Let G be a solvable Lie group. Let $\chi$ be a character on G with values in $\mathbb R$. Let $\mathfrak B$ be a Banach space on which G is made to act by bounded operators, with the equality $\Vert gb\Vert=\chi(g)\Vert b\Vert$ satisfied for all $g\in G$ and $b\in\mathfrak B$. A criterion is given on the pair $(G,\chi)$ for the cohomology group $H^1(G,\mathfrak B)$ to vanish. Using this criterion, we classify up to Cr conjugacy all the volume-preserving locally free Cr actions of G on closed manifolds with dimension $\dim(G)+1$, for G belonging to a wide class of solvable Lie groups.

We study injective locally contracting maps of the interval. After giving an upper-bound on the number of ergodic components, we show that generically finitely many periodic orbits attract the whole dynamics and that this picture is stable under perturbation. In relation to the problem of maximizing measures for regular maps, we then consider a class of probability measures on the circle invariant by × p generalizing the family of Sturm measures and show its generic periodic character. We detail the structure of order-preserving locally contracting maps on the circle. The rotation number is shown to be generically rational and the transformations having a given rational rotation number are shown explicitly. We also count the periodic attractors. We then deduce the existence of measurable conjugacies with three-interval exchange transformations in non-periodic cases for a model with three pieces on the interval.

In contrast with a result of Burago, Ferleger and Kononenko stating that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite, we give an example of a (degenerate) semi-dispersing billiard with infinite topological entropy.

Let V be a finite-dimensional vector space over a field ${\mathbb{K}}$ and let G be a countable amenable group. Let $\tau \colon V^G \to V^G$ be a linear cellular automaton. We prove that $\tau$ is surjective if and only if it is pre-injective.

We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map ${\mathcal I}$ from the parameter space of possible shapes of tiles to H1 of a model tiling space, with values in ${\mathbb R}^d$. Two tiling spaces that have the same image under ${\mathcal I}$ are mutually locally derivable (MLD). When the difference of the images is ‘asymptotically negligible’, then the tiling dynamics are topologically conjugate, but generally not MLD. For substitution tilings, we give a simple test for a cohomology class to be asymptotically negligible, and show that infinitesimal deformations of shape result in topologically conjugate dynamics only when the change in the image of ${\mathcal I}$ is asymptotically negligible. Finally, we give criteria for a (deformed) substitution tiling space to be topologically weakly mixing.

If S(t) is a C0-semigroup on a Banach space X with a hypercyclic vector x, then x is a hypercyclic vector for each discrete semigroup $\lbrace S(kT)\mid k \in \mathbb{N}\rbrace$ with T > 0 fixed.

We introduce and study, from a combinatorial-topological viewpoint, some semigroups of continuous non-deterministic dynamical systems. Combinatorial stability, i.e. the persistence of the combinatorics of the attractors, is characterized and its genericity established. Some implications on topological (deterministic) dynamics are drawn.

There is a natural action of SL$(2,\mathbb{R})$ on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = \big\{\big(\begin{smallmatrix}1 & * \\ 0 & 1\end{smallmatrix}\big)\big\}$. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL$(2,\mathbb{R})$-orbit of the set of branched covers of a fixed Veech surface.) For the U-action on these submanifolds, this is an analogue of Ratner's theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-Veech rational triangles, namely, the isosceles triangles in which exactly one angle is $2 \pi/n$, with $n \ge 5$ and n odd.

We prove the equidistribution of Hecke points for any connected non-compact $\mathbb{Q}$-simple real algebraic group G and an arithmetic subgroup $\Gamma\subset G(\mathbb{Q})$, generalizing a theorem of Clozel, Oh and Ullmo. The main tool is a theorem of Mozes and Shah on unipotent flows.

In this paper we discuss whether germs of holomorphic self maps of $\mathbb{C}^n$ can have a dense orbit. We give both a positive and a negative answer depending on how the problem is interpreted more precisely.

Consider a tiling $\mathcal T$ of the two-dimensional Euclidean space made with copies up to translation of a finite number of polygons meeting each other full edge to full edge. In this paper, we prove that, associated with $\mathcal T$, there exists a tiling of a (compact) translation surface made with copies up to translation of some of the polygons used to construct $\mathcal T$. Furthermore, when $\mathcal T$ is repetitive, there exists a tiling of a translation surface, made with copies up to translation of arbitrarily large polygons chosen in a finite collection of patches of $\mathcal T$; each of these patches contain copies of all the polygons used to construct $\mathcal T$.

We study the spectral properties of the Ruelle–Perron–Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the ${\mathcal C}^\infty$ case, the essential spectral radius is arbitrarily small, which yields a description of the correlations with arbitrary precision. Moreover, we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the Sinai–Ruelle–Bowen measure, the variance for the central limit theorem, the rates of decay for smooth observable, etc.).

For a given factor map $\pi:X\longrightarrow Y$ between two topological dynamical systems and a Borel cover ${\mathcal U}$, two notions of measure-theoretical conditional entropy $h_\mu^+(T,{\mathcal U}\mid Y)$ and $h_\mu^-(T,{\mathcal U}\mid Y)$ for an invariant Borel probability measure $\mu$ are introduced. It is shown that $h_\mu^+(T,{\mathcal U}\mid Y)=h_\mu^-(T,{\mathcal U}\mid Y)$. Moreover, $\max_{\mu}h_\mu^+(T,{\mathcal U}\mid Y)=h_{{\rm top}}(T,{\mathcal U}\mid Y)$ when $\mathcal U$ is an open cover. The relative variational principle is a consequence of the results.

If a closed manifold M possesses two Riemannian metrics which have the same unparameterized geodesics and are not strictly proportional at each point, then the topological entropy of both geodesic flows is zero. This is the main result of the paper and it has many dynamical and topological corollaries. In particular, such a manifold M should be finitely covered by the product of a rationally elliptic manifold and a torus.

We generalize the notion of reversible systems in symbolic dynamics, and investigate their properties. It is shown that a reversible topological Markov shift can be represented by a pair of matrices of special types. This enables us to classify the invariant measures of the reversible systems. Necessary and/or sufficient conditions for the existence of a reversal of finite order are established in terms of the adjacency matrices. We also prove that a topological Markov shift with a reversal of order two admits reversals of all orders.

Let $f:M\to M$ be a partially hyperbolic diffeomorphism, ${\it TM}=E^{ss}\oplus E^c\oplus E^{uu}$ such that the stable foliation $\mathcal{F}^{ss}(f)$ is minimal. We give a sufficient condition so that this foliation remains minimal after perturbations, i.e. $\mathcal{F}^{ss}(g)$ is minimal for every g sufficiently close to f.

We obtain a local topological and dynamical description of expansive attractors on surfaces. The main result is that expansive attractors on surfaces are hyperbolic and have local product structure, except possibly at a finite number of periodic points, which can be either sinks, singularities or épines. Some open questions concerning this kind of dynamics are posed.