On hyperbolic metric and asymptotically finite invariant differentials in holomorphic dynamics

TL;DR

This paper investigates conditions under which invariant Beltrami differentials can exist on subsets of the postcritical set of a rational map, linking geometric properties to the dynamics and showing non-existence in many cases.

Contribution

It establishes new criteria based on hyperbolic geometry and asymptotic behavior that determine when invariant Beltrami differentials cannot be supported on certain invariant subsets.

Findings

Invariant Beltrami differentials do not exist on subsets with finite hyperbolic area unless the map is a Lattès map.

Geometric restrictions on the subset influence the existence of invariant differentials.

The paper connects hyperbolic geometry with the dynamics of rational maps in complex analysis.

Abstract

Given a rational map $R$, we consider the complement of the postcritical set $S_R$. In this paper we discuss the existence of invariant Beltrami differentials supported on a $R$ invariant subset $A$ of $S_R$. Under some geometrical restrictions, either on the hyperbolic geometry of $A$ or on the asymptotic behavior of infinitesimal geodesics of the Teichm\"uller space of $S_R$, we show the absence of invariant Beltrami differentials supported on $A$. In particular, we show that if $A$ has finite hyperbolic area, then $A$ can not support invariant Beltrami differentials except in the case where $R$ is a Latt\`es map.