Real bounds and Lyapunov exponents

Edson de  Faria; Pablo  Guarino

doi:10.3934/dcds.2016.36.1957

Article Contents

2016, Volume 36, Issue 4: 1957-1982. Doi: 10.3934/dcds.2016.36.1957

This issue Previous Article Global solutions to a one-dimensional non-conservative two-phase model Next Article A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes

Real bounds and Lyapunov exponents

Edson de Faria^1, and
Pablo Guarino^2,

1.
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo SP
2.
Instituto de Matemática e Estatstica, Universidade Federal Fluminense, Rua Mário Santos Braga S/N, 24020-140, Niterói, Rio de Janeiro

Received: June 30, 2014

Revised: June 30, 2015

Published: May 2017

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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 satisfies 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.

Keywords:

Lyapunov exponents,
real bounds,
critical circle maps,
infinitely renormalizable unimodal maps,
neutral measures on Julia sets.

Mathematics Subject Classification: Primary: 37E10; Secondary: 37A05, 37E20, 37F10.

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References

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