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

Volume 216 / 2012

Lorenzo J. Díaz, Katrin Gelfert Fundamenta Mathematicae 216 (2012), 55-100 MSC: 37D35, 37D25, 37E05, 37D30, 37C29. DOI: 10.4064/fm216-1-2


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.


  • Lorenzo J. DíazDepartamento de Matemática PUC-Rio
    Marquês de São Vicente 225
    Gávea, Rio de Janeiro 225453-900, Brazil
  • Katrin GelfertInstituto de Matemática UFRJ
    Av. Athos da Silveira Ramos 149
    Cidade Universitária
    Ilha do Fundão
    Rio de Janeiro 21945-909, Brazil

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