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Multiplicative constants are a fundamental tool in the study of maximal representations. In this paper we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable $\textup{PU}(m,1)$-cocycles of complex hyperbolic lattices., Comment: 35 pages; Major corrections along the paper following the referee's suggestions. To appear in Ergod. Theory Dyn. Syst

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms. In former papers all examples for actions enjoying the local smooth rigidity phenomenon are higher rank and have no rank-one factors. In this paper we give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.

Suppose that a countable group $G$ admits a cusp-uniform action on a hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal'bo-Otal-Peign\'e. For geometrically finite Cartan-Hadamard manifolds with pinched negative curvature this condition ensures the finiteness of Bowen-Margulis-Sullivan measures. In this case, our result recovers a theorem of Roblin (in a weaker form). Our main tool is the Patterson-Sullivan measures on the Gromov boundary of $X$, and a variant of the Sullivan shadow lemma called partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the condition of Dal'bo-Otal-Peign\'e. These results are further used in the paper \cite{YANG7}., Comment: Version 2: 34 pages, 2 figures. Sections 4 and 5 was rewritten following suggestions of the referee. Paper accepted by Ergodic Theory and Dynamical Systems

This paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.

In this paper we prove bounds for ergodic averages for nilflows on general higher-step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become equidistributed at polynomial speed. We analyze the rate of decay which is determined by the number of steps and structure of general nilpotent Lie algebras. Our main result follows from the technique of controlling scaling operators in irreducible representations and measure estimation on close return orbits on general nilmanifolds.

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$, then the number of equivalence classes is uncountable (Theorem A). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems B and C)., Comment: 38 pages, 5 figures. To appear in Ergodic Theory and Dynamical Systems

A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are those for which the postsingular set is a compact subset of the Fatou set. Equivalenty, they are characterized as being expanding. Mihaljevi\'c-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalise these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics., Comment: V3: Author accepted manuscript. To appear in Ergod. Theory Dyn. Syst

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving (p.m.p.) actions of countablegroups introduced in Part I [B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310]. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semicontinuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.

In this paper we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona’s theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformationsGis hyperbolic then the extremal conformal measures and the hyperbolic boundary ofGcoincide. We then interpret these results in terms of the asymptotic behavior of cutting sequences of geodesics on a regular cover of a compact hyperbolic surface.

It is known that the positiveness of topological entropy, the existence of a horseshoe and the existence of a homoclinic trajectory are mutually equivalent, for interval maps. The aim of the paper is to investigate the relations between the properties for continuous maps of trees, graphs and dendrites. We consider three different definitions of a horseshoe and two different definitions of a homoclinic trajectory. All the properties are mutually equivalent for tree maps, whereas not for maps on graphs and dendrites. For example, positive topological entropy and the existence of a homoclinic trajectory are independent and neither of them implies the existence of any horseshoe in the case of dendrites. Unfortunately, there is still an open problem, and we formulate it at the end of the paper.

Althoughpiecewise isometries(PWIs) are higher-dimensional generalizations of one-dimensionalinterval exchange transformations(IETs), their generic dynamical properties seem to be quite different. In this paper, we consider embeddings of IET dynamics into PWI with a view to better understanding their similarities and differences. We derive some necessary conditions for existence of such embeddings using combinatorial, topological and measure-theoretic properties of IETs. In particular, we prove that continuous embeddings of minimal 2-IETs into orientation-preserving PWIs are necessarily trivial and that any 3-PWI has at most one non-trivially continuously embedded minimal 3-IET with the same underlying permutation. Finally, we introduce a family of 4-PWIs, with an apparent abundance of invariant non-smooth fractal curves supporting IETs, that limit to a trivial embedding of an IET.

In this paper we extend the results of an earlier paper, which deal with a description of the smooth untwisted cohomology for Z k -actions by hyperbolic automorphisms of a torus, to the partially hyperbolic case. Along the way we correct an error found in one of the steps in the proof for the hyperbolic case.

