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It has been observed that the famous Feigenbaum–Coullet–Tresser period-doubling universality has a counterpart for area-preserving maps of ℝ2. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in Eckmannet al[Existence of a fixed point of the doubling transformation for area-preserving maps of the plane.Phys. Rev. A 26(1) (1982), 720–722; A computer-assisted proof of universality for area-preserving maps.Mem. Amer. Math. Soc. 47(1984), 1–121]. As is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period-doubling universality exists to date. We argue that the period-doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Hénon-like (after a coordinate change) map:where ϕ solvesWe then give a ‘proof’ of existence of solutions of small analytic perturbations of this one-dimensional problem, and describe some of the properties of this solution. The ‘proof’ consists of an analytic argument for factorized inverse branches of ϕ together with verification of several inequalities and inclusions of subsets of ℂ numerically. Finally, we suggest an analytic approach to the full period-doubling problem for area-preserving maps based on its proximity to the one-dimensional case. In this respect, the paper is an exploration of possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.

Given an integer nonsingular $n\times n$ matrix $M$ and a point $y \in \mathbb{R}^n/\mathbb{Z}^n$, consider the set $\tilde E(M,y)$ of vectors $x\in \mathbb{R}^n$ such that $y$ is not a limit point of the sequence $\{M^k x \mod \mathbb{Z}^n: k\in\mathbb{N}\}$. S.G. Dani showed in 1988 that whenever $M$ is semisimple and $y \in \mathbb{Q}^n/\mathbb{Z}^n$, the set $\tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary $y \in \mathbb{R}^n/\mathbb{Z}^n$ and integer nonsingular $M$, and in fact replacing the sequence of powers of $M$ by any lacunary sequence of (not necessarily integer) $m\times n$ matrices. Furthermore, we show that sets of the form $\tilde E(M,y)$ and their generalizations always intersect with `sufficiently regular' fractal subsets of $\mathbb{R}^n$. As an application we give an alternative proof of a recent result of Einsiedler and Tseng on badly approximable systems of affine forms., 13 pages; a slightly modified version, to appear in Ergodic Theory Dynamical Systems

Let X be a metrizable space with a continuous group or semi-group action G. Let D be a non-empty subset of X. Our problem is how to choose a fixed number of sets in {g−1D; g∈G}, say σ−1D with σ∈τ, to maximize the cardinality of the partition ℙ({σ−1D; σ∈τ}) generated by them. Let An infinite subset Σ of G is called an optimal position of the triple (X,G,D) if holds for any k=1,2,… and τ⊂Σ with #τ=k. In this paper, we discuss examples of the triple (X,G,D) admitting or not admitting an optimal position. Let X=G=ℝn (n≥1) , where the action g∈G to x∈X is the translation x−g. If D is the n-dimensional unit ball, then holds and the triple (X,G,D) admits an optimal position. In fact, if n≥2 and Σ is an infinite subset of G such that for some δ with 0<δx∈ℝn ; ‖x‖=δ}, and that any subset of Σ with cardinality n+1 is not on a hyperplane, then Σ is an optimal position of the triple (X,G,D) . We have determined the primitive factor of the uniform sets coming from these optimal positions. We also show that in the above setting with n=2 and the unit square D′ in place of the unit disk D, the maximal pattern complexity is unchanged and p*X,G,D′(k)=k2 −k+2 , but there is no optimal position.

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 presents an example of Riemann surface lamination with at least two ergodic invariant measures. The generic leaves for those measures are of different growth and have different numbers of ends.

In this paper, we study perturbations of multidimensional shifts of finite type. Specifically, for any ℤd shift of finite type X with d>1 and any finite pattern w in the language of X, we denote by Xw the set of elements of X not containing w. For strongly irreducible X and patterns w with shape a d-dimensional cube, we obtain upper and lower bounds on htop (X)−htop (Xw) dependent on the size of w. This extends a result of Lind for d=1 . We also apply our methods to an undecidability question in ℤd symbolic dynamics.

