关于谱是康托尔集的不可数图族
On an uncountable family of graphs whose spectrum is a Cantor set
摘要：
对于每个$p\geq 1$，星形自动机群$\mathcal{G}_{S_p}$是可以从$p+1$顶点上的星形图开始定义的自动机群。研究了群$\mathcal{G}_{S_p}$在$p+1$度的正则根树$T_{p+1}$上及其边界$\partial T_{p+1}$上的作用下的Schreier图. $T_{p+1}$的$n$-级上的传递作用与有限Schreier图$Γ^p_n$相关联，而边界上存在不可数的许多作用轨道，由Gromov-Hausdorff拓扑中序列$\{Γ_n^p\}_{n\geq 1}$的极限获得的无限Schreier图表示。我们得到了图$\{Γ_n^p\}_{n\geq 1}$的谱的一个明确的描述。然后，利用$\mathcal{G}_{S_p}$的顺从性，证明了无限Schreier图的谱是二次映射$f_p(z) = z^2-2(p-1)z -2p$的Julia集，即零Lebesgue测度的康托尔集与支持KNS谱测度的可数孤立点集的并.我们还给出了无限Schreier图的一个完全分类，直到无根图的同构，表明它们可以有$1$，$2$或$2p$结束，和$1$结束的情况是一般的关于$\partial T_{p+1}$上的一致测度。
英文摘要：
For each $p\geq 1$, the star automaton group $\mathcal{G}_{S_p}$ is an automaton group which can be defined starting from a star graph on $p+1$ vertices. We study Schreier graphs associated with the action of the group $\mathcal{G}_{S_p}$ on the regular rooted tree $T_{p+1}$ of degree $p+1$ and on its boundary $\partial T_{p+1}$. With the transitive action on the $n$-th level of $T_{p+1}$ is associated a finite Schreier graph $Γ^p_n$, whereas there exist uncountably many orbits of the action on the boundary, represented by infinite Schreier graphs which are obtained as limits of the sequence $\{Γ_n^p\}_{n\geq 1}$ in the Gromov-Hausdorff topology. We obtain an explicit description of the spectrum of the graphs $\{Γ_n^p\}_{n\geq 1}$. Then, by using amenability of $\mathcal{G}_{S_p}$, we prove that the spectrum of each infinite Schreier graph is the union of a Cantor set of zero Lebesgue measure, which is the Julia set of the quadratic map $f_p(z) = z^2-2(p-1)z -2p$, and a countable collection of isolated points supporting the KNS spectral measure. We also give a complete classification of the infinite Schreier graphs up to isomorphism of unrooted graphs, showing that they may have $1$, $2$ or $2p$ ends, and that the case of $1$ end is generic with respect to the uniform measure on $\partial T_{p+1}$.
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