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

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) which we call shift-like. The properties of interest include chaotic properties such as Li-Yorke chaos, hypercyclicity, frequent hypercyclicity as well as tame properties 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 prove some results concerning when a property is shared by a linear dynamical system and its factors.

Expansivity and strong structural stability for composition operators on $$L^p$$ spaces

“…Although some dynamical properties have been fully understood for composition operators [5,9], unfortunately, many other notions require the extra hypotheses of dissipativity and bounded distortion to be characterized for such operators: this is the case, among others, of generalized hyperbolicity and shadowing property [12,13]. More is true: recently, in [11], focusing on chaotic properties as well as hyperbolic properties (such as shadowing, expansivity and generalized hyperbolicity), the authors established that dissipative composition operators with bounded distortion are shifts-like operators, in the sense that they behave similarly to weighted shifts. On the other hand, up to now, no characterization of structural stability and strong structural stability is known for this large class of operators.…”

Expansivity and strong structural stability for composition operators on $L^p$ spaces

“…Although some dynamical properties have been fully understood for composition operators [5,7], unfortunately, many other notions require the extra hypotheses of dissipativity and bounded distortion in order to be characterized for such operators: this is the case, among others, of generalized hyperbolicity and shadowing property [11,13]. More is true: recently, in [12], focusing on chaotic properties as well as hyperbolic properties (such as shadowing, expansivity and generalized hyperbolicity), the authors established that dissipative composition operators with bounded distortion are shifts-like operators, in the sense that they behave similarly to weighted shifts. On the other hand, up to now, no characterization of structural stability and strong structural stability is known for this large class of operators.…”