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Supercyclic Weighted Composition Operators on the Space of Smooth Functions | Synapse

September 12, 2025Open Access

Supercyclic Weighted Composition Operators on the Space of Smooth Functions

Key Points

Supercyclic weighted composition operators are found to be weakly mixing, establishing a link to chaos.
Mixing operators also demonstrate chaotic behavior, reinforcing their connection to supercyclic properties.
In one dimension, supercyclicity requires both mixing and chaotic characteristics to be present.
These findings suggest that the nature of mixing operators can reveal deeper properties within function spaces.

Abstract

A weighted composition operator on the space of scalar-valued smooth functions on an open subset of a 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 such mixing 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.

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Cite This Study

Bès et al. (Thu,) studied this question.

synapsesocial.com/papers/68d44f6931b076d99fa565d9 https://doi.org/https://doi.org/10.3390/math13182944