Strong transitivity properties for operators

Abstract

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

Figures (2)

• Figure 1: Densities and transitivity properties
• Figure 2: Known relations

Theorems & Definitions (49)

• Definition 1.1
• Definition 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof