We consider C 2 Hénon-like families of diffeomorphisms of R 2 and study the boundary of the region of parameter values for which the nonwandering set is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency and that uniform hyperbolicity estimates hold uniformly in the parameter up to this bifurcation parameter and even, to some extent, at the bifurcation parameter.

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

In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in [12]. Despite the fact that the non-wandering set is a horseshoe, it contains intervals. We prove that every recurrent point has non-zero Lyapunov exponents and all ergodic invariant measures are hyperbolic. As a consequence, we obtain the existence of equilibrium measures for any continuous potential. We also obtain an example of a family of C ∞ potentials with phase transition.

We study the destruction of hyperbolic sets (horseshoes) in parametrized families of diffeomorphisms through homoclinic tangencies taking place inside the limit set. If the limit set at the tangency parameter has small dimension (limit capacity) then hyperbolicity prevails after the bifurcation (full Lebesgue density). We also prove that, if that limit set is thick then the system exhibits homoclinic tangencies for a whole parameter interval across the bifurcation. These results are based on a geometric analysis of the limit set at the tangency, including a statement of bounded distortion.

In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated to Hölder continuous potentials.