Shift-like Operators on $L^p(X)$

TL;DR

This paper introduces a technique to extend known properties of weighted backward shifts to a broad class of operators called shift-like operators on $L^p(X)$, covering chaos, hypercyclicity, and hyperbolic dynamics.

Contribution

It develops a general method to characterize complex dynamical properties for shift-like operators on $L^p$ spaces, expanding understanding of their chaotic and hyperbolic behaviors.

Findings

Characterization of properties for shift-like operators on $L^p(X)$

Conditions under which properties are shared by systems and their factors

Application to composition operators on dissipative measure systems

Abstract

In this article we develop a general technique which takes a known characterization of a property for weighted backward shifts and lifts it up to a characterization of that property for a large class of operators on $L^p(X)$. We call these operators ``shift-like''. The properties of interest include chaotic properties such as Li-Yorke chaos, hypercyclicity, frequent hypercyclicity as well as properties related to hyperbolic dynamics such as shadowing, expansivity and generalized hyperbolicity. Shift-like operators appear naturally as composition operators on $L^p(X)$ when the underlying space is a dissipative measure system. In the process of proving the main theorem, we provide some results concerning when a property is shared by a linear dynamical system and its factors.