论文标题
在表面上均匀扩展随机动力学系统的固定措施和轨道封闭
Stationary measures and orbit closures of uniformly expanding random dynamical systems on surfaces
论文作者
论文摘要
我们研究了具有给定平滑不变措施的表面上的非亚洲作用的固定度量和轨道封闭的问题。利用棕色和罗德里格斯·赫兹（Rodriguez Hertz）的结果，我们表明，在一定有限的平均生长条件下，唯一的非原子固定度量是给定的平滑不变度量，并且每个轨道闭合都是有限的或密集的。此外，相对于平稳的不变测度，表面上有无限轨道等分的每个点。这类似于均匀环境中Benoist Quint和Eskin-Lindenstrauss的结果，以及Eskin-Mirzakhani在翻译表面模量空间中的结果。然后，我们将此结果应用于两个具体设置，即标准映射的离散扰动和OUT（$ f_2 $） - 在某个字符上的操作。我们在以前的环境中分析验证生长条件，并在后一种环境中进行数值验证。
We study the problem of classifying stationary measures and orbit closures for non-abelian action on a surface with a given smooth invariant measure. Using a result of Brown and Rodriguez Hertz, we show that under a certain finite verifiable average growth condition, the only nonatomic stationary measure is the given smooth invariant measure, and every orbit closure is either finite or dense. Moreover, every point with infinite orbit equidistributes on the surface with respect to the smooth invariant measure. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homogeneous setting, and the result of Eskin-Mirzakhani in the setting of moduli spaces of translation surfaces. We then apply this result to two concrete settings, namely discrete perturbation of the standard map and Out($F_2$)-action on a certain character variety. We verify the growth condition analytically in the former setting, and verify numerically in the latter setting.
