Preprint C84/2009
Ergodic measures far away from tangencies
Yang Jiagang
Keywords: dominated splitting | far from tangencies
In this thesis we study the dynamics of C^1 diffeomorphisms far away from homoclinic tangencies, that is, such that no diffeomorphism in a neighborhood exhibits a non-transverse intersection between the stable manifold and the unstable manifold of some periodic point. There are two main sets of results, having in common the general theme that diffeomorphisms far away from tangencies resemble hyperbolic diffeomorphisms. In the first part of the work we study the ergodic measures of diffeomorphisms far away from homoclinic tangencies. We show that every ergodic measure has at most one vanishing Lyapunov exponent, and the Osledets splitting corresponding to positive, zero, and negative exponents is dominated. In fact we prove that Pesin theory (existence of smooth local stable and local unstable manifolds) holds in this C1 setting: the usual C1+Holder regularity assumption can be replaced by the condition that this system is fay away from homoclinic tangencies. Morever, some shadowing lemma holds, and every hyperbolic ergodic measure is the weak limit of a sequence of atomic invariant measures supported on periodic orbits belonging to the same homoclinic class. By means of a result announced recently by D{i}az and Gorodetski, we deduce that for C^1 generic diffeomorphisms far away from tangencies, every chain recurrent class C is either hyperbolic or has a non-hyperbolic ergodic invariant measure. In particular, if C is an aperiodic class, then every ergodic measure supported in it is non-hyperbolic. That gives a partial answer to a conjecture given by Diaz and Gorodetski in the case of diffeomorphisms far away from tangencies. In the second part of the work we study the so-called C^1 Newhouse phenomenon: existance of infinitely many periodic sinks or sources for a residual subset of some C^1 open set of diffeomorphisms. We prove that if the C^1 Newhouse phenomenon occurs for diffeomorphisms far away from tangencies, then those periodic sinks/sources must be related to some homoclinic class of codimension 1. In fact, the homoclinic class is the Hausdorff limit of a sequence of periodic sinks/sources. This is in contrast with the only known example of C1 Newhouse phenomenon, due to Bonatti and Diaz, which correspond to diffeomorphisms close to homoclinic tangencies, and for which the periodic sinks/sources are often related to aperiodic classes.