Let A be a finite set and let ϕ:Aℤ→ℝ be a locally constant potential. For each β>0 (‘inverse temperature’), there is a unique Gibbs measure μβϕ. We prove that as β→+∞, the family (μβϕ)β>0 converges (in the weak-* topology) to a measure that we characterize. This measure is concentrated on a certain subshift of finite type, which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron–Frobenius theorem for matrices à la Birkhoff. The crucial idea we bring is a ‘renormalization’ procedure which explains convergence and provides a recursive algorithm for computing the weights of the ergodic decomposition of the limit.

Let A be a locally compact abelian group, and H a locally compact group acting on A. Let G=H⋉A be the semidirect product, assumed σ-compact. We prove that the pair (G,A) has Kazhdan’s property T if and only if the only countably approximable H-invariant mean on the Borel subsets of the Pontryagin dual , supported at the neighbourhood of the trivial character, is the Dirac measure.

We present a class of subshifts ZN,N=1,2,…, whose associated C*-algebras 𝒪ZN are simple, purely infinite and not stably isomorphic to any Cuntz–Krieger algebra nor to the Cuntz algebra 𝒪∞. The class of the subshifts is the first example whose associated C*-algebras are not stably isomorphic to any Cuntz–Krieger algebra nor to the Cuntz algebra 𝒪∞. The subshifts ZN are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that a KMS-state (a state satisfying the Kubo–Martin–Schwinger condition) for gauge action on the associated C*-algebra 𝒪ZN exists if and only if the logarithm of the inverse temperature is the topological entropy for the subshift ZN, and the corresponding KMS-state is unique.

The hull Ω of an aperiodic repetitive Delone set P in ℝd is a compact metric space on which ℝd acts continuously by translation. Let G be ℝm or 𝕋m and α be a continuous G-cocycle over the dynamical system (Ω,ℝd) . In this paper we study conditions under which the cohomological equation α(ω,x)=ψ(ω−x)−ψ(ω) has continuous solutions. We give a sufficient condition for general continuous G-cocycles and a necessary condition for transversally locally constant G-cocycles. These conditions are given in terms of the set of first return vectors associated with a tower system for Ω. For linearly repetitive Delone sets we give a necessary and sufficient condition for solving the cohomological equation in the class of transversally Hölder G-cocycles.

We prove that for a minimal rotation T on a two-step nilmanifold and any measure μ, the push-forward Tn⋆μ of μ under Tn tends toward Haar measure if and only if μ projects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density one. These results strengthen Parry’s result [Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math.91 (1968), 757–771] that such systems are uniquely ergodic. Extending the work of Furstenberg [Strict ergodicity and transformations of the torus. Amer. J. Math.83 (1961), 573–601], we get the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally we characterize limits of Tn⋆μ for some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties that strengthen unique ergodicity in a way analogous to that in which mixing and weak mixing strengthen ergodicity for measure-preserving systems.

We show that Bowen’s equation, which characterizes the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.

We will consider the local dimension spectrum of a weak Gibbs measure on a C1 non-uniformly hyperbolic system of Manneville–Pomeau type. We will present the spectrum in three ways: using invariant measures, ergodic invariant measures supported on hyperbolic sets and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.

Uniform sets and super-stationary sets over the binary alphabet have been extensively studied. In this paper, they are generalized to general alphabets. We generalize the fact that any uniform set contains a super-stationary set so that any uniform complexity is realized by a super-stationary set. This gives a formula to calculate the uniform complexity functions. We also give characterizations of the class of super-stationary sets in general settings in two somewhat different ways than in the binary case. Super-stationary sets are considered as phenomena which are independent of the time scale, but sensitive only to the direction of time, or dependent just on the order of events in time series. Hence, characterizations of super-stationary sets give insights into what is time, what looks like a history without a description of time duration, or what remains meaningful after we lose the quantitative sense of time.

We study the growth and divergence of quotients of Kleinian groups G (i.e. discrete, torsionless groups of isometries of a Cartan–Hadamard manifold with pinched negative curvature). Namely, we give general criteria ensuring the divergence of a quotient group of G and the ‘critical gap property’ . As a corollary, we prove that every geometrically finite Kleinian group satisfying the parabolic gap condition (i.e. δP<δG for every parabolic subgroup P of G) is growth tight. These quotient groups naturally act on non-simply connected quotients of a Cartan–Hadamard manifold, so the classical arguments of Patterson–Sullivan theory are not available here; this forces us to adopt a more elementary approach, yielding as by-product a new elementary proof of the classical results of divergence for geometrically finite groups in the simply connected case. We construct some examples of quotients of Kleinian groups and discuss the optimality of our results.

Let 𝒟r+[0,1], r≥1, denote the group of orientation-preserving 𝒞r diffeomorphisms of [0,1]. We show that any two representations of ℤ2 in 𝒟r+[0,1], r≥2, are connected by a continuous path of representations of ℤ2 in 𝒟1+[0,1] . We derive this result from the classical works by G. Szekeres and N. Kopell on the 𝒞1 centralizers of the diffeomorphisms of [0,1) that are at least 𝒞2 and fix only 0 .

Exact regularity was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider the analog of exact regularity for arbitrary tiling spaces. Let T be a d-dimensional repetitive tiling, and let Ω be its hull. If Ȟd(Ω,ℚ)=ℚk, then there exist k patches each of whose appearances governs the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling T comes from a substitution, then we can quantify that convergence rate. If T is also one dimensional, we put constraints on the measure of any cylinder set in Ω.

The pattern complexity of a word for a given pattern S, where S is a finite subset of {0,1,2,…}, is the number of distinct restrictions of the word to S+n (with n=0,1,2,…). The maximal pattern complexity of the word, introduced in the paper of T. Kamae and L. Zamboni [Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys.22(4) (2002), 1191–1199], is the maximum value of the pattern complexity of S with #S=k as a function of k=1,2,…. A substitution of constant length on an alphabet is a mapping from the alphabet to finite words on it of constant length not less than two. An infinite word is called a fixed point of the substitution if it stays the same after the substitution is applied. In this paper, we prove that the maximal pattern complexity of a fixed point of a substitution of constant length on {0,1} (as a function of k=1,2,…) is either bounded, a linear function of k, or 2k.

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.

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 introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset as the set of essential values of the multiplicity function for the Koopman operator of a mixing ergodic infinite measure preserving transformation, (ii) construct mixing power weakly mixing infinite measure preserving transformations, and (iii) construct mixing Poissonian automorphisms with a simple spectrum, etc.

We determine all the normal subgroups of the group of Cr diffeomorphisms of ℝn, 1≤r≤∞, except when r=n+1 or n=4, and also of the group of homeomorphisms of ℝn ( r=0). We also study the group A0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with non-empty boundary, then the quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

We give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville–Pomeau family of maps.

In this article we consider twist maps that are non-periodic (and hence are defined on the plane rather than on the cylinder) and have small twist at infinity. Under natural assumptions the existence of infinitely many bounded orbits is established, and furthermore it is proved that unbounded orbits follow bounded orbits for long times. An application is given to the Fermi–Ulam ping-pong model with a non-periodic moving wall.

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 Eckmann et 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 solves We 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.