Stochastic adding machines based on Bratteli diagrams

TL;DR

This paper introduces Markov chains derived from Vershik maps on Bratteli diagrams, exploring their spectral properties and linking spectra to higher-dimensional Julia sets, revealing complex topological structures.

Contribution

It establishes a novel connection between Markov chains on Bratteli diagrams and complex dynamics via Julia sets, expanding understanding of spectral properties in this context.

Findings

Spectra of transition operators are connected to higher-dimensional Julia sets

Topological properties of spectra are characterized

Spectral analysis reveals complex dynamical structures

Abstract

In this paper, we define some Markov Chains associated to Vershik maps on Bratteli diagrams. We study probabilistic and spectral properties of their transition operators and we prove that the spectra of these operators are connected to Julia sets in higher dimensions. We also study topological properties of these spectra.