M. Lyubich
On the Lebesgue Measure of the Julia Set of a Quadratic Polynomial
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 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.