Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I

TL;DR

This paper explores how graph representations via operators on Hilbert space, especially automata-based graphs, exhibit fractal properties, with applications in signal processing and symbolic dynamics, using spectral and representation theory.

Contribution

It introduces a framework connecting automata, graph groupoids, and Hecke operators to identify fractal characteristics and construct operator representations on Hilbert spaces.

Findings

Automata-based graphs can have fractal-like structures.

Spectral properties of Hecke operators are explicitly computed.

Automata generate fractaloids with specific spectral features.

Abstract

We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with the use of Representation Theory and of Spectral Theory of a certain family of Hecke operators. Let G be a directed graph. We begin by building the graph groupoid G induced by G, and representations of G. Our main application is to the groupoids defined from automata. By assigning weights to the edges of a fixed graph G, we give conditions for G to acquire fractal-like properties, and hence we can have fractaloids or G-fractals. Our standing assumption on G is that it is locally finite and connected, and our labeling of G is determined by the "out-degrees of vertices". From our labeling, we arrive at a family of Hecke-type operators whose spectrum is computed. As applications, we are able to build representations by operators on Hilbert spaces (including the Hecke operators); and we further show that automata built on a finite alphabet generate fractaloids. Our Hecke-type operators, or labeling operators, come from an amalgamated free probability construction, and we compute the corresponding amalgamated free moments. We show that the free moments are completely determined by certain scalar-valued functions.