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Lyapunov exponent of random dynamical systems on the circle

Part of: Smooth dynamical systems: general theory Low-dimensional dynamical systems Random dynamical systems

Published online by Cambridge University Press:  31 May 2021

DOMINIQUE MALICET Open the ORCID record for DOMINIQUE MALICET [Opens in a new window]
DOMINIQUE MALICET*
Affiliation:
Laboratoire d’Analyse et de Mathématiques Appliquées, Université Gustave Eiffel, 5 Boulevard Descartes, 77420Champs-sur-Marne, France
*
e-mail: mdominique@crans.org
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Abstract

We consider products of an independent and identically distributed sequence in a set $\{f_1,\ldots ,f_m\}$ of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent $\lambda $ . Under few assumptions, it is known that $\lambda \leq 0$ and that the equality holds if and only if $f_1,\ldots ,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots ,f_m$ are $C^k$ perturbations of rotations with rotation numbers $\rho (f_1),\ldots ,\rho (f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of $\lambda $ (Taylor expansion) and we prove that there exist a diffeomorphism g and rotations $r_i$ such that $\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$ for $i=1,\ldots , m$ . We also state analogous results for random products of $2\times 2$ matrices, without any diophantine condition.

Keywords

random dynamicsone dimensional dynamicsKAM theoryLyapunov exponents

MSC classification

Primary: 37C05: Smooth mappings and diffeomorphisms 37C75: Stability theory 37E10: Maps of the circle 37H15: Multiplicative ergodic theory, Lyapunov exponents
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 2080 - 2107
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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