A horseshoe with a discontinuous entropy spectrum

TL;DR

This paper constructs a family of horseshoes demonstrating a discontinuity in the entropy spectrum at the boundary of Lyapunov exponents, challenging the typical regularity properties known for hyperbolic maps.

Contribution

It introduces a novel example of horseshoes with a discontinuous entropy spectrum, revealing new irregularities in the spectral properties of hyperbolic dynamical systems.

Findings

Entropy spectrum can be discontinuous at the boundary of Lyapunov exponents

The constructed horseshoes exhibit non-analytic entropy spectra at boundary points

Challenges existing assumptions about the regularity of entropy spectra in hyperbolic maps

Abstract

We study the regularity of the entropy spectrum of the Lyapunov exponents for hyperbolic maps on surfaces. It is well-known that the entropy spectrum is a concave upper semi-continuous function which is analytic on the interior of the set Lyapunov exponents. In this paper we construct a family of horseshoes with a discontinuous entropy spectrum at the boundary of the set of Lyapunov exponents.