  We show that contrary to anticipation suggested by the dictionary between
rational maps and Kleinian groups and by the ``hairiness phenomenon'', there
exist many Feigenbaum Julia sets $J(f)$ whose Hausdorff dimension is strictly
smaller than two. We also prove that for any Feigenbaum Julia set, the
Poincar\'e critical exponent $\de_\crit$ is equal to the hyperbolic dimension
$\HD_\hyp(J(f))$. Moreover, if $\area J(f)=0$ then $\HD_\hyp (J(f))=\HD(J(f))$.
In the stationary case, the last statement can be reversed: if $\area J(f)> 0$
then $\HD_\hyp (J(f))< 2$. We also give a new construction of conformal
measures on $J(f)$ that implies that they exist for any $\de\in [\de_\crit,
\infty)$, and analyze their scaling and dissipativity/conservativity
properties.
