Operators of stochastic adding machines and Julia sets

TL;DR

This paper extends the concept of stochastic adding machines to a Cantor numeration system, analyzing the spectrum of the associated transition operator in relation to a fibered Julia set, with implications for understanding complex stochastic processes.

Contribution

It introduces a new model of stochastic adding machines based on Cantor numeration and characterizes the spectrum of the transition operator using fibered Julia sets.

Findings

Spectrum of the transition operator equals the fibered Julia set when the process halts finitely.

Spectrum equals the boundary of the Julia set if the process may continue indefinitely.

Provides a mathematical link between stochastic processes and complex dynamical systems.

Abstract

A stochastic adding machine is a Markov chain on the set of non-negative integers $\mathbb{Z}_{+}$ that models the process of adding one by successively updating the digits of a number's expansion in a given numeration system. At each step, random failures may occur, interrupting the procedure and preventing it from continuing beyond a certain point. The first model of such a stochastic adding machine, constructed for the binary base, was introduced by Killeen and Taylor. Their work was motivated by applications to biological clocks, aiming to model phenomena related to time discrimination and/of psychological judgment. From a mathematical perspective, they characterized the spectrum of the associated transition operator in terms of a filled Julia set. In this paper, we consider a stochastic adding machine based on a bounded Cantor numeration system and extend its definition to a continuous state space--namely, the closure of $\mathbb{Z}_+$ with respect to the topology induced by the Cantor numeration system. This stochastic process naturally induces a transition operator $S$ acting on the Banach space of continuous complex-valued functions over the continuous state space, as well as a fibered filled Julia set $\mathcal{E}$. Our main result describes the spectrum of $S$ in terms of the fibered filled Julia set $\mathcal{E}$. Specifically, if the stochastic adding machines halts with probability one after a finite number of steps, then the spectrum of $S$ coincides with $\mathcal{E}$; otherwise, the spectrum coincides with the boundary $\partial \mathcal{E}$.