论文标题
随机步行，单词指标和飞机上的轨道分布
Random walks, word metric and orbits distribution on the plane
论文作者
论文摘要
鉴于在某些空间$ x $上作用的无限集团$ g $，一个有限的子集$ g_n $和$ x \ in x $中的$ x $的家族，一个自然的问题是，一个自然的问题是套装$ g_nx $表单是什么渐近分布。更正式的是，我们为$ x $上的功能定义$ f $ yms $ s_n（f，x）= \ sum_ {g_n} f（gx）$ in g_n} f（gx）$，并询问是否存在一个函数$ψ（n）：\ mathbb {n} \ to \ to \ mathbb {r} $ sequence $ $ sequence $ n（n）（n）（n）（n）（n）。这是一个在各种环境下研究的微妙问题。当选择$ sl（2，\ mathbb {z}）$中的特定晶格中选择元素的精心选择的单词指标选择元素时，我们首先显示一个完整的解决方案。此外，证明所得的度量相对于一定的随机行走是固定的，并且与二聚体近似领域的良好研究功能有紧密的连接。然后，当选择$ sl（2，\ mathbb {r}）$上选择元素时，我们继续研究渐近分布问题。我们提供了我们最初问题的一种变体，这会产生一些令人惊讶且有趣的结果。
Given a countably infinite group $G$ acting on some space $X$, an increasing family of finite subsets $G_n$ and $x\in X$, a natural question to ask is what asymptotical distribution the sets $G_nx$ form. More formally, we define for a function $f$ over $X$ the sums $S_n(f,x)=\sum_{g\in G_n}f(gx)$ and ask whether exists a function $Ψ(n):\mathbb{N}\to\mathbb{R}$ such that the sequence $Ψ(n)S_n(f,x)$ converges. This is a delicate problem that was studied under various settings. We first show a full solution when elements are chosen using a carefully chosen word metric from a specific lattice in $SL(2,\mathbb{Z})$ acting on the circle. In addition, it is proven that the resulting measure is stationary with respect to a certain random walk and has a tight connection to a well studied function from the field of Diophantine approximations. We then proceed to study the asymptotic distribution problem when elements are chosen using a random walk over $SL(2,\mathbb{R})$ acting on $\mathbb{R}^2$. We offer a variant of our initial problem which yields some surprising and interesting results.