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.

In this paper, using the Kolmogorov–Arnold–Moser method we prove reducibility of a class of nonlinear quasi-periodic differential equation with degenerate equilibrium point under small perturbation and obtain a quasi-periodic solution near the equilibrium point.

In this paper we prove that the Poincaré map associated to a Lorenz-like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τr(x,x0) is the time needed for the orbit of a point x to enter a ball Br(x0) centered at x0, with small radius r, for the first time. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the physical measure at x0: for each x0 such that the local dimension dμ (x0) exists, holds for μ almost each x. In a similar way, it is possible to consider a quantitative recurrence indicator quantifying the speed of an orbit to come back to its starting point. Similar results hold for this recurrence indicator.

In this paper we deal with the existence of periodic orbits of geodesible vector fields on closed 3-manifolds. A vector field is geodesible if there exists a Riemannian metric on the ambient manifold making its orbits geodesics. In particular, Reeb vector fields and vector fields that admit a global section are geodesible. We will classify the closed 3-manifolds that admit aperiodic volume-preservingCωgeodesible vector fields, and prove the existence of periodic orbits forCωgeodesible vector fields (not volume preserving), when the 3-manifold is not a torus bundle over the circle. We will also prove the existence of periodic orbits ofC2geodesible vector fields on some closed 3-manifolds.

This paper deals with non-singular Smale flows on oriented 3-manifolds. We shall show a relation between the properties of invariant sets of a Smale flow and a kind of Whitehead torsion of the underlying manifold.

The purpose of this paper is to exhibit highly structured subdynamics for a class of non-expansive algebraic ℤd-actions based on the closed orbits of elements of an action. This is done using dynamical Dirichlet series to encode orbit counts. It is shown that there is a distinguished group homomorphism from ℤd onto a finite abelian group that controls the form of the Dirichlet series of elements of an action and that these series have common analytic properties. Corresponding orbit growth asymptotics are subsequently investigated.

This paper introduces Markov chains and processes over non-abelian free groups and semigroups. We prove a formula for the f-invariant of a Markov chain over a free group in terms of transition matrices that parallels the classical formula for the entropy a Markov chain. Applications include free group analogues of the Abramov–Rohlin formula for skew-product actions and Yuzvinskii’s addition formula for algebraic actions.

Considering the Potts model on diamond-like hierarchical lattices, in this paper, we show that the Julia sets of the singularities of the free energy may be a Feigenbaum Julia set. Furthermore, we prove that these Julia sets are locally connected if they are connected.

In this paper, we analyse the dynamics encoded in the spectral sequence (Er,dr) associated with certain Conley theory connection maps in the presence of an ‘action’ type filtration. More specifically, we present an algorithm for finding a chain complex C and its differential; the method uses a connection matrix Δ to provide a system that spans Er in terms of the original basis of C and to identify all of the differentials drp:Erp→Erp−r. In exploring the dynamical implications of a non-zero differential, we prove the existence of a path that joins the singularities generating E0p and E0p−r in the case where a direct connection by a flow line does not exist. This path is made up of juxtaposed orbits of the flow and of the reverse flow, and proves to be important in some applications.

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit circle, in various parameters, and applications to the study of quantum dynamics.

Consider an invertible measure-preserving transformation of a probability space. A finite partition of the space is called weakly independent if there are infinitely many images of this partition under powers of the transformation that are jointly independent. Krengel proved that a transformation is weakly mixing if and only if weakly independent partitions of the underlying space are dense among all finite partitions. Using the tools developed in the later papers of del Junco-Reinhold-Weiss and del Junco-Begun we obtain Krengel- type results for weakly mixing random dynamical systems (or equivalently, skew products that are relatively weakly mixing).

In the present paper, we introduce a natural extension of Adler, Konheim and MacAndrew topological entropy for proper maps of locally compact separable metrizable spaces and prove a variational principle which states that this topological entropy, the supremum of the Kolmogorov–Sinai entropies and the minimum of the Bowen entropies always coincide. We apply this variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the Bowen formula for the Bowen entropy of an automorphism of a non-compact Lie group with respect to some invariant metric is just an upper bound for its topological entropy.

In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincaré’s theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.

The aim of the present paper is to reveal an unforeseen link between the classical vanishing theorems of Matsushima and Weil, on the one hand, and rigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank non-compact Lie group, on the other. The connection is established via ‘transverse extension theorems’: roughly speaking, they claim that a tangential 1-form of the orbit foliation of the Weyl chamber flow that is tangentially closed (and satisfies a certain mild additional condition) can be extended to a closed 1-form on the whole space in a canonical manner. In particular, infinitesimal rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

Tao has recently proved that if $T_1,...,T_l$ are commuting, invertible, measure-preserving transformations on a dynamical system then for any $L^\infty$ functions $f_1,...,f_l$, the average $\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i\leq l}f_i\circ T^n_i$ converges in the $L^2$ norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence "backwards". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties., Comment: References fixed

This paper concerns the problem of embedding, in the flow of an autonomous vector field, a local diffeomorphism near a hyperbolic fixed point in a Banach space. To solve the problem, we first extend Floquet theory to Banach spaces, and then prove that two C∞ hyperbolic diffeomorphisms are formally equivalent if and only if they are C∞-equivalent. The latter result is a version, in the Banach space context, of a classical theorem by Chen.

In this paper we discuss the dimension-theoretical properties of rational maps on the Riemann sphere. In particular, we study the existence and uniqueness of generalized physical measures for several classes of maps including hyperbolic, parabolic, non-recurrent and topological Collet–Eckmann maps. These measures have the property that their typical points have maximal Hausdorff dimension. On the other hand, we prove that the set of divergence points (the set of points which are non-typical for any invariant measure) also has maximal Hausdorff dimension. Finally, we prove that if (fa)ais a holomorphic family of stable rational maps, then the dimensiond(fa) is a continuous and plurisubharmonic function of the parametera. In particular,d(f) varies continuously and plurisubharmonically on an open and dense subset ofRatd, the space of all rational maps with degreed≥2.

We produce arbitrarily large equivalence classes of matings with the aeroplane polynomial. These are obtained by a slight generalisation of the technique of proof of a similar result for Wittner captures., Comment: 20 pages, 7 figures. A dedication has been added to the memory of Adrien Douady. Some corrections have been made, mainly to later lemmas. One figure has been added. The paper has been accepted for publication in ``Ergodic Theory and Dynamical Systems''

Let ℳ={f(z)=(zm/sinhm z) for z∈ℂ∣ either m or m/2 is an odd natural number}. For eachf∈ℳ, the set of singularities of the inverse function offis an unbounded subset of the real line ℝ. In this paper, the iteration of functions in one-parameter family 𝒮={fλ(z)=λf(z)∣λ∈ℝ∖{0}} is investigated for eachf∈ℳ. It is shown that, for eachf∈ℳ, there is a critical parameterλ*>0 depending onfsuch that a period-doubling bifurcation occurs in the dynamics of functionsfλin 𝒮 when the parameter |λ| passes throughλ*. The non-existence of Baker domains and wandering domains in the Fatou set offλis proved. Further, it is shown that the Fatou set offλis infinitely connected for 0λ∣≤λ*whereas for ∣λ∣≥λ*, the Fatou set offλconsists of infinitely many components and each component is simply connected.

