Disjoint strong transitivity of composition operators

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 $\{n\in \mathbb{N}:\ U_0 \cap T_1^{-n}(U_1)\cap...\cap T_N^{-n}(U_N)\neq\emptyset\}$ belongs to $\mathcal{F}$. In this paper, depending on the topological properties of $\Omega$, we characterize the disjoint $\mathcal{F}$-transitivity of $N\geq2$ 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 $\phi_1,...,\phi_N$.