Ultra hypercyclicity and its connection to mixing properties

2023/12/05 by Martin Liu, Liu, Martin, D. Walmsley +2
Mathematics · #Advanced Banach Space Theory #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2312.02899

openalex publication_date 2023/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Recently, two new topological properties for operators acting on a topological vector space were introduced: strong hypercyclicity and hypermixing. We introduce a new property called ultra hypercyclicity and compare it to strong hypercyclicity and hypermixing, as well as the classical notions of mixing, weak mixing, and hypercyclicity. We show that every ultra hypercyclic operator on Fréchet space must be weakly mixing, and that there exists a strongly hypercyclic operator which is not ultra hypercyclic. We also characterize, in terms of the weight sequence, the ultra hypercyclic weighted backward shifts on c₀ and ℓ^p, 1≤ p<∞. Finally, we improve upon a necessary condition for strongly hypercyclic weighted backward shifts.

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• Strong topological transitivity, hypermixing, and their relationships with other dynamical properties
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• Disjoint Hypercyclicity along filters