A Contracting Fractal Group, Schreier Graphs and The Spectra

2023/10/22 by Bozorgmehr Vaziri, Vaziri, Bozorgmehr, Frahad Rahmati +1
Biochemistry, Genetics and Molecular Biology · Computer Science · #Cellular Automata and Applications #DNA and Biological Computing #FOS: Mathematics #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2310.14145

openalex publication_date 2023/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper explores a previously uncharted automaton group generated by a 5-state automaton (Π, A) acts by self-similarity on the regular rooted tree A^* over a 2-letter alphabet set A. Group G has been subjected to several observations that reveal the self-similar G-action possesses several notable characteristics: it is a weak branch, contracting, fractal group, and satisfies the open set condition with an exponential growth rate in activity. The corresponding Schreier graphs Γ of the group action on the binary rooted tree, for n≤ 12, are presented. Furthermore, we delve into a numerical approximation of the spectrum of the discrete Laplacian operator, which is defined on the limit Schreier graph Γ=lim_n→∞ Γ_n.

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