Nice sets and invariant densities in complex dynamics

TL;DR

This paper constructs nice sets in complex dynamics to facilitate the analysis of invariant measures, proving a key equivalence between measure finiteness and an integrability condition for certain meromorphic maps.

Contribution

It introduces a method to construct nice sets around points in Julia sets, simplifying the study of invariant measures and providing a converse to a recent theorem.

Findings

Construction of nice sets around Julia set points

Equivalence between measure finiteness and integrability condition

Simplification of invariant measure analysis in complex dynamics

Abstract

In complex dynamics, we construct a so-called nice set (one for which the first return map is Markov) around any point which is in the Julia set but not in the post-singular set, adapting a construction of Juan Rivera-Letelier. This simplifies the study of absolutely continuous invariant measures. We prove a converse to a recent theorem of Kotus and Swiatek, so for a certain class of meromorphic maps the absolutely continuous invariant measure is finite if and only if an integrability condition is satisfied.