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 Poincaré critical exponent $\delta _{\mathrm {cr}}$ is equal to the hyperbolic dimension $\mathrm {HD}_{\mathrm {hyp}}(J(f))$. 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.