On an uncountable family of graphs whose spectrum is a Cantor set

Abstract

For each $p\ge 1$, 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 ${\Gamma }_n^p$, whereas there exist uncountably many orbits of the action on the boundary, represented by infinite Schreier graphs which are obtained as the limits of the sequence $\{{\Gamma }_n^p{\}}_{n\ge 1}$ in the Gromov–Hausdorff topology. We obtain an explicit description of the spectrum of the graphs $\{{\Gamma }_n^p{\}}_{n\ge 1}$. 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 Kesten–Neumann–Serre 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 $\partial T_{p+1}$.