On commuting holomorphic maps in the unit disc of

Abstract

A connection between the study of commuting holomorphic maps of the unit disc Δ of into itself and the study of holomorphic maps of Δ into itself that belong to the same pseudo-iteration semigroup (in the sense of Cowen) is investigated. The open (and difficult) case in which the maps have derivative 1 at the Wolff point τ ∂ e Δ is considered in this paper. Among other results, it is proved that the set of all holomorphic maps of the unit disc Δ into itself which commute with a given holomorphic map f coincides with the set of all holomorphic maps which belong to the pseudo-iteration semigroup of f. The results are obtained under the hypothesis—which appears naturally in the context—that the iterates of the involved maps converge to the (common) Wolff point “non-tangentially”.