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

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.

45 pages, 4 figures

References in corpus (3)

• Non-hyperbolic ergodic measures with large support
• Rich phase transitions in step skew-products
• Symbolic extensions for partially hyperbolic diffeomorphisms