On the spectrum of stochastic perturbations of the shift and Julia sets

TL;DR

This paper investigates the spectral properties of stochastic perturbations of the shift operator related to stochastic adding machines in base 2 and Fibonacci base, linking spectra to Julia sets of quadratic maps and endomorphisms.

Contribution

It extends prior work by analyzing spectra in various Banach spaces and connecting them to complex dynamical systems involving Julia sets.

Findings

Spectra in base 2 relate to Julia sets of quadratic maps.

Spectra in Fibonacci base involve Julia sets of endomorphisms of ^2.

Different Banach spaces reveal diverse spectral behaviors.

Abstract

We extend the Killeen-Taylor study in \cite{KT} by investigating in different Banach spaces ($\ell^\alpha(\N), c_0(\N),c_c(\N)$) the point, continuous and residual spectra of stochastic perturbations of the shift operator associated to the stochastic adding machine in base 2 and in Fibonacci base. For the base 2, the spectra are connected to the Julia set of a quadratic map. In the Fibonacci case, the spectra involve the Julia set of an endomorphism of $\C^2$.