It is shown that, for the von Neumann algebra $A$ obtained from a principal measured groupoid $\mathcal{R}$ with the diagonal subalgebra $D$ of $A$, there exists a natural ‘bijective’ correspondence between coactions on $A$ that fix $D$ pointwise and Borel 1-cocycles on $\mathcal{R}$. As an application of this result, we classify a certain type of coactions on approximately finite-dimensional type II factors up to cocycle conjugacy. By using our characterization of coactions mentioned above, we are also able to generalize to some extent those results of Zimmer concerning 1-cocycles on ergodic equivalence relations into compact groups.

Nous introduisons la catégorie des espaces boréliens-topologiques, appelée BT-catégorie : c'est un cadre naturel pour la classification mesurable des feuilletages et des laminations usuels. Nous nous intéressons au cas des laminations de dimension deux. Nous prouvons les deux résultats principaux suivants :

1. Une lamination borélienne par plans est la suspension d'une action de $\mathbb{Z}^2$ sur un espace de Borel standard si et seulement si elle est hyperfinie.

2. Toute lamination par surfaces admettant une structure complexe parabolique mesurable est moyennable.

1. Une lamination par plans, cylindres et tores est $\mu$-moyennable si et seulement si elle admet une métrique de Riemann plate le long de $\mu$-presque chaque feuille.

We consider horizontal iterated function systems in the Heisenberg group $\mathbb{H}^1$, i.e. collections of Lipschitz contractions of $\mathbb{H}^1$ with respect to the Heisenberg metric. The invariant sets for such systems are so-called horizontal fractals. We study questions related to connectivity of horizontal fractals and regularity of functions whose graph lies within a horizontal fractal. Our construction yields examples of horizontal BV (bounded variation) surfaces in $\mathbb{H}^1$ that are in contrast with the non-existence of horizontal Lipschitz surfaces which was recently proved by Ambrosio and Kirchheim (Rectifiable sets in metric and Banach spaces. Math. Ann.318(3) (2000), 527–555).

Let $\{h_{a,b} : a,b \in \mathbb{R}, b \neq 0\}$ denote the Hénon family of quadratic polynomial diffeomorphisms of $\mathbb{R}^2$, with $b$ equal to the Jacobian of $h_{a,b}$. In this paper, we describe the locus of parameter values $(a,b)$ such that $0 < |b| < 0.06$, and the restriction of $h_{a,b}$ to its non-wandering set is topologically conjugate to the horseshoe map. The boundary of the horseshoe locus is shown to be characterized by a homoclinic tangency which is part of a generic unfolding.

We consider a $C^\infty$ Lagrangian function $L: T^*M\rightarrow \mathbb{R}$ which is superlinear and convex in the fibers and has one antisymplectic symmetry. We prove that:

1. in every energy level strictly above the critical one, there exists a Mather set which is the union of some periodic orbits;

2. for $L$ generic, these orbits are hyperbolic;

3. on the torus, these orbits have one homoclinic orbit.

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 establish the existence of center manifolds for a class of non-uniformly partially hyperbolic trajectories of non-autonomous ordinary differential equations $v'=A(t)v+f(t,v)$ on Banach spaces. In particular, we allow the stable and unstable components of $v'=A(t)v$ to exhibit non-uniform contraction and expansion along the trajectory. We also allow the center component to exhibit non-uniform behavior. To the best of our knowledge, we establish the first center manifold theorem in the non-uniform setting.

It is well known that on any Veech surface, the dynamics in any minimal direction is uniquely ergodic. In this paper it is shown that for any genus-2 translation surface which is not a Veech surface there are uncountably many minimal but not uniquely ergodic directions. The theorem can be applied to certain billiard tables.

We consider actions of the free semigroup with two generators on the real line, where the generators act as affine maps, one contracting and one expanding, with distinct fixed points. Then every orbit is dense in a half-line, which leads to the question whether it is, in some sense, uniformly distributed. We present answers to this question for various interpretations of the phrase ‘uniformly distributed’.

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.

