Multidimensional nonhyperbolic attractors
Ergodic theory of interval exchange maps.
A unified introduction to the dynamics of interval exchange maps and related topics, such as the geometry of translation surfaces, renormalization operators, and Teichmüller flows, starting from the basic definitions and culminating with the proof that almost every interval exchange map is uniquely ergodic. Great emphasis is put on examples and geometric interpretations of the main ideas.
Random perturbations and statistical properties of Hénon-like maps
Uniform (projective) hyperbolicity or no hyperbolicity : a dichotomy for generic conservative maps
Strong stochastic stability and rate of mixing for unimodal maps
Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows
We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.
The entropy conjecture for diffeomorphisms away from tangencies
We prove that every diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive.
A probabilistic and geometric perspective.
Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices
Dynamics in the moduli space of Abelian differentials.
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