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 Feigenbaum Julia set, the Poincaré critical exponent $\de_{\mathrm{cr}}$ is equal to the hyperbolic dimension $\HD_{\mathrm{hyp}}(J(f))$. Moreover, if $\operatorname{area} J(f)=0$, then $\operatorname{HD}_{\mathrm{hyp}} (J(f))=\operatorname{HD}(J(f))$. In the stationary case, the last statement can be reversed: if $\operatorname{area} J(f)> 0$, then $\operatorname{HD}_{\mathrm{hyp}} (J(f))< 2$. We also give a new construction of conformal measures on $J(f)$ that implies that they exist for any $\delta\in [\delta_{\mathrm{cr}}, \infty)$, and analyze their scaling and dissipativity/conservativity properties.