Lyapunov exponent of random dynamical systems on the circle

TL;DR

This paper provides a quantitative analysis of the Lyapunov exponent for random products of circle diffeomorphisms close to rotations, establishing estimates and conjugacy results under Diophantine conditions.

Contribution

It introduces a Taylor expansion estimate for the Lyapunov exponent and demonstrates conjugacy closeness to rotations for perturbations satisfying Diophantine conditions.

Findings

Explicit Taylor expansion of Lyapunov exponent near rotations

Existence of conjugacy to rotations with controlled distortion

Results extend to random matrix products without Diophantine assumptions

Abstract

We consider products of a i.i.d. sequence in a set $\{f_1,\ldots,f_m\}$ of preserving orientation 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: we give a precise estimate on $\lambda$ (Taylor expansion) and we prove that there exists a diffeomorphism $g$ and rotations $r_i$ such that $\mbox{dist}(gf_ig^{-1},r_i)\ll |\lambda|^{\frac{1}{2}}$ for $i=1,\ldots m$. We also state analog results for random products of matrices $2\times 2$, without diophantine condition.