Disjoint strong transitivity of composition operators

Abstract

A Furstenberg family $\mathcal {F}$ is a collection of infinite subsets of the set of positive integers such that if $A\subset B$ and $A\in \mathcal {F}$, then $B\in \mathcal {F}$. For a Furstenberg family $\mathcal {F}$, finitely many operators $T_1,...,T_N$ acting on a common topological vector space X are said to be disjoint $\mathcal {F}$-transitive if for every non-empty open subsets $U_0,...,U_N$ of X the set $\{n\in \mathbb {N}:\ U_0 \cap T_1^{-n}(U_1)\cap ...\cap T_N^{-n}(U_N)\ne \emptyset \}$ belongs to $\mathcal {F}$. In this paper, depending on the topological properties of $\Omega$, we characterize the disjoint $\mathcal {F}$-transitivity of $N\ge 2$ composition operators $C_{\phi _1},\ldots ,C_{\phi _N}$ acting on the space $H(\Omega )$ of holomorphic maps on a domain $\Omega \subset \mathbb {C}$ by establishing a necessary and sufficient condition in terms of their symbols $\phi _1,...,\phi _N$.