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

Lorenzo J. Díaz¹ ; Katrin Gelfert²

¹ Departamento de Matemática PUC-Rio Marquês de São Vicente 225 Gávea, Rio de Janeiro 225453-900, Brazil
² Instituto de Matemática UFRJ Av. Athos da Silveira Ramos 149 Cidade Universitária Ilha do Fundão Rio de Janeiro 21945-909, Brazil

Fundamenta Mathematicae, Tome 216 (2012) no. 1, pp. 55-100.

Résumé

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 $ \Lambda $ 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 $F|_{{\Lambda }}$ contains a gap and hence gives rise to a first order phase transition. A major part of the proofs relies on the analysis of an associated iterated function system that is genuinely non-contracting.

Détail

DOI : 10.4064/fm216-1-2

Keywords: study partially hyperbolic topologically transitive local diffeomorphism skew product horseshoe map system derived homoclinic class contains infinitely many hyperbolic periodic points different indices hence hyperbolic associated transitive invariant set lambda possesses rich fiber structure contains uncountably many trivial uncountably many non trivial fibers moreover spectrum central lyapunov exponents lambda contains gap hence gives rise first order phase transition major part proofs relies analysis associated iterated function system genuinely non contracting