Entropy spectrum of Lyapunov exponents for nonhyperbolic step skew-products and elliptic cocycles

TL;DR

This paper analyzes the spectrum of Lyapunov exponents in nonhyperbolic skew-product systems, revealing a multifractal structure and entropy characteristics of level sets, especially for elliptic cocycles.

Contribution

It provides a multifractal analysis of Lyapunov exponents in nonhyperbolic systems, including a detailed entropy characterization and the existence of measures of maximal entropy.

Findings

Entropy of level sets varies continuously with Lyapunov exponent.

Zero exponent level set has positive but not maximal entropy.

Existence of two unique ergodic measures of maximal entropy for proximal systems.

Abstract

We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of $N$, $N\ge2$, symbols and with $C^1$ diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. Examples of such systems arise from the projective action of $2\times 2$ matrix cocycles and our results apply to an open and dense subset of elliptic $\mathrm{SL}(2,\bR)$ cocycles. We derive a multifractal analysis for the topological entropy of the level sets of Lyapunov exponent. The results are formulated in terms of Legendre-Fenchel transforms of restricted variational pressures, considering hyperbolic ergodic measures only, as well as in terms of restricted variational principles of entropies of ergodic measures with a given exponent. We show that the entropy of the level sets is a continuous function of the Lyapunov exponent. The level set of the zero exponent has positive, but not maximal, topological entropy. Under the additional assumption of proximality, as for example for skew-products arising from certain matrix cocycles, there exist two unique ergodic measures of maximal entropy, one with negative and one with positive fiber Lyapunov exponent.