Real bounds and Lyapunov exponents

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.

27 pages, 3 figures. Comments are welcome

References in corpus (1)

• Quasisymmetric rigidity in one-dimensional dynamics

Cited by in corpus (7)

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• Laminar chaos in systems with quasiperiodic delay
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• Dynamics of asymptotically holomorphic polynomial-like maps
• Dynamics of multicritical circle maps
• Quasisymmetric orbit-flexibility of multicritical circle maps
• Suppression of Quasiperiodicity in Circle Maps with Quenched Disorder