[GIKN] and [BBD1] propose two very different ways for building non hyperbolic measures, [GIKN] building such a measure as the limit of periodic measures and [BBD1] as the $\omega$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in [GIKN] was essentially used in a generic setting, as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in [BBD1] are accumulated by periodic measures. In this paper we use a shadowing lemma from [G]: $\bullet$for getting the periodic orbits in [GIKN] without perturbing the dynamics, $\bullet$for recovering the compact set in [BBD1] with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset $\mathcal{U}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in\mathcal{U}$, there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset $\mathcal{V}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in\mathcal{V}$, there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures., Comment: 37 pages

The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles.Nonlinearity27(4) (2014), 647] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota–Yorke maps under a variety of perturbations, including ‘Ulam’s method’, a popular numerical method for approximating acims. The expansivity requirements of Froylandet alwere that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froylandet al, requiring only that the cocycle be eventually expanding on average, and importantly,allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

In this work we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such class is $C^r$-open, $r>1$, among the partially hyperbolic diffeomorphisms (in the narrow sense) and we prove that the mostly expanding condition guarantee the existence of physical measures and provide more information about the statistics of the system. Ma����'s classical derived-from-Anosov diffeomorphism on $\mathbb{T}^3$ belongs to this set., We improve the paper incorporating the referee's suggestions

In this paper we study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. We show that multi-sensitivity implies thick sensitivity, and the converse holds true for transitive systems. Our main result is an analog of the Auslander-Yorke dichotomy theorem: a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. Furthermore, we refine it by introducing the concept of syndetically equicontinuous points: a transitive system is either thickly sensitive or contains syndetically equicontinuous points., Comment: This version is accepted by Ergod. Th. and Dynam. Sys. arXiv admin note: substantial text overlap with arXiv:1504.00587

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length n relates to the one for words of length n+1 for each n = 1,2,3..., 30 pages, 6 figures

Let $X$ be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group $\Gamma$ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $\Gamma$-action. Although the construction of Bowen-Margulis measures for rank one nonpositively curved manifolds and for CAT(-1) spaces is well-known, the construction for CAT(0) spaces hinges on establishing a new structural result of independent interest: Almost no geodesic, under the Bowen-Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in $\partial_\infty X$, under the Patterson-Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen-Margulis measure is finite, as it is in the cocompact case). Finally, we precisely characterize mixing when $X$ has full limit set: A finite Bowen-Margulis measure is not mixing under the geodesic flow precisely when $X$ is a tree with all edge lengths in $c \mathbb Z$ for some $c > 0$. This characterization is new, even in the setting of CAT(-1) spaces. More general (technical) versions of these results are also stated in the paper., Comment: v2: 26 pages, 1 figure. Theorems stated in much more generality (in particular, the cocompactness hypothesis was removed almost everywhere), also a number of proofs dropped. This is the July 2015 version that was accepted for publication in Ergodic Theory and Dynamical Systems. v1: 39 pages, 1 figure

Bratteli–Vershik systems have been widely studied. In the context of general zero-dimensional systems, Bratteli–Vershik systems are homeomorphisms that have Kakutani–Rohlin refinements. Bratteli diagrams are well suited to analyzing such systems. Besides this approach, general graph covers can be used to represent any zero-dimensional system. Indeed, all zero-dimensional systems can be described as certain kinds of sequences of graph covers that may not be brought about by Kakutani–Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable zero-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani–Rohlin refinements. We demonstrate this for the general zero-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general zero-dimensional systems. Unlike previous studies, in our case of general graph covers there arises the possibility of the linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous research also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices. Finally, we present examples that are linked with a set of simple Bratteli diagrams having the equal path number property.

We present new examples of open sets of diffeomorphisms such that a generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that except from one particular class, the other classes are contained in periodic surfaces. This study allows us to obtain existence of Milnor attractors as well as studying ergodic properties of the diffeomorphisms in those open sets by using the ideas and results from [BV] and [BF]., Comment: 30 pages, 4 figures. This paper contains the results of http://arxiv.org/abs/1003.2280 and some more. It has been completely rewritten

In this paper, we introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context., Comment: 28 pages. Minor changes. Title changed. To appear in Ergod. Th. Dynam. Sys

In this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski., Comment: 35 pages

For SFTs, any equilibrium measure is Gibbs, as long a $f$ has $d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobru{\v{s}}in states that for strongly-irreducible subshifts, shift-invariant Gibbs-measures are equilibrium measures. Here we prove a generalization of the Lanford-Ruelle theorem: for all subshifts, any equilibrium measure for a function with $d$-summable variation is "topologically Gibbs". This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs. In the second part of the paper, we study Gibbs and equilibrium measures for some interesting families of subshifts: $\beta$-shifts, Dyck-shifts and Kalikow-type shifts (defined below). In all of these cases, a Lanford-Ruelle type theorem holds. For each of these families we provide a specific proof of the result., Comment: 19 pages

