Supercyclic weighted composition operators on the space of smooth functions

Juan Bès; C. Foster

doi:10.48550/arxiv.2507.16425

July 22, 2025Open Access

Supercyclic weighted composition operators on the space of smooth functions

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Key Points

A weighted composition operator is supercyclic if and only if it is weakly mixing, revealing intrinsic properties related to function behavior.
The study establishes that a strongly supercyclic operator must be mixing, highlighting the interplay between these concepts in function spaces.
Every mixing operator is chaotic, indicating that chaotic behavior is an inherent quality of this type of operator for smooth functions.
In one-dimensional cases, supercyclic behavior directly correlates with mixing and chaotic properties, simplifying analysis in this context.

Implication

This study demonstrates conditions for supercyclic behavior in weighted composition operators, indicating connections to mixing and chaos.

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.

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Cite This Study

Bès et al. (2025) studied this question.

synapsesocial.com/papers/68d4757f31b076d99fa6cfc7 https://doi.org/10.48550/arxiv.2507.16425

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Supercyclic weighted composition operators on the space of smooth functions

Key Points

Implication

Abstract

Discussion

Cite This Study

Supercyclic weighted composition operators on the space of smooth functions

Key Points

Implication

Abstract

Discussion

Cite This Study

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