Strong transitivity of composition operators

Abstract

A Furstenberg family $\mathcal{F}$ is a collection of infinite subsets ofthe set of positive integers such that if $A\subset B$ and $A\in \mathcal{F}$, then $B\in \mathcal{F}$. For aFurstenberg family $\mathcal{F}$, an operator $T$ on a topological vector space $X$ is said tobe $\mathcal{F}$-transitive provided that for each non-empty open subsets $U$, $V$ of $X$ the set$\{n \in \mathbb{N}: T^n (U) \cap V \neq\emptyset\}$ belongs to $\mathcal{F}$. In this paper, we characterize the $\mathcal{F}$-transitivityof composition operator $C_\phi$ on the space $H(\Omega)$ of holomorphic functionson a domain $\Omega\subset \mathbb{C}$ by providing a necessary and sufficient condition in terms ofthe symbol $\phi$.