Dynamics of composition operators induced by odometers

TL;DR

This paper investigates the linear dynamics of composition operators induced by odometers, revealing unique properties such as the equivalence of various chaos notions and a notable divergence from classical weighted shift behaviors.

Contribution

It demonstrates that, for odometer-induced operators, many dynamical properties coincide and differ from classical cases, highlighting a distinct rigidity and separation in their behavior.

Findings

Supercyclicity implies hypercyclicity on finite measure spaces.

For odometer-induced operators, supercyclicity, chaos, and hypercyclicity are equivalent.

A mixing, chaotic operator is constructed that does not satisfy the Frequent Hypercyclicity Criterion.

Abstract

We study the linear dynamics of composition operators induced by measurable transformations on finite measure spaces, with particular emphasis on operators induced by odometers. Our first main result shows that, on a finite measure space, supercyclicity of a composition operator implies hypercyclicity. This phenomenon has no analogue in several classical settings and highlights a rigidity specific to the finite-measure context. We then focus on composition operators induced by odometers and show that many dynamical properties that are distinct for weighted backward shifts collapse in this setting. In particular, for such operators, supercyclicity, Li-Yorke chaos, hypercyclicity, weak mixing, and Devaney chaos are all equivalent. In contrast to this collapse, we show that the classical equivalence between Devaney chaos and the Frequent Hypercyclicity Criterion for weighted backward shifts fails for odometers. Specifically, we construct a mixing, chaotic, and distributionally chaotic composition operator that does not satisfy the Frequent Hypercyclicity Criterion. This combination of rigidity and separation demonstrates that the dynamical behavior of composition operators induced by odometers differs sharply from that of weighted backward shifts.