Ricardo Mañé

Lagrangian flows: The dynamics of globally minimizing orbits

The objective of this note is to present some results, to be proved in a forthcoming paper, about certain special solutions of the Euler-Lagrange equations on closed manifolds. Our main results extend to time dependent periodic Lagrangians with minor modifications.We have chosen the autonomous case because this formally simpler framework allows to reach more easily the core of our concepts and results. Moreover the autonomous case exhibits certain special features involving the energy as a first integral that deserve special attention. They are closely related to the link found by Carneiro [C] between the energy and Mathers action function [Ma].

On a theorem of Fatou

We prove a result on the backward dynamics of a rational function nearby a point not contained in the ω-limit set of a recurrent critical point. As a corollary we show that a compact invariant subset of the Julia set, not containing critical or parabolic points, and not intersecting the ω-limit set ofrecurrent critical points, is expanding, thus extending a classical criteria of Fatou. We also prove that the boundary of a Siegel disk is always contained in the ω-limit set of arecurrent critical point.

The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces

Letf be aCr diffeomorphism,r≥2, of a two dimensional manifoldM2, and let Λ be a horseshoe off (i.e. a transitive and isolated hyperbolic set with topological dimension zero). We prove that there exist aCr neighborhoodU off and a neighbourhoodU of Λ such that forg∈U, the Hausdorff dimension of ∩ngn(U) is aCr−1 function ofg.

The explosion of singular cycles

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On the creation of homoclinic points

Soit M une variete compacte sans bord lisse, f:M← un diffeomorphisme C r et p un point fixe hyperbolique de f. Quand il existe des points presque homoclines, on etudie la possibilite de creer des points homoclines par une petite deformtion du diffeomorphisme