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Lyapunov exponent of random dynamical systems on the circle
- Source :
- Ergodic Theory and Dynamical Systems. 42:2080-2107
- Publication Year :
- 2021
- Publisher :
- Cambridge University Press (CUP), 2021.
-
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.
- Subjects :
- Sequence
Mathematics::Commutative Algebra
Applied Mathematics
General Mathematics
Diophantine equation
010102 general mathematics
Dynamical Systems (math.DS)
State (functional analysis)
Lyapunov exponent
Computer Science::Computational Geometry
Lambda
01 natural sciences
Combinatorics
Orientation (vector space)
symbols.namesake
0103 physical sciences
FOS: Mathematics
symbols
Taylor series
Computer Science::Symbolic Computation
010307 mathematical physics
Diffeomorphism
Mathematics - Dynamical Systems
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14694417 and 01433857
- Volume :
- 42
- Database :
- OpenAIRE
- Journal :
- Ergodic Theory and Dynamical Systems
- Accession number :
- edsair.doi.dedup.....5a25ec554e646b743e5fb90ece7c867a