## Institute for Mathematical Sciences

## Preprint ims91-10

** 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 {\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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