Flexibility of Lyapunov exponents for expanding circle maps

Alena Erchenko

doi:10.3934/dcds.2019098

Abstract

Let \begin{document}$ g $\end{document} be a smooth expanding map of degree \begin{document}$ D $\end{document} which maps a circle to itself, where \begin{document}$ D $\end{document} is a natural number greater than \begin{document}$ 1 $\end{document} . It is known that the Lyapunov exponent of \begin{document}$ g $\end{document} with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to \begin{document}$ \log D $\end{document} which, in addition, is less than or equal to the Lyapunov exponent of \begin{document}$ g $\end{document} with respect to the measure of maximal entropy. Moreover, the equalities can only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree \begin{document}$ D $\end{document} on a circle.

Highlights

IntroductionLet g be a smooth expanding circle map of degree D, where D is a natural number greater than 1
We concentrate our attention on the Lyapunov exponent λabs(g) with respect to the unique measure μabs ∈ M(g) that is absolutely continuous with respect to the Lebesgue measure and the Lyapunov exponent λmax(g) with respect to the unique measure μmax ∈ M(g) of maximal entropy
We have seen that the equalities in λabs(g) ≤ log D ≤ λmax(g) can only hold simultaneously. It is a natural question if these inequalities are the only restrictions on the pair of values of the considered Lyapunov exponents. We answer this question affirmatively by constructing smooth expanding circle maps of degree D that take on all possible values of pairs of Lyapunov exponents corresponding to measures μabs and μmax, respectively

Summary

Introduction

Let g be a smooth expanding circle map of degree D, where D is a natural number greater than 1. Let M(g) denote the set of g-invariant Borel probability measures. For any μ ∈ M(g) that is ergodic, its Lyapunov exponent is defined by λμ(g) = S1 log |g |dμ, and we denote by h(μ, g) the metric entropy of g with respect to μ. For the ×D-map on S1 given by x → Dx (mod 1), we have that λabs(×D) = htop(×D) = λmax(×D) = log D and the Lebesgue measure is μabs and μmax. Flexibility, expanding maps, Lyapunov exponents, circle maps, invariant measures

ALENA ERCHENKO

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