Regularity of Dynamical Green Functions

TL;DR

This paper investigates the regularity properties of dynamical Green's functions in complex dynamics, simplifying existing results, extending them, and providing new insights and examples on their limitations.

Contribution

It offers a comprehensive analysis of Green's function regularity, extending previous results and introducing new findings in the context of complex dynamical systems.

Findings

Simplified and extended known regularity results

Proved new theorems on Green's function properties

Provided examples illustrating the limits of regularity-based approaches

Abstract

For meromorphic maps of complex manifolds, ergodic theory and pluripotential theory are closely related. In nice enough situations, dynamically defined Green's functions give rise to invariant currents which intersect to yield measures of maximal entropy. `Nice enough' is often a condition on the regularity of the Green's function. In this paper we look at a variety of regularity properties that have been considered for dynamical Green's functions. We simplify and extend some known results and prove several others which are entirely new. We also give some examples indicating the limits of what one can hope to achieve in complex dynamics by relying solely on the regularity of a dynamical Green's function.