A class of adding machine and Julia sets

TL;DR

This paper introduces a stochastic Fibonacci-based adding machine modeled as a Markov chain, exploring its probabilistic properties and linking its spectrum to fibered Julia sets in complex dynamics.

Contribution

It defines a new class of stochastic adding machines tied to Fibonacci bases and connects their spectral properties to fibered Julia sets, advancing understanding in complex dynamics and probabilistic processes.

Findings

Markov chain transience and recurrence analyzed

Spectrum connected to fibered Julia sets

New stochastic Fibonacci adding machine model

Abstract

In this work we define a stochastic adding machine associated to the Fibonacci base and to a probabilities sequence $\overline{p}=(p_i)_{i\geq 1}$. We obtain a Markov chain whose states are the set of nonnegative integers. We study probabilistic properties of this chain, such as transience and recurrence. We also prove that the spectrum associated to this Markov chain is connected to the fibered Julia sets for a class of endomorphisms in $\mathbb{C}^2$.