Let β∈(1,2). Each x∈[0,1/(β−1)] can be represented in the form where εk∈{0,1} for all k (a β-expansion of x). If , then, as is well known, there always exist x∈(0,1/(β−1)) which have a unique β-expansion. We study (purely) periodic unique β-expansions and show that for each n≥2 there exists such that there are no unique periodic β-expansions of smallest period n for β≤βn and at least one such expansion for β>βn. Furthermore, we prove that βk<βm if and only if k is less than m in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps.

Gibbs measures μ on cookie-cutter sets are the archetype of multifractal measures on Cantor sets. We compute the singularity spectrum of the inverse measure of μ. Such a measure is discrete (it is constituted only by Dirac masses), it satisfies a multifractal formalism and its Lq-spectrum possesses one point of non-differentiability. The results rely on heterogeneous ubiquity theorems.

For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of n variables, K:[0,1]n→ℝ, which are separately Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We also present applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.

Let 0<α<1 and β>1. We show that every x∈[0,1] has an expansion of the form where hi=hi(x)∈{0,α/β}, and pi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values of α, we give the relationship of this expansion with the greedy β-expansion.

Let (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

We introduce a new method for estimating the growth of various quantities arising in dynamical systems. Applying our method to polygonal billiards on surfaces of constant curvature, we obtain estimates almost everywhere on direction complexities and position complexities. As a byproduct, we recover the power bounds of Boshernitzan on the number of billiard orbits between almost all pairs of points in a planar polygon [M. Boshernitzan. Private letter to H. Masur (1986)].

For a large class of non-uniformly expanding maps of ℝm, with indifferent fixed points and unbounded distortion and that are non-necessarily Markovian, we construct an absolutely continuous invariant measure. We extend previously used techniques for expanding maps on quasi-Hölder spaces to our case. We give general conditions and provide examples to which our results apply.

We study asymptotic distributions of the sums yn(x)=∑ k=0n−1ψ(x+kα) with respect to the Lebesgue measure, where α∈ℝ−ℚ and where ψ is the 1-periodic function of bounded variation such that ψ(x)=1 if x∈[0,1/2[ and ψ(x)=−1 if x∈[1/2,1[. For every α∈ℝ−ℚ, we find a sequence (nj)j⊂ℕ such that is asymptotically normally distributed. For n≥1, let zn∈(ym)m≤n be such that ‖zn‖L2=max m≤n‖ym‖L2. If α is of constant type, we show that zn/‖zn‖L2 is also asymptotically normally distributed. We give a heuristic link with the theory of expanding maps of the interval.

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)a is a holomorphic family of stable rational maps, then the dimension d(fa) is a continuous and plurisubharmonic function of the parameter a. In particular, d(f) varies continuously and plurisubharmonically on an open and dense subset of Ratd, the space of all rational maps with degree d≥2.

Given a code from a shift space to an irreducible sofic shift, any two of the three conditions—open, constant-to-one and (right or left) closing—imply the third. If the range is not sofic, then the same result holds when bi-closingness replaces closingness. Properties of open mappings between shift spaces are investigated in detail. In particular, we show that a closing open (or constant-to-one) extension preserves the structure of a sofic shift.

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.

If the β-function of a time-periodic Lagrangian on a manifold M has a vertex at a k-irrational homology class h, then 2k<dim M, so if dim M=2, then h is rational.

Tao has recently proved that if T1,…,Tl are commuting, invertible, measure-preserving transformations on a dynamical system, then for any L∞ functions f1,…,fl, the average (1/N)∑ n=0N−1∏ i≤lfi∘Tin converges in the L2 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.

We consider piecewise invertible systems exhibiting intermittency and establish a generalized variational principle adapted to a non-stationary process in the following sense; the supremum is attained by non-singular (not necessarily invariant) probability measures and if the system exhibits hyperbolicity, then it reduces to the usual variational principle for the pressure. Our method relies on Ruelle’s program in the study of non-equilibrium statistical mechanics to analyze dissipative phenomena. We show non-positivity of entropy production at weak Gibbs measures and clarify when it indeed vanishes. We also discuss a generalized variational principle in the context of σ-finite invariant measures.

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.

We give three corrections to the paper [K. Schmidt and A. M. Vershik, Algebraic polymorphisms, Ergod. Th. & Dynam. Sys.28 (2008), 633–642].