Fathi and Franks showed that a pseudo-Anosov diffeomorphism $f$ with orientable foliations and dilation coefficient $\lambda$ with no conjugates (over ${\mathbb Q}$) in the unit circle factors onto a (homologically non-trivial) invariant subset of a hyperbolic toral automorphism. After recounting this result, we show that the factor map is either almost everywhere one-to-one or almost everywhere $m$-to-one for some $m>1$ and the pseudo-Anosov map $f$ is an $m$-to-one ramified covering of another pseudo-Anosov (or Anosov) map on a surface of smaller genus. As a corollary, any pseudo-Anosov diffeomorphism with orientable foliations and hyperbolic action on the first homologies almost everywhere embeds into a hyperbolic toral automorphism.

We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and of the Lyapunov exponents. These formulas are obtained with a direct approach, pertaining to the fundamentals of ergodic theory, which in particular does not require Pesin theory. This allows us to show, for those systems, that the Hausdorff dimension of a (non-ergodic) invariant measure is equal to the essential supremum of the Hausdorff dimensions of the measures in an ergodic decomposition. We also establish corresponding results for non-uniformly hyperbolic diffeomorphisms on surfaces, now using Pesin theory. To the best of our knowledge, the formula for the Hausdorff dimension of an invariant measure along an ergodic decomposition is new even in the case of uniformly hyperbolic dynamics.

We show that given an adding machine of type $\alpha$, for a dense set of parameters $s$ in the interval $[\sqrt2,2]$, if $f$ is the tent map with slope $s$, then the restriction of $f$ to the closure of the orbit of the turning point is topologically conjugate to the adding machine map of type $\alpha$.

We consider properties of Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. This is a slight generalization of the situation first considered by Fornaess and Sibony in Random iterations of rational functions. Ergod. Th. & Dynam. Sys.11 (1991), 687–708. The classical definition of hyperbolicity has a natural extension to our setting, and we show that many of the classical results concerning hyperbolic Julia sets can indeed be generalized. We introduce two important ideas in order to prove these results—the use of Cantor diagonalization, and how the Julia set moves in the Hausdorff topology as the sequence generating it changes.

Ruelle proved that for quasiconformal deformations of cocompact Fuchsian groups, the Hausdorff dimension of the limit set is an analytic function of the deformation. In this paper, we give a criterion for the failure of analyticity for certain infinitely generated groups. In particular, we show that it fails for any infinite abelian cover of a compact surface, answering a question of Astala and Zinsmeister.

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.

We show that a periodic orbit of large period of a diffeomorphism or flow either admits a dominated splitting of a prescribed strength or can be turned into a sink or a source by a $C^1$-small perturbation along the orbit. As a consequence we show that the linear Poincaré flow of a $C^1$-vector field admits a dominated splitting over any robustly transitive set.

We obtain explicit formulas for the average density, also called order-two density, of an arbitrary conformal repeller of a $C^{1+\epsilon}$ transformation. We note that unlike with the pointwise densities, the average densities exist almost everywhere on each repeller, and are in fact constant almost everywhere. Thus, they are natural parameters to describe the geometric structure of the corresponding invariant sets, and as such it is of interest to obtain explicit formulas for their almost constant value. We obtain simultaneously formulas without using symbolic dynamics, and formulas based on the symbolic dynamics generated by a given Markov partition. The proofs strongly depend on the use of appropriate Markov partitions, and are also based on methods of Falconer and of Zähle.

In this paper we obtain different types of random ergodic theorems for dynamical systems or continuous semi-flows. These results recover and extend previous works on dynamical systems and are completely new in the case of semi-flows. The proofs are based on uniform estimates on random almost periodic polynomials that we obtained very recently and on an improvement of a tool introduced by Talagrand in 1995 and further developed by Fernique in 1996. In the course of this, we partially recover results of Marcus and Pisier from the 1980s on almost sure uniform convergence of random almost periodic series.

We present a generalization of the functional equation for the Weierstrassk $\wp$-function for hyperelliptic surfaces of infinite genus arising from iteration of the horseshoe and baker maps. The ramified cover of these infinite genus surfaces over the complex plane are associated to a quadratic differential of finite norm with simple poles accumulating to infinity. We study the geometry of its critical trajectories emanating from these poles and their rate of accumulation.