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

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

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

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

Let G be a linear Lie group. We define the G-reducibility of a continuous or discrete cocycle modulo N. We show that a G-valued continuous or discrete cocycle which is GL(n,ℂ)-reducible is in fact G-reducible modulo two if G=GL(n,ℝ),SL(n,ℝ),Sp(n,ℝ) or O(n) and modulo one if G=U(n) .

This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that established by Bergelson, Host and Kra for one single transformation. We use the machinery of ‘magic systems’ established recently by B. Host for the proof.

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.

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 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.

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.

A renewal system is a shift space consisting of all bi-infinite free concatenations of some finitely many words. It is known that the zeta function of a renewal system generated by a circular set coincides with the generating function of the circular set. We extend this result to the systems generated by almost circular sets. Also we establish the formulas for the zeta functions of the systems generated by codes and pure codes, respectively. Using the formulas, we show that pure codes generating shifts of finite type are circular.

We study how parallel chip-firing on the complete graph Kn changes behavior as we vary the total number of chips. Surprisingly, the activity of the system, defined as the average number of firings per time step, does not increase smoothly in the number of chips; instead it remains constant over long intervals, punctuated by sudden jumps. In the large n limit we find a ‘devil’s staircase’ dependence of activity on the number of chips. The proof proceeds by reducing the chip-firing dynamics to iteration of a self-map of the circle S1, in such a way that the activity of the chip-firing state equals the Poincaré rotation number of the circle map. The stairs of the devil’s staircase correspond to periodic chip-firing states of small period.

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 Hölder 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⊂Wuε(Λ) with λ(A)=λ(Wuε(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 note is a companion paper to ‘Monodromie et classification topologique des germes de feuilletages holomorphes’ (in French) by Marin and Mattei. In their paper the authors develop a very complete theory about topologically conjugate singularities of foliations. However, their techniques require the topological conjugacies to be transversely holomorphic. The purpose of the present note is to show that this assumption can be made without restricting the generality of their results.

We use the results of Neshveyev and Størmer to show that for a generic shift on a C*-algebra associated with a bitstream the Voiculescu topological entropy is strictly larger that the supremum of topological entropies of its classical subsystems.