Ergodicity in the dynamics of holomorphic correspondences

Abstract

This paper studies ergodic properties of certain measures arising in the dynamics of holomorphic correspondences. These measures, in general, are not invariant in the classical sense of ergodic theory. We define a notion of ergodicity, and prove a version of Birkhoff's ergodic theorem in this setting. We also show the existence of ergodic measures when a holomorphic correspondence is defined on a compact complex manifold. Lastly, we give an explicit class of dynamically interesting measures that are ergodic as in our definition.