自相似群极限空间上的图定向系统与自相似测度
Graph-directed systems and self-similar measures on limit spaces of self-similar groups
摘要：
设$G$是一个群，$ϕ:H\to G$是有限指数子群$H<G$的收缩同态. V.Nekrashevych [25]将极限动力系统$(\lims,\si)$和极限$G$ -空间$\limGs$以及由瓦片$\tile$覆盖的$\cup_{g\in G}\tile\cdot g$与$(G,ϕ)$对相关联。我们在这些极限空间上发展了自相似测度$μ$理论。证明了$(\lims,\si,μ)$与单侧Bernoulli位移共轭。利用sofic子移位证明了$\tile$图具有整数测度，并给出了求其整数测度的算法，同时给出了求$g\in G$图$\tile\cap (\tile\cdot g)$的交测度的算法.我们目前的应用程序的黎曼球面上的有理函数的不变测度和积分自仿射瓷砖的勒贝格测度的评价。
英文摘要：
Let $G$ be a group and $ϕ:H\to G$ be a contracting homomorphism from a subgroup $H<G$ of finite index. V.Nekrashevych [25] associated with the pair $(G,ϕ)$ the limit dynamical system $(\lims,\si)$ and the limit $G$-space $\limGs$ together with the covering $\cup_{g\in G}\tile\cdot g$ by the tile $\tile$. We develop the theory of self-similar measures $μ$ on these limit spaces. It is shown that $(\lims,\si,μ)$ is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile $\tile$ has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles $\tile\cap (\tile\cdot g)$ for $g\in G$. We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles.
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