We consider the irrational Aubry-Mather sets of an exact symplectic monotone C 1 twist map, introduce for them a notion of " C 1-regularity " (related to the notion of Bouligand paratingent cone) and prove that : • a Mather measure has zero Lyapunov exponents iff its support is almost everywhere C 1-regular; • a Mather measure has non zero Lyapunov exponents iff its support is almost everywhere...

Let Λ be a compact locally maximal invariant set of a C2diffeomorphism f : M → M on a smooth Riemannian manifold M . In this paper we study the topological pressure Ptop(φ) (with respect to the dynamical system f |Λ) for a wide class of Hölder continuous potentials and analyze its relation to dynamical, as well as geometrical, properties of the system. We show that under a mild non-uniform hype...

This paper is based on the author’s previous work [S4]. We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a com...

Since my graduate work and throughout my postdoctoral career, I have been interested in hyperbolic dynamical systems, more precisely Anosov flows. Roughly speaking, a smooth flow ft on a closed Riemannian manifold M is called Anosov if the tangent bundle of M splits into three continuous sub-bundles, TM = E⊕E⊕E, such that E is tangent to the flow direction, E is exponentially contracted and E i...

We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular gro...

We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those belong a certain prescribed set. establish two types sufficient conditions under non\-auto\-nomous ordinary differential equations have such property. The first criterion applies semilinear equation provided that its linear part is hyperbolic and nonlinearity small in neig...