In this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

Attractors for hyperbolic diffeomorphisms are known to possess unique Sinai–Ruelle–Bowen measures with interesting properties. In this paper we investigate the case of non-invertible maps (endomorphisms) which have repellers Λ. We work with preimages of points in a neighbourhood of the repeller (assumed to be non-expanding); the situation here is different than the one for diffeomorphisms or positive iterates of endomorphisms. We give two methods to obtain invariant measures from local inverse iterates. We show that if Λ is a hyperbolic s-conformal repeller for f, not necessarily expanding, and if f is d-to-1 on Λ then for Lebesgue almost every x in the repelling basin of Λ there are histories of x asymptotically distributed like the equilibrium measure μs of the Holder continuous potential Φs, with Φs (y):=log ∣Dfs (y)∣ for y∈Λ. The measure μs plays the role of an inverse Sinai–Ruelle–Bowen measure on the non-invertible repeller. We prove also that there exists a set A⊂Wue(Λ) with λ(A)=λ(Wue(Λ)) (where λ(⋅) is the Lebesgue measure) such that for any z∈A and any real continuous function g, with In particular, we obtain the asymptotic distribution of preimages of Lebesgue almost all points for a class of hyperbolic toral endomorphisms on 𝕋m,m≥2 .

This paper deals with random perturbations of diffeomorphisms on n-dimensional Riemannian manifolds with distributions supported on k-dimensional disks, where k

In this paper, we study some skew product diffeomorphisms with non-uniformly hyperbolic structure along fibers and show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers. For such diffeomorphisms, our result gives an affirmative answer to the question posed by Herman as to whether a smooth diffeomorphism of positive topological entropy would fail to be uniquely ergodic. The proof is based on some techniques that are analogous to those developed by Pesin and Katok, together with an investigation of certain combinatorial properties of the projected return map on the base.

The Ruelle operator theorem has been studied extensively both in dynamical systems and iterated function systems. In this paper we study the Ruelle operator theorem for non-expansive systems. Our theorems give some sufficient conditions for the Ruelle operator theorem to be held for a non-expansive system.

Mortiss introduced the notion of critical dimension of a non-singular action, a measure of the order of growth of sums of Radon derivatives. The critical dimension was shown to be an invariant of metric isomorphism; this invariant was calculated for two-point product odometers and shown to coincide, in certain cases, with the average coordinate entropy. In this paper we extend the theory to apply to all product odometers, introduce upper and lower critical dimensions, and prove a Katok-type covering lemma.

The period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space ofC2+αunimodal maps, forα>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space ofC2unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to geta prioribounds. In this smoother class, calledC2+∣⋅∣, the failure of hyperbolicity is tamer than inC2. Things get much worse with just a bit less smoothness thanC2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space ofC1+Lipunimodal maps has infinite topological entropy.

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

In Bunimovich and Del Magno [Semi-focusing billiards: hyperbolicity. Comm. Math. Phys.262 (2006), 17–32], we proved that billiards in certain three-dimensional convex domains are hyperbolic. In this paper, we continue the study of these systems, and prove that they enjoy the Bernoulli property. This result answers affirmatively a long-standing question on the existence of ergodic billiards in convex domains in dimensions greater than two. Besides, it shows that the chaotic components of the first rigorously investigated three-dimensional billiards with mixed phase space (mushroom billiards), introduced in Bunimovich and Del Magno, are in fact Bernoulli.

It is well known that a generic polynomial foliation of ℂ2 is minimal and ergodic. In this paper we prove an analogous result for analytic foliations.

The representation of Cantor minimal systems by Bratteli–Vershik diagrams has been extensively used to study particular aspects of their dynamics. A main role has been played by the symbolic factors induced by the way vertices of a fixed level of the diagram are visited by the dynamics. The main result of this paper states that Cantor minimal systems that can be represented by Bratteli–Vershik diagrams with a uniformly bounded number of vertices at each level (called finite-rank systems) are either expansive or topologically conjugate to an odometer. More precisely, when expansive, they are topologically conjugate to one of their symbolic factors.

A Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A dominated splitting is a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of an adapted metric for a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub. Invariant Manifolds(Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.

In this paper we use infinite ergodic theory to study limit sets of essentially free Kleinian groups which may have parabolic elements of arbitrary rank. By adapting a method of Adler, we construct a section map S for the geodesic flow on the associated hyperbolic manifold. We then show that this map has the Markov property and that it is conservative and ergodic with respect to the invariant measure induced by the Liouville–Patterson measure. Furthermore, we obtain that S is rationally ergodic with respect to different types of return sequences (an), which are governed by the exponent of convergence . We then give applications to number theory and to the statistics of cuspidal windings. Also, as a corollary we obtain a special case of Sullivan's result that the geodesic flow on a geometrically finite hyperbolic manifold is ergodic with respect to the Liouville–Patterson measure.

