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 \(\phi _1,...,\phi _N\).

Furstenberg 族\(\mathcal {F}\)是正整数集的无限子集的集合，如果\(A\subset B\)和\(A\in \mathcal {F}\)，则\(B\in \mathcal {F}\)。对于 Furstenberg 族\(\mathcal {F}\) ，作用在公共拓扑向量空间X上的有限多个算子\(T_1,...,T_N\)被认为是不相交的\(\mathcal {F}\) -如果对于X的每个非空开子集\(U_0,...,U_N\)集合\(\{n\in \mathbb {N}:\ U_0 \cap T_1^{-n}(U_1 )\cap ...\cap T_N^{-n}(U_N)\ne \emptyset \}\)属于\(\数学{F}\)。在本文中，根据\(\Omega\)的拓扑性质，我们刻画了\(N\ge 2\)组合算子\(C_{\phi _1 )的不相交\(\mathcal {F}\) -传递性},\ldots ,C_{\phi _N} \ )通过建立一个必要的和其符号\(\phi _1,...,\phi _N\)的充分条件。