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

Let f be a homeomorphism of a compact manifold M. The Krylov–Bogoloubov theorem guarantees the existence of a measure that is invariant with respect to f. The set of all invariant measures ℳ(f) is convex and compact in the weak topology. The goal of this paper is to construct the set ℳ(f). To obtain an approximation of ℳ(f), we use the symbolic image with respect to a partition C={M(1),M(2),…,M(n)} of M. A symbolic image G is a directed graph such that a vertex i corresponds to the cell M(i) and an edge i→j exists if and only if f(M(i))∩M(j)≠0̸. This approach lets us apply the coding of orbits and symbolic dynamics to arbitrary dynamical systems. A flow on the symbolic image is a probability distribution on the edges which satisfies Kirchhoff’s law at each vertex, i.e. the incoming flow equals the outgoing one. Such a distribution is an approximation to some invariant measure. The set of flows on the symbolic image G forms a convex polyhedron ℳ(G) which is an approximation to the set of invariant measures ℳ(f). By considering a sequence of subdivisions of the partitions, one gets sequence of symbolic images Gk and corresponding approximations ℳ(Gk) which tend to ℳ(f) as the diameter of the cells goes to zero. If the flows mk on each Gk are chosen in a special manner, then the sequence {mk} converges to some invariant measure. Every invariant measure can be obtained by this method. Applications and numerical examples are given.

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 aim of this paper is to show that the dynamics of $L^p$ heat semigroups ($p>2$) on a symmetric space of non-compact type is very different from the dynamics of the $L^p$ heat semigroups if $p\leq 2$. To see this, it is shown that certain shifts of the $L^p$ heat semigroups have a chaotic behavior if $p>2$ and that such a behavior is not possible in the cases $p\leq 2$. These results are compared with the corresponding situation for euclidean spaces and symmetric spaces of compact type where such a behavior is not possible., 14 pages; final version; to appear in Ergodic Theory and Dynamical Systems

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.

In this paper we study characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group G. Projection against the Iwasawa decomposition G=KAN defines an additive cocycle over the flow with values in 𝔞=log A. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. A symmetric property of these spectral sets is proved, namely invariance under the Weyl group. We also prove that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.

In this paper the global attractors of analytic and polynomial plane flows are characterized up to homeomorphisms. Following on from previous results for continuous and differentiable dynamical systems, our theorem completes the characterization of the global attractors of plane flows.

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy–Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmüller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely linear involutions. These maps are related to general measured foliations on surfaces (whether orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with ℤ/2ℤ linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy–Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows us, in particular, to classify the connected components of all exceptional strata.

In this paper we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. To the best of our knowledge this is the first time that conditions of this type have been given. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then becomes equal to zero. We also discuss the existence and uniqueness of equilibrium measures. In order to do so, we introduce a family of countable Markov shifts that can be thought of as a generalization of the renewal shift.

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.

In this paper, the boundedness of the dual billiard orbits of trapezoids is obtained. This gives an example showing that there are also polygons, which are neither rational nor quasi-rational, with all of their dual billiard orbits bounded. Moreover, most of the orbits are Poisson-stable, which means that most points will come sufficiently near to their original positions under iterations of the dual billiard map.

In this paper, we propose a model for the destruction of three-dimensional horseshoes via heterodimensional cycles. This model yields some new dynamical features. Among other things, it provides examples of homoclinic classes properly contained in other classes and it is a model of a new sort of heteroclinic bifurcations we callgenerating.

In this paper we prove smooth conjugate theorems of Sternberg type for almost periodic differential systems, based on the Lyapunov exponents of the corresponding reduced systems.

The purpose of the present paper is to provide a link between skew-product systems and linear dynamics. In particular, we give a criterion for skew-products of linear operators to be topologically transitive. This is then applied to certain families of linear operators including scalar multiples of the backward shift, backward unilateral weighted shifts, composition, translation and differentiation operators. We also prove the existence of common hypercyclic vectors for certain families of skew-product systems.

