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Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions

Lorenzo J. DíazKatrin Gelfert — 2012

Fundamenta Mathematicae

We study a partially hyperbolic and topologically transitive local diffeomorphism F that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set Λ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many non-trivial fibers. Moreover, the spectrum of the central Lyapunov exponents of...

Generic diffeomorphisms on compact surfaces

Flavio AbdenurChristian BonattiSylvain CrovisierLorenzo J. Díaz — 2005

Fundamenta Mathematicae

We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly exhibits homoclinic tangencies. In the course of our discussion,...

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