Equidistribution of graphs of holomorphic correspondences

TL;DR

This paper proves that the graphs of iterates of certain holomorphic correspondences on compact Riemann surfaces become uniformly distributed rapidly, under specific degree or modularity conditions.

Contribution

It establishes exponential equidistribution of iterated graphs of holomorphic correspondences with new conditions on degrees and modularity.

Findings

Graphs of iterates become equidistributed exponentially fast.

Results depend on the relation between dynamical degrees and modularity.

Provides a new understanding of the distribution of iterates in complex dynamics.

Abstract

Let $X$ be a compact Riemann surface. Let $f$ be a holomorphic self-correspondence of $X$ with dynamical degrees $d_1$ and $d_2$. Assume that $d_1\neq d_2$ or $f$ is non-weakly modular. We show that the graphs of the iterates $f^n$ of $f$ are equidistributed exponentially fast with respect to a positive closed current in $X\times X$.