We say that the geometric dimension of a countable group G is equal to n if any free Borel action of G on a standard Borel probability space (X,μ), induces an equivalence relation of geometric dimension n on (X,μ) in the sense of Gaboriau. Let ℬ be the set of all finitely generated amenable groups all of whose subgroups are also finitely generated, and let 𝒜 be the subset of ℬ consisting of finite groups, torsion-free groups and their finite extensions. In this paper we study finite free products K of groups in 𝒜. The geometric dimension of any such group K is one: we prove that also geom-dim(Gf(K))=1 for any finite extension Gf(K) of K, applying the results of Stallings on finite extensions of free product groups, together with the results of Gaboriau and others in orbit equivalence theory. Using results of Karrass, Pietrowski and Solitar we extend these results to finite extensions of free groups. We also give generalizations and applications of these results to groups with geometric dimension greater than one. We construct a family of finitely generated groups {Kn}n∈ℕ,n>1, such that geom-dim(Kn)=n and geom-dim(Gf(Kn))=n for any finite extension Gf(Kn) of Kn. In particular, this construction allows us to produce, for each integer n>1, a family of groups {K(s,n)}s∈ℕ of geometric dimension n, such that any finite extension of K(s,n) also has geometric dimension n, but the finite extensions Gf(K(s,n)) are non-isomorphic, if s≠s′.

A set Λ isinternally chain transitiveif for anyx,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ betweenxandy. In this paper we characterize allω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type,X, then there is a pointz∈Xwithω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet ℬ is theω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent mapf, a closed, stronglyf-invariant, internally chain transitive subset of the interval is theω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift spaceZ𝒢(a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not theω-limit set of any point inZ𝒢.

We consider C∞-diffeomorphisms on a Banach space with a fixed point 0 and linear part L. Suppose that these diffeomorphisms have C∞ non-contracting and non-expanding invariant manifolds, and formally conjugate along their intersection (the center). We prove that they admit local C∞ conjugation. In particular, subject to non-resonance conditions, there exists a local C∞ linearization of the diffeomorphisms. It also follows that a family of germs with a hyperbolic linear part admits a C∞ linearization, which has C∞ dependence on the parameter of the linearizing family. The results are proved under the assumption that the Banach space allows a special extension of the maps. We discuss corresponding properties of Banach spaces. The proofs of this paper are based on the technique, developed in the works of Belitskii [Funct. Anal. Appl.18 (1984), 238–239; Funct. Anal. Appl.8 (1974), 338–339].

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 this paper, preimage pressure, which is based on the preimage structure of the system, is defined and studied for random transformations. We obtain analogs of many known results of preimage entropy and preimage pressure for deterministic cases in Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113] and Zeng et al [Pre-image pressure and invariant measures. Ergod. Th. & Dynam. Sys.27 (2007), 1037–1052]. In particular, a variational principle is given and some applications of preimage pressure, such as the investigation of the invariant measures and the equilibrium states, are obtained.

The main purpose of this paper is to show that ideas of deformation theory can be applied to "infinite dimensional geometry". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves., Comment: 18 pages

LetNbe an integer withN≥2 and letXbe a compact subset of ℝd. If$\mathsf {S}=(S_{1},\ldots ,S_{N})$is a list of contracting similaritiesSi:X→X, then we will write$K_{\mathsf {S}}$for the self-similar set associated with$\mathsf {S}$, and we will writeMfor the family of all lists$\mathsf {S}$satisfying the strong separation condition. In this paper we show that the maps(1)and(2)are continuous; here$\dim _{\mathsf {H}}$denotes the Hausdorff dimension, ℋsdenotes thes-dimensional Hausdorff measure and 𝒮sdenotes thes-dimensional spherical Hausdorff measure. In fact, we prove a more general continuity result which, amongst other things, implies that the maps in (1) and (2) are continuous.

In the Michael Herman Memorial Volume, we stated and proved some rather simple invariant-manifold theorems, having many old and new applications. This paper presents the final state of the Lipschitzian part of the theory: the results are put into a more general framework and new properties are established. Both the hypotheses and the proofs of all our statements are quite simple. Smoothness will be treated as in Chaperon [Stable manifolds and the Perron–Irwin method. Ergod. Th. & Dynam. Sys.24 (2004), 1359–1394] in a forthcoming book.

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.