This article is devoted to the development of a general theory of transformations of $\mathbb{R}^n$, called dimension-preserving (DP) transformations, which preserve the Hausdorff–Besicovitch dimension of arbitrary subsets.

The main attention is given to continuous transformations of $\mathbb{R}$ and [0, 1]. A class of distribution functions of random variables with independent s-adic digits is studied in detail. It is proved that any absolutely continuous function from the previously mentioned class is a DP function, despite the fact that it may have a very complicated local structure. Necessary, respectively, sufficient conditions for dimension preservation are also given for singular functions. Relations between the entropy of transformations and their DP properties are investigated.

Examples and counterexamples are provided,and some applications are discussed.

Bonatti and Viana introduced a robust (non-empty interior) class of partially hyperbolic attractors of C2-diffeomorphisms on a compact manifold, for which they construct Sinai–Ruelle–Bowen measures. For some such robust examples, we prove the exponential decay of correlations and the central limit theorem, in the space of Hölder continuous functions. For the proof, we adapt the techniques (backward inducing, redundancy elimination algorithm) we have previously developed.

We show that for any finitely generated solvable group of exponential growth one can find a measure-preserving action for which the multiple recurrence theorem fails and a measure-preserving action for which the ergodic Roth theorem fails. This is in contrast with the positive results established by Leibman (Geom. Funct. Anal.8 (1998), 853–931) and Bergelson and Leibman (Invent. Math.147 (2002), 429–470) for nilpotent group actions.

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 show that any surface in $\mathbb{R}^3$ can be modified by gluing on small ‘focusing caps’ so that its geodesic flow becomes ergodic. A new concept, finite horizon cap geometry, is what makes the construction work.

The dynamics of a system of two diffusively coupled identical unimodal maps is investigated. The unimodal maps have simple kneading sequences which imply that their orbits are asymptotically periodic with period a power of two. Analogous results are obtained for the coupled system for all coupling parameters. We prove the eventual periodicity of the position of orbits with respect to the diagonal of the square phase space and the asymptotic periodicity for orbits whose coordinates have the same sign. Finally, we give a global condition for the existence of symmetric orbits that are examples of orbits with this asymptotic property.

Let $\Gamma$ be a lattice in SU(n, 1). For each loxodromic element $\gamma_0\in\Gamma$ we define a closed curve $\{\gamma_0\}$ on $\Gamma\backslash SU(n,1)$ that projects to the closed geodesic on the factor of the complex hyperbolic space $\Gamma\backslash{\mathbb H}^n_{{\mathbb C}}$ associated with $\gamma_0$. We prove that the cohomological equation $\mathfrak{D} F=f$ has a solution if f is the lift of a holomorphic cusp form to SU(n, 1) under the following condition: for each restriction of f to $\{\gamma_0\}$ a finite number of Fourier coefficients vanish, and this finite number grows linearly with the length of the curve. This is a generalization of the classical Livshitz theorem for SU(1, 1) (A. Livshitz. Mat. Zametki10 (1971), 555–564) where the curves are the closed geodesics themselves and the vanishing of the integrals of f over them, i.e. the zeroth Fourier coefficients, is both necessary and sufficient. An application of our result to the construction of spanning sets for spaces of holomorphic cusp forms on complex hyperbolic spaces is given in Appendix A.

We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these groups our revised formalisms give access to a description of the spectrum of ‘homological growth rates’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ‘thermodynamic phase transitions’.

The volume entropy, h(g), of a compact Riemannian manifold (M,g) measures the growth rate of the volume of a ball of radius R in its universal cover. Under the Ricci flow, g evolves along a certain path $(g_t, t\geq0)$ that improves its curvature properties. For a compact surface of variable negative curvature we use a Katok–Knieper–Weiss formula to show that h(gt) is strictly decreasing.

We prove that the billiard map in a Bunimovich stadium has polynomial decay of correlations of the order of n-1. Ergodic Bunimovich-type billiards (two arcs not bigger than half a circle) that do not have trajectories with infinitely many consecutive bounces on straight lines satisfy the same property. We also prove that a very particular class of these billiards have polynomial decay of correlations of the order of n-2.

Symbolic matrix systems are generalizations of finite symbolic matrices for sofic systems to subshifts. We prove that if two symbolic matrix systems are strong shift equivalent, then the gauge actions of the associated C*-algebras are stably outer conjugate. The proof given here is based on the construction of an imprimitivity bimodule from a bipartite $\lambda$-graph system, so that an equivariant version of the Brown–Green–Rieffel Theorem proved by Combes is used, together with its proof. As a corollary, if two subshifts are topologically conjugate, then the gauge actions of the associated C*-algebras are stably outer conjugate.

Let M be a complete Riemannian locally symmetric space of non-positive curvature and of finite volume. We show that there are only finitely many compact maximal flats in M of volume bounded by a given number. As a corollary in the case $M=\mathrm{SL}_n(\mathbb{Z})\backslash\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n$, we give a different proof of a theorem of Remak that for any $n\in \mathbb{N}$, there are only finitely many totally real number fields of degree n whose regulator is less than a given number.

If M is a compact or convex-cocompact negatively curved manifold, we associate to any Gibbs measure on T1M a quasi-invariant transverse measure for the horospherical foliation, and prove geometrically that this measure is uniquely determined by its Radon–Nikodym cocycle. (This extends the Bowen–Marcus unique ergodicity result for this foliation.) We shall deduce from this result some equidistribution properties for the leaves of the foliation with respect to these Gibbs measures. We use it also in the study of invariant measures for horospherical foliations on regular covers of M.

For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Young's axioms (Lai-Sang Young, Ann. Math.147 (1998), 585–650), which imply exponential decay of correlations and the central limit theorem (CLT), a local CLT is proven. In fact, a unified version of the local CLT is found, covering, among others, the absolutely continuous and arithmetic cases. For planar Lorentz process with a finite horizon, this result implies (a) a local CLT and (b) recurrence. For the latter case (d = 2, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt (C. R. Acad. Sci. Paris Ser. 1 Math. 372(9) (1998), 837–842) and J.-P. Conze (Ergod. Th. & Dynam. Sys.19(5) (1999), 1233–1245) could already establish recurrence.

We deal with all the mappings $f_\lambda(z)=\lambda e^z$ that have an attracting periodic orbit. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ that do not escape to infinity under positive iterates of $f_\lambda$. Our ultimate result is that the function $\lambda\mapsto{\rm HD}(J_r(f_\lambda))$ is real-analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form $-t\log|F_\lambda'|$, where $F_\lambda$ is the natural map associated with $f_\lambda$ closely related to the corresponding map introduced in Urbański and Zdunik (2001). The formalism includes appropriately defined topological pressure, Perron–Frobenius operators, and geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron–Frobenius operators are quasicompact and that they embed into a family of operators depending holomorphically on an appropriate parameter, and we obtain several other properties of these operators. We prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for the topological pressure is independent of the set Jr(f), Bowen's formula also indicates that Jr(f) is the right set to deal with. We also study in detail the properties of quasiconformal conjugacies between the maps $f_\lambda$. As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and exponential decay of correlations.