## Institute for Mathematical Sciences

## Preprint ims04-05

** A. Avila and M. Lyubich**
* Hausdorff dimension and conformal measures of Feigenbaum Julia sets*

Abstract: 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.

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