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September 23, 20250 citationsOpen Access

Supercyclic weighted composition operators on the space of smooth functions

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JBJuan BèsCFC. Foster

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.

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/68d4757f31b076d99fa6cfc7https://doi.org/10.48550/arxiv.2507.16425
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