Ergodicity in the dynamics of holomorphic correspondences

TL;DR

This paper explores ergodic properties of measures in holomorphic correspondence dynamics, introducing a new ergodicity concept, proving a Birkhoff-type theorem, and identifying conditions for ergodic measures on compact complex manifolds.

Contribution

It introduces a novel notion of ergodicity for holomorphic correspondences, proves a Birkhoff-type ergodic theorem, and constructs explicit ergodic measures in this context.

Findings

Existence of ergodic measures on compact complex manifolds.

A version of Birkhoff's ergodic theorem for these measures.

Identification of a class of ergodic measures in holomorphic correspondence dynamics.

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.