Supercyclic weighted composition operators on the space of smooth functions

TL;DR

This paper characterizes supercyclic and strongly supercyclic weighted composition operators on smooth functions, establishing their equivalence to weakly mixing and mixing properties, and linking mixing to chaos, especially in one dimension.

Contribution

It provides a complete characterization of supercyclicity and chaos for weighted composition operators on smooth functions, connecting these properties to mixing behaviors.

Findings

Supercyclicity is equivalent to weakly mixing for these operators.

Strong supercyclicity coincides with mixing.

All mixing operators in this class are chaotic.

Abstract

A weighted composition operator on the space of scalar-valued smooth functions on an open set of d-dimensional Euclidean space is supercyclic if and only if it is weakly mixing, and it is strongly supercyclic if and only if it is mixing. Every mixing such operator is chaotic. In the one-dimensional case, it is supercyclic if and only if it is mixing and if and only if it is chaotic.