Real bounds and Lyapunov exponents

TL;DR

This paper proves that certain critical circle maps have zero Lyapunov exponent and do not satisfy Collet-Eckmann conditions, using real bounds, with implications for rational maps.

Contribution

It establishes zero Lyapunov exponents for critical circle maps without periodic points directly from real bounds, avoiding Pesin's theory, and extends results to unimodal maps.

Findings

Critical circle maps without periodic points have zero Lyapunov exponent.

No critical point of such maps satisfies Collet-Eckmann condition.

Method applies to infinitely renormalizable unimodal maps and informs rational map studies.

Abstract

We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponent with respect to its unique invariant Borel probability measure. Moreover, no critical point of such a map satisfy the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds, without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizable unimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutral measures of certain rational maps of the Riemann sphere.