2 1 A ug 2 00 4 HAUSDORFF DIMENSION AND CONFORMAL MEASURES OF FEIGENBAUM JULIA SETS

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 Feigen-baum Julia set, the Poincaré critical exponent δcr 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 δ ∈ [δcr, ∞), and analyze their scaling and dissipativity/conservativity properties.