In this paper, we establish analytic conjugacy theorems of Poincare type and Poincare–Dulac type for analytic random dynamical systems based on their Lyapunov exponents.

In this paper we consider the family of rational maps of the complex plane given by z 2 + λ z 2 where λ is a complex parameter. We regard this family as a singular perturbation of the simple function z 2 . We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the correspondingmaps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However,we also showthatparameterscorrespondingto differentopensets havedynamics that are not conjugate.

In this paper we show that a continuous map f from a connected graph G to itself is pointwise-recurrent if and only if one of the following two statements holds: (1) G is a circle, and f is a homeomorphism topologically conjugate to an irrational rotation of the unit circle S1; (2) f is a periodic homeomorphism.

Krieger's embedding theorem for subshifts into square-filling mixing SFTs which will be carried out in a subsequent paper.

We consider one-dimensional dynamical systems with indifferent fixed points (fixed points with derivative one). Many such maps have absolutely continuous ergodic infinite invariant measures. We study the limit of the ratio of the ergodic sum of f A to that of f B , where the integrals of f A and f B are infinite with respect to the absolutely continuous ergodic infinite invariant measure. If f A and f B are analytic functions on [0, 1], the result in this paper makes it clear whether the ratio of the ergodic sum of f A to that of f B converges in the Lebesgue measure or not.

A singular-hyperbolic attractor for vector fields is a partially hyperbolic attractor with singularities (that are hyperbolic) and volume expanding central direction. The geometric Lorenz attractor is the most representative example of a singular-hyperbolic attractor. In this paper, we prove that if $\Lambda$ is a singular-hyperbolic attractor of a three-dimensional vector field X , then there is a neighborhood U of $\Lambda$ in M such that every attractor in U of a C r vector field C r close to X is singular, i.e. it contains a singularity. With this result we prove the following corollaries. There are neighborhoods U of $\Lambda$ (in M ) and $\mathcal U$ of X (in the space of C r vector fields) such that if n denotes the number of singularities of X in $\Lambda$ , then $\#\{A\subset U:A$ is an attractor of $Y\in\mathcal U\}\leq n$ . Every three-dimensional vector field C r close to one exhibiting a singular-hyperbolic attractor has a singularity non-isolated in the non-wandering set. A singularity of a three-dimensional C r vector field Y is stably non-isolated in the non-wandering set if it is the unique singularity of a singular-hyperbolic attractor of Y . These results generalize well-known properties of the geometric Lorenz attractor.

We prove several new results about AF-equivalence relations and relate these to Cantor minimal systems (i.e. to minimal Z-actions). The results we obtain turn out to be crucial for the study of the topological orbit structure of more general countable group actions (as homeomorphisms) on Cantor sets, which will be the topic of a forthcoming paper. In all this, Bratteli diagrams and their dynamical interpretation are indispensable tools.

It has been conjectured that a generic diffeomorphism on a compact manifold will have trivial centralizer. In this paper we show that the conjecture is true within certain classes of partially hyperbolic diffeomorphisms.

In this paper we study the behavior of diffeomorphisms, contained in the closure (A) over bar (alpha) (in the inductive limit topology) of the set A(alpha) of real-analytic diffeomorphisms of the t ...

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.

Under some weak hyperbolicity conditions, we establish C^0- and C^\infty-local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0-local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0-local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty-local rigidity is deduced from the C^0-local rigidity following a procedure outlined by Katok and Spatzier.Using similar considerations, we also establish C^0-global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

This paper consists of two nearly independent parts, both of which discuss the common theme of biaccessible points in the Julia set $J$ of a quadratic polynomial $f:z\mapsto z^2+c$.In Part I, we assume that $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points (through the basin of attraction of infinity) in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev map for which the corresponding measure is one. As a corollary, we show that a locally-connected quadratic Julia set is not a countable union of embedded arcs unless it is a straight line or a Jordan curve.In Part II, we assume that $f$ has an irrationally indifferent fixed point $\alpha$. If $z$ is a biaccessible point in $J$, we prove that the orbit of $z$ eventually hits the critical point of $f$ in the Siegel case, and the fixed point $\alpha$ in the Cremer case. As a corollary, it follows that the set of biaccessible points in $J$ has Brolin measure zero.