  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.
