On the Lebesgue measure of the Julia set of a quadratic polynomial

TL;DR

This paper proves that for certain quadratic polynomials without irrational indifferent points or infinite renormalization, the Julia set has zero Lebesgue measure, linking critical point recurrence to dynamical properties.

Contribution

It establishes that under specified conditions, the Julia set's Lebesgue measure is zero and relates critical point recurrence to minimality and entropy in the dynamics.

Findings

Julia set has zero Lebesgue measure for specified quadratic polynomials.

Persistent recurrence of the critical point implies topological minimality and zero entropy.

Results connect complex dynamics with real one-dimensional map properties.

Abstract

The goal of this note is to prove the following theorem: Let $p_a(z) = z^2+a$ be a quadratic polynomial which has no irrational indifferent periodic points, and is not infinitely renormalizable. Then the Lebesgue measure of the Julia set $J(p_a)$ is equal to zero. As part of the proof we discuss a property of the critical point to be {\it persistently recurrent}, and relate our results to corresponding ones for real one dimensional maps. In particular, we show that in the persistently recurrent case the restriction $p_a|\omega(0)$ is topologically minimal and has zero topological entropy. The Douady-Hubbard-Yoccoz rigidity theorem follows this result.