论文标题
在圆环上仿射随机行走
Affine random walks on the torus
论文作者
论文摘要
我们考虑由仿射转化组的作用引起的圆环随机行走。我们在这次随机行走的情况下给出了定量的等分分配结果，假设线性部分生成的Zariski闭合在$ \ Mathbb {r}^d $上强烈不可总结，并且是连接的Zariski或包含一个近端元素。具体而言，除非初始点和仿射变换的翻译部分可能会扰动，否则我们对随机行走等分的速度进行定量估计（仅取决于随机步行的线性部分），以使随机行走被困在小基数有限的轨道中。特别是，我们证明，当且仅当随机行走未被困在有限轨道中时，随机行走等法在HAAR措施中。
We consider random walks on the torus arising from the action of the group of affine transformations. We give a quantitative equidistribution result for this random walk under the assumption that the Zariski closure of the group generated by the linear part acts strongly irreducibly on $\mathbb{R}^d$ and is either Zariski connected or contains a proximal element. Specifically, we give quantitative estimates (depending only on the linear part of the random walk) for how fast the random walk equidistributes unless the initial point and the translation part of the affine transformations can be perturbed so that the random walk is trapped in a finite orbit of small cardinality. In particular, we prove that the random walk equidistributes in law to the Haar measure if and only if the random walk is not trapped in a finite orbit.
