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arXiv · 1011.6294

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

L. J. Díaz·K. Gelfert
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Abstract

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.

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L. J. Díaz, K. Gelfert. 2010-11-29. Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions. https://arxiv.org/abs/1011.6294

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