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

TL;DR

This paper investigates a complex dynamical system derived from a horseshoe map, revealing rich fiber structures, a spectrum gap, and phase transitions through analysis of non-contracting iterated function systems.

Contribution

It introduces a new class of partially hyperbolic skew-product systems with intricate fiber and Lyapunov spectrum structures, demonstrating phase transitions.

Findings

Presence of uncountably many trivial and non-trivial fibers.

Existence of a gap in the central Lyapunov spectrum.

Identification of a first order phase transition.

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.