论文标题
在当地紧凑型组的密集亚组上随机步行
Random walks on dense subgroups of locally compact groups
论文作者
论文摘要
令$γ$为一个可计数的离散组，$ h $ a lcsc完全断开的组，$ρ：γ\ rightarrow h $ a是具有密集图像的同构。我们开发了一种一般且明确的技术，为每一个紧凑的开放子组$ l <h $和bi- $ l $ invariant的概率度量$θ$上的$ h $，fufstenberg ovelstenberg oveltization $θ$ $θ$，以便$（h，θ）$的poisson边界是$τ$ $τ$ -Bungary。除其他外，这项技术使我们能够在某些灯塔组和可解决的baumslag-solitar群组上构建有限支撑的随机步行的示例，这些小组的泊松边界是典型的，但对于任何$ p \ geq 1 $ by bader-muchnik的猜测。此外，我们举了一个可计数的离散组$γ$的示例，以及$γ$上的两个扩展概率度量$τ_1$和$τ_2$，使得$（γ，τ_1）$的边界熵光谱是一个间隔，而边界熵谱是$（γ，τ_2）$的边界熵谱。
Let $Γ$ be a countable discrete group, $H$ a lcsc totally disconnected group and $ρ: Γ\rightarrow H$ a homomorphism with dense image. We develop a general and explicit technique which provides, for every compact open subgroup $L < H$ and bi-$L$-invariant probability measure $θ$ on $H$, a Furstenberg discretization $τ$ of $θ$ such that the Poisson boundary of $(H,θ)$ is a $τ$-boundary. Among other things, this technique allows us to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not $L^p$-irreducible for any $p \geq 1$, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we give an example of a countable discrete group $Γ$ and two spread-out probability measures $τ_1$ and $τ_2$ on $Γ$ such that the boundary entropy spectrum of $(Γ,τ_1)$ is an interval, while the boundary entropy spectrum of $(Γ,τ_2)$ is a Cantor set.
