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

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 $\Gamma^p_n$, 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 $\{\Gamma_n^p\}_{n\geq 1}$ in the Gromov-Hausdorff topology. We obtain an explicit description of the spectrum of the graphs $\{\Gamma_n^p\}_{n\geq 1}$. 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 $\partial T_{p+1}$.