The Lyapunov spectrum of some parabolic systems

TL;DR

This paper investigates the Hausdorff dimension spectrum of Lyapunov exponents in certain interval maps, including non-hyperbolic cases, revealing that zero exponent level sets have full Hausdorff dimension but no topological entropy.

Contribution

It extends the analysis of Lyapunov spectra to non-hyperbolic interval maps and characterizes the dimension and entropy of specific level sets.

Findings

Zero Lyapunov exponent level set has full Hausdorff dimension

Level sets with zero exponent carry no topological entropy

The study includes non-hyperbolic dynamical systems

Abstract

We study the Hausdorff dimension spectrum for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.