论文标题
在自我相似集中随机步行和正常数字
Random walks on tori and normal numbers in self similar sets
论文作者
论文摘要
我们通过仿射线性零件通勤的仿射图来研究随机步行。假设其翻译部分存在非理性条件，我们证明HAAR度量是独特的固定度量。我们推断出，如果$ k \ subset \ mathbb {r}^d $是$ n \ geq 2 $ geq 2 $地图的有限迭代功能系统的吸引子这些地图和$ r_ {i} \在\ mathbb {n} $中，在翻译部分的非理性条件下，几乎每个点在$ k $中（W.R.T. w.r.t. w.r.t.任何bernoulli措施）在地图下都有一个等式分布式轨道，地图$ x \ mapsto dx $ dx $ dx $ $ x $ $ \ mathbb}在一维情况下，这个结论等于$ d $。因此，例如，中间三分之二的cantor集的非理性扩张中的几乎每个点都是正常的基础3。
We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K \subset \mathbb{R}^d$ is an attractor of a finite iterated function system of $n\geq 2$ maps of the form $x \mapsto D^{-r_i} x + t_i \ (i=1, \ldots, n)$, where $D$ is an expanding $d\times d$ integer matrix, and is the same for all the maps, and $r_{i} \in\mathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $x\mapsto Dx$ (multiplication mod $\mathbb{Z}^{d}$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base 3.
