No abstract

We show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mañé example and one obtained through a Hopf bifurcation.

We study the dynamics of a type of nonconservative billiards where the ball is "kicked" by the wall giving a new impulse in the direction of the normal. For different types of billiard tables we study the existence of attractors with dominated splitting.

Let f : M → M be a C 2 diffeomorphism of a compact surface. We give a complete description of the dynamics of any compact invariant set having dominated splitting. In particular, we prove a Spectral Decomposition Theorem for the limit set L(f ) under the assumption of dominated splitting. Moreover, we describe all the bifurcations that these systems can exhibit and the different types of dynamics that could follow for small C r −perturbations.

Link to this article: http://journals.cambridge.org/abstract_S0143385708080346How to cite this article: L. J. DÍAZ, V. HORITA, I. RIOS and M. SAMBARINO (2009