Remarks on Ruelle operator and Invariant Line Field Problem

TL;DR

This paper investigates conditions under which invariant conformal structures exist on Julia sets, specifically relating to the measure of the postcritical set and measure convergence criteria.

Contribution

It establishes necessary and sufficient conditions for invariant conformal structures on Julia sets when the postcritical set has Lebesgue measure zero, using measure convergence.

Findings

Invariant conformal structures exist under specific measure convergence conditions.

Necessary and sufficient conditions are characterized for Lebesgue measure zero postcritical sets.

The results connect measure-theoretic properties with geometric structures on Julia sets.

Abstract

In case of Lebesgue measure zero of postcritical set the necessary and sufficient conditions (in terms of convergence of sequences of measures) of existence of invariant conformal structures on J(R) are obtained.