论文标题
在Schreier和Cayley图的光谱和光谱测量上
On spectra and spectral measures of Schreier and Cayley graphs
论文作者
论文摘要
我们对有限生成的组的Cayley和Schreier图的光谱刚度的各个方面感兴趣。对于每对整数$ d \ geq 2 $和$ m \ ge 1 $，我们考虑了一个无数的固定$ d $ regratular树一组无数的自动形态，其中提供了以下有趣现象的示例。对于$ d = 2 $和任何$ m \ geq 2 $，我们获得了一个不可数的非准等级cayley图，并具有相同的laplacian频谱，在两个间隔的结合中绝对连续计算，我们明确地计算了。其中一些组提供了示例，其中将Cayley图的频谱连接为一个生成集，并为另一个生成设置而言。 对于每个$ d \ geq 3，m \ geq 1 $，我们都会显示这些组的无限schreier图，并用频谱一组lebesgue cantor Measure Measure Measure Union一组可计数的孤立点积聚在其上。这些Schreier图上Laplacian的Kesten光谱度量是离散的，并集中在孤立点上。此外，我们构建了一个完整的本征函数系统，这些系统是强烈本地化的。
We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the rooted $d$-regular tree which provide examples of the following interesting phenomena. For $d=2$ and any $m\geq 2$, we get an uncountable family of non quasi-isometric Cayley graphs with the same Laplacian spectrum, absolutely continuous on the union of two intervals, that we compute explicitly. Some of the groups provide examples where the spectrum of the Cayley graph is connected for one generating set and has a gap for another. For each $d\geq 3, m\geq 1$, we exhibit infinite Schreier graphs of these groups with the spectrum a Cantor set of Lebesgue measure zero union a countable set of isolated points accumulating on it. The Kesten spectral measures of the Laplacian on these Schreier graphs are discrete and concentrated on the isolated points. We construct moreover a complete system of eigenfunctions which are strongly localized.
