2025/06/27 by Szymon Glab, Glab, Szymon, Paolo Leonetti +1
Mathematics · #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2506.22341
openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a continuous linear operator T:X→ X, where X is a topological vector space, let UFHC(T) be the set of upper frequently hypercyclic vectors, that is, the set of vectors x ∈ X such that \n ∈ ω: Tnx ∈ U\ has positive upper asymptotic density for all nonempty open sets U⊆ X. It is known that UFHC(T) is a Gδσδ-set which is either empty or contains a dense Gδ-set. Using a purely topological proof, we improve it by showing that UFHC(T) is always a Gδσ-set. Bonilla and Grosse-Erdmann asked in [Rev. Mat. Complut. 31 (2018), 673--711] whether UFHC(T) is always a Gδ-set. We answer such question in the negative, by showing that there exists a continuous linear operator T for which UFHC(T) is not a Fσδ-set (hence not Gδ). In addition, we study the [non-]equivalence between (the ideal versions of) upper frequently hypercyclicity in the product topology and upper frequently hypercyclicity in the norm topology.