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Chaos and frequent hypercyclicity for composition operators

Part of: General theory of linear operators Special classes of linear operators Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  04 June 2021

Udayan B. Darji  and
Benito Pires
Udayan B. Darji
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY40292, USA ([email protected])
Benito Pires
Affiliation:
Departamento de Computaçao e Matemática, Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, Ribeirão Preto, SP14040-901, Brazil ([email protected])
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Abstract

The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Ruzsa in 2015 that for backward weighted shifts on $\ell _p(\mathbb {Z})$, the notions of chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^{p}$-spaces, the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class, an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible operator frequently hypercyclic on $\ell _1$ whose inverse is not frequently hypercyclic is constructed.

Keywords

Frequently hypercyclicchaotic operatorcomposition operatorLp-space

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 47B33: Composition operators
Secondary: 37D45: Strange attractors, chaotic dynamics
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 513 - 531
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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