We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is mixing and, in particular, the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, by the average rate at which typical points start to exhibit exponential growth of the derivative.

The untimely death of Michael Robert Herman in November 2000 deprived the scientific community of one of its deepest mathematical minds, who had a profound impact on the theory of dynamical systems over the last 30 years.

Born in New York, he was educated in France. He was a student at École Polytechnique before being one of the first members of the Centre de Mathématiques created there by Laurent Schwartz. For more than 20 years, his seminar had a major influence worldwide and was the main vector of the development of the theory of dynamical systems in France. All of his students remember with thankfulness and emotion the passion with which he led them into the wonderful mathematical world. He maintained through the years strong connections with the Instituto de Matemática Pura e Aplicada in Rio de Janeiro.

His interests covered most aspects of the modern theory of dynamical systems and much beyond that, from economics to arts and philosophy. However, it is fair to say that from the start the so-called small divisors problems, related in particular to the stability of quasiperiodic motions, were closest to his heart. His epoch-making theorem on the linearization of circle diffeomorphisms [1–4], his two volumes [5, 6] on invariant curves for twist diffeomorphisms, which are still the standard reference 20 years later, his very many deep contributions on the existence and geometry of invariant tori all bear witness to that interest [7–10].

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.

Using the techniques developed in Alves and Sousa Ramos (Comm. Math. Phys.204(1) (1999), 89–114), we generalize the Milnor and Thurston results about the zeta function, semiconjugacy and topological entropy of interval maps to tree maps.

In this paper we consider homeomorphisms $f:T^2 \rightarrow T^2$ homotopic to the identity and their rotation sets $\rho (\tilde{f})$, which are compact convex subsets of the plane. We show that if $\rho (\tilde{f})$ has an extremal point $(t,\omega )$ which is not a rational vector, then arbitrarily C0 close to f we can find a homeomorphism g such that $\rho (\tilde{g})\cap \rho (\tilde{f})^c\neq \emptyset$. So in this case, we have instability for the rotation set.

Certain topological dynamical systems that arise from actions of $\sigma$-compact locally compact Abelian groups on compact spaces of translation bounded measures are considered. Such a measure dynamical system is shown to have a pure point dynamical spectrum if and only if its diffraction spectrum is pure point.

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 introduce the notion of a space with measured walls, generalizing the concept of a space with walls due to Haglund and Paulin (Simplicité de groupes d'automorphismes d'espaces à courbure négative. Geom. Topol. Monograph1 (1998), 181–248). We observe that if a locally compact group G acts properly on a space with measured walls, then G has the Haagerup property. We conjecture that the converse holds and we prove this conjecture for the following classes of groups: discrete groups with the Haagerup property, closed subgroups of SO(n, 1), groups acting properly on real trees, SL2(K) where K is a global field and amenable groups.

We consider analytic maps $f_j:D\to D$ of a domain D into itself and ask when does the sequence $f_1\circ\dotsb\circ f_n$ converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence $\{w_1,w_2,\dotsc\}$ in D whose values are not taken by any fj in D, and which is homogeneous in the sense that it comes within a fixed hyperbolic distance of any point of D. The situation for several complex variables is also discussed.

In this paper, we give the explicit construction of certain components of the space of holomorphic foliations of codimension one, in complex projective spaces. These components are associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. Some of them, the so-called exceptional or Klein–Lie components, are rigid in the sense that all generic foliations in the component are equivalent (Example 1). In particular, we obtain rigid foliations of all degrees. Some generalizations and open problems are given at the end of §1.

We show that if $\varphi$ is a lacunary Fourier series and the equation $\psi (x) -\psi (x + \alpha) = \varphi(x), x \bmod 1$ has a measurable solution $\varphi$, then in fact the equation has a solution in L2.

We consider dynamical systems on compact manifolds that are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that a (strong) integrability of the first hyperbolic time map implies a positive frequency of hyperbolic times. We also present an example of a map with a positive frequency of hyperbolic times at Lebesgue almost every point, but whose first hyperbolic time map is not integrable with respect to the Lebesgue measure.

We study the existence, structure and topological entropy of almost automorphic arrays in symbolic lattice dynamical systems. In particular, we show that almost automorphic arrays with arbitrarily large entropy are typical in symbolic lattice dynamical systems. Applications to pattern formation and spatial chaos in infinite-dimensional lattice systems are considered and the construction of chaotic almost automorphic signals is discussed.

We present new invariants for substitutional dynamical systems. Our main contribution is a flow invariant which is strictly finer than, but related and akin to, the dimension groups of Herman et al. We present this group as a stationary inductive limit of a system associated to an integer matrix defined from combinatorial data based on the class of special words of the dynamical system.

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.

For any ergodic endomorphism of a non-atomic probability space and any Borel generating set of a locally compact Polish (LCP) amenable group, we construct a measurable function taking values in the generating set so that the skew product generated is ergodic.

We study C1 invariant, two-dimensional tori of constant energy without singularities of the Euler–Lagrange flow of a convex, superlinear Lagrangian defined in the tangent space of a closed surface. We show that in regular energy levels, such tori are graphs of the canonical projection if and only if they are minimizing. This result was known for C3 invariant, Lagrangian tori of Finsler Lagrangians, which implied the statement for C3 invariant Lagrangian tori of unimodal Lagrangians with a high-energy level. We also show that graphs have energy $E \geq c_0(L)$, where c0(L) is Mañé's strict critical value; and that the presence of Reeb components in such tori implies that the energy is c0(L). Moreover, if we assume that the Lagrangian is unimodal, we show that Lagrangian invariant tori with no singularities have no Reeb components, regardless of the energy level.

Let $(X,\mathcal{B},\mu, T)$ be an ergodic dynamical system on the finite measure space $(X,\mathcal{B},\mu,T)$ and $\mathcal{K}$ its Kronecker factor. We denote by U the restriction of T onto $\mathcal{K}^{\perp}$ the orthocomplement of $\mathcal{K}$. We give a spectral characterization in L2 of Wiener–Wintner functions in terms of the capacity of the support of the maximal spectral type of U and the almost everywhere continuity of the fractional rotated ergodic Hilbert transform. The study of the L2 case leads to new classes of dynamical systems.

This paper explores the effect of various graphical constructions upon the associated graph C*-algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that out-splittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C*-algebras. We generalize the notion of a delay as defined in (D. Drinen, Preprint, Dartmouth College, 2001) to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C*-algebras. We provide examples which suggest that our results are the most general possible in the setting of the C*-algebras of arbitrary directed graphs.

Let f be a generalized eigenfunction of an interval exchange transformation, T, on the unit interval, which satisfies the infinite distinct orbit condition (IDOC). We assume that the minimum spacing, $\epsilon_n(T)$, of the partition defined by Tn is of the order 1/n for infinitely many n. This assumption is generic. Given $\delta>0$ we prove that for sufficiently large n and for every interval J satisfying $\vert J \vert =\epsilon_n(T)$, there exists $x_0 \in J$ such that \[\vert \{ x \in J : \vert f(x) - f(x_0) \vert \geq \delta \} \vert< \delta \vert J \vert.\] This provides a specific sufficient generic diophantine condition for Veech's result [V2, Lemma 7.3]. Above $\vert \cdot\vert$ denotes the linear measure of the set. Moreover, if T is uniquely ergodic, then any bounded variation continuous f must be of the form $f(x)= \gamma \exp(2 \pi iax)$ almost everywhere, where $\gamma \in \mathbb{C}$ and $a \in \mathbb{R}$.