  Let $f:\bar\bold C\to\bar\bold C$ be a rational map on the Riemann sphere ,
such that for every $f$-critical point $c\in J$ which forward trajectory does
not contain any other critical point, $|(f^n)'(f(c))|$ grows exponentially fast
(Collet--Eckmann condition), there are no parabolic periodic points, and else
such that Julia set is not the whole sphere. Then smooth (Riemann) measure of
the Julia set is 0.
  For $f$ satisfying additionally Masato Tsujii's condition that the average
distance of $f^n(c)$ from the set of critical points is not too small, we prove
that Hausdorff dimension of Julia set is less than 2. This is the case for
$f(z)=z^2+c$ with $c$ real, $0\in J$, for a positive line measure set of
parameters $c$.
