Spectrum of stochastic adding machines and fibered Julia sets

TL;DR

This paper studies a family of Markov chains derived from a stochastic variation of the digit-by-digit addition algorithm, revealing their spectra as fibered Julia sets and analyzing their properties.

Contribution

It introduces a novel stochastic algorithm for digit addition and characterizes the spectra of the resulting Markov chains as fibered Julia sets, linking probability and complex dynamics.

Findings

Spectra are fibered Julia sets of fibered polynomials.

Topological properties depend on Markov chain parameters.

Analytical properties are characterized through complex dynamics.

Abstract

Consider the basic algorithm to perform the transformation n--> n+1 changing digits of the d-adic expansion of n one by one. We obtain a family of Markov chains on the non-negative integers through sucessive and independent applications of the algorithm modified by a parametrized stochastic rule that randomly prevents one of the steps in the algorithm to finish. The objects of study in this paper are the spectra of the transition operators of these Markov chains. The spectra of these Markov chains turn out to be fibered Julia sets of fibered polynomials. This enable us to analyze their topological and analytical properties with respect to the underlying parameters of the Markov chains.