normal forms with a new method for obtaining generic Nekhoroshev estimates developed by the author and L. Niederman in another paper. We will focus mainly on the neighbourhood of elliptic fixed points, since with our approach the other cases can be treated in a very similar way.

In this article semigroups in a general metric space V, which have pointwise exponentially attracting local unstable manifolds of compact invariant sets, are considered. We show that under a suitable set of assumptions these semigroups possess strong exponential dissipative properties. In particular, there exists a compact global attractor which exponentially attracts each bounded subset of V. Applications of abstract results to ordinary and partial differential equations are given.

We consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Ω* of orbits which never falls into a fixed hole. We study various aspects of the symbolic representation of Ω* and of its non-wandering set Ωnw. Our results are on the cardinality of the set of topologically transitive components of Ωnw and their structure. We also prove that Ω* is generically a subshift of finite type in several senses.

Large deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. We use a method employed previously by the first author [Large deviations bound for semiflows over a non-uniformly expanding base. Bull. Braz. Math. Soc. (N.S.)38(3) (2007), 335–376], which follows that of Young [Some large deviation results for dynamical systems. Trans. Amer. Math. Soc.318(2) (1990), 525–543]. As a corollary of the main results, we obtain a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, extending earlier work of Athreya [Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geom. Dedicata119 (2006), 121–140].

We introduce a property of measures on Euclidean space, termed ‘uniform scaling scenery’. For these measures, the empirical distribution of the measure-valued time series, obtained by rescaling around a point, is (almost everywhere) independent of the point. This property is related to existing notions of self-similarity: it is satisfied by the occupation measure of a typical Brownian motion (which is ‘statistically’ self-similar), as well as by the measures associated to attractors of affine iterated function systems (which are ‘exactly’ self-similar). It is possible that different notions of self-similarity are unified under this property. The proofs trace a connection between uniform scaling scenery and Furstenberg’s ‘CP processes’, a class of natural, discrete-time, measure-valued Markov processes, useful in fractal geometry.

We show how multifractal properties of a measure supported by a fractal F⊆[0,1] may be expressed in terms of complementary intervals of F and thus in terms of spectral triples and the Dixmier trace of certain operators. For self-similar measures this leads to a non-commutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.

We prove a formula for the sofic entropy of expansive principal algebraic actions of residually finite groups, extending recent work of Deninger and Schmidt.

We establish two results under which the topology of a compact hyperbolic set constrains ambient dynamics. First, if Λ is a transitive, codimension-one, expanding attractor for some diffeomorphism, then Λ is a union of transitive, codimension-one attractors (or contracting repellers) for any diffeomorphism such that Λ is hyperbolic. Secondly, if Λ is a locally maximal nonwandering set for a surface diffeomorphism, then Λ is locally maximal for any diffeomorphism for which Λ is hyperbolic.

The main theorem of this paper establishes conditions under which the ‘chaos game’ algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non-contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.