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

  • Matteo Cavaleri

    Università degli Studi Niccolò Cusano, Roma, Italy
  • Daniele D’Angeli

    Università degli Studi Niccolò Cusano, Roma, Italy
  • Alfredo Donno

    Università degli Studi Niccolò Cusano, Roma, Italy
  • Emanuele Rodaro

    Politecnico di Milano, Italy
On an uncountable family of graphs whose spectrum is a Cantor set cover

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Abstract

For each , the star automaton group is an automaton group which can be defined starting from a star graph on vertices. We study Schreier graphs associated with the action of the group on the regular rooted tree of degree and on its boundary . With the transitive action on the -th level of is associated a finite Schreier graph , 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 in the Gromov–Hausdorff topology. We obtain an explicit description of the spectrum of the graphs . Then, by using amenability of , 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 , 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 , or ends, and that the case of end is generic with respect to the uniform measure on .

Cite this article

Matteo Cavaleri, Daniele D’Angeli, Alfredo Donno, Emanuele Rodaro, On an uncountable family of graphs whose spectrum is a Cantor set. Groups Geom. Dyn. (2024), published online first

DOI 10.4171/GGD/770