Ergodic Properties of Composition Semigroups on the Disc Algebra

Abstract

Every semigroup $$\{\varphi _t\}_{t\ge 0}$$ of self-maps of the disc defines a semigroup $$\{C_{\varphi _t}\}_{t\ge 0}$$ of compositions operators on the space of holomorphic functions on the disc. We characterize the (uniform) mean ergodicity (in the sense of continuous means) and the asymptotic behaviour of these operators when they define a $$C_0$$ -semigroup on the disc algebra, in terms of the Denjoy–Wolff point and the associated planar domain in the sense of Berkson and Porta. Finally we deal with the case of Hardy and Bergman spaces.