Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroups

TL;DR

This paper investigates recurrence in the dynamics of multi-valued holomorphic maps, establishing an analogue of Poincaré recurrence for meromorphic correspondences and addressing measure-theoretic challenges using descriptive set theory.

Contribution

It introduces a Poincaré recurrence theorem analogue for meromorphic correspondences and handles measure-theoretic obstacles with the Measurable Projection Theorem.

Findings

Proved recurrence properties for meromorphic correspondences.

Addressed measure-theoretic issues using descriptive set theory.

Analyzed invariance of measure supports.

Abstract

This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincar\'e recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance properties of the supports of the measures mentioned.