Strong transitivity properties for operators

Abstract

Given a Furstenberg family $\mathcal{F}$ of subsets of $\mathbb{N}$, an operator $T$ on a topological vector space $X$ is called $\mathcal{F}$-transitive provided for each non-empty open subsets $U$, $V$ of $X$ the set $\{n\in {\mathbb{Z}}_{+}:T^n(U)\cap V\ne \varnothing \}$ belongs to $\mathcal{F}$. We classify the topologically transitive operators with a hierarchy of $\mathcal{F}$-transitive subclasses by considering families $\mathcal{F}$ that are determined by various notions of largeness and density in ${\mathbb{Z}}_{+}$.

References in corpus (1)

• Recurrence properties of hypercyclic operators

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