Disjoint strong transitivity of composition operators

A Furstenberg family F is a collection of infinite subsets of the set of positive integers such that if A ⊂ B and A ∈ F , then B ∈ F . For a Furstenberg family F , finitely many operators T 1 , . . . , T N acting on a common topological vector space X are said to be disjoint F -transitive if for every non-empty open subsets U 0 , . . . , U N of X the set { n ∈ N : U 0 ∩ T 1 - n ( U 1 ) ∩ . . . ∩ T N - n ( U N ) ≠ ∅ } belongs to F . In this paper, depending on the topological properties of Ω , we characterize the disjoint F -transitivity of N ≥ 2 composition operators C ϕ 1 , … , C ϕ N acting on the space H ( Ω ) of holomorphic maps on a domain Ω ⊂ C by establishing a necessary and sufficient condition in terms of their symbols ϕ 1 , . . . , ϕ N .