Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions
Volume 216 / 2012
Fundamenta Mathematicae 216 (2012), 55-100
MSC: 37D35, 37D25, 37E05, 37D30, 37C29.
DOI: 10.4064/fm216-1-2
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.
