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arXiv 2006.15397math.DS
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圆周上随机动力系统的李雅普诺夫指数

Lyapunov exponent of random dynamical systems on the circle

Dominique Malicet

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摘要：

我们考虑i.i.d.的产品。序列在一个集合$\{f_1,\ldots,f_m\}$的保持方向同胚的圆。我们可以自然地关联李雅普诺夫指数$λ$。在很少的假设下，已知$λ\leq 0$和等式成立当且仅当$f_1,\ldots,f_m$同时与旋转共轭。在本文中，我们陈述了这一事实的定量版本的情况下，$f_1,\ldots,f_m$是$C^k$扰动的旋转与旋转数$ρ(f_1),\ldots,ρ(f_m)$满足同时丢番图条件的Moser：我们给出了一个精确的估计$λ$（泰勒展开），我们证明了存在一个同构$g$和旋转$r_i$，使$\mbox{dist}(gf_ig^{-1},r_i)\ll |λ|^{\frac{1}{2}}$为$i=1,\ldots m$。我们还陈述了矩阵$2\times 2$的随机乘积的模拟结果，没有丢番图条件。

英文摘要：

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 $λ$. Under few assumptions, it is known that $λ\leq 0$ 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 $ρ(f_1),\ldots,ρ(f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser: we give a precise estimate on $λ$ (Taylor expansion) and we prove that there exists a diffeomorphism $g$ and rotations $r_i$ such that $\mbox{dist}(gf_ig^{-1},r_i)\ll |λ|^{\frac{1}{2}}$ for $i=1,\ldots m$. We also state analog results for random products of matrices $2\times 2$, without diophantine condition.

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