论文标题
圆上随机动力学系统的Lyapunov指数
Lyapunov exponent of random dynamical systems on the circle
论文作者
论文摘要
我们考虑I.I.D.的产品$ \ {f_1，\ ldots，f_m \} $中的序列保留了圆圈的定向差异性。我们自然可以将Lyapunov指数$λ$关联。在少数假设下，众所周知，$λ\ leq 0 $，并且当且仅当$ f_1，\ ldots，f_m $同时将相等性保持在旋转时。在本文中，我们陈述了这一事实的定量版本，在这种情况下，$ f_1，\ ldots，f_m $是$ c^k $ tingting ting ting ting ting ting th Rotation Number $ρ（f_1），\ ldots，ρ（f_m）$的旋转数量$ c^k $存在差异$ g $和旋转$ r_i $，因此$ \ mbox {dist}（gf_ig^{ - 1}，r_i），r_i）\ ll |λ|^{\ frac {1} {1} {1} {2}}}} $ for $ i = 1，\ ldots m $。我们还陈述了矩阵随机产物的模拟结果$ 2 \ times 2 $，没有二磷剂条件。
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.
