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

Mikhail Lyubich
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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.

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Mikhail Lyubich. 1991-05-28. On the Lebesgue measure of the Julia set of a quadratic polynomial. https://arxiv.org/abs/math/9201285

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