Hausdorff dimension and conformal measures of Feigenbaum Julia sets

Avila, Artur¹ ; Lyubich, Mikhail^2, 3

¹ CNRS UMR 7599, Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie–Boîte courrier 188, 75252–Paris Cedex 05, France
² Department of Mathematics, University of Toronto, Ontario, Canada M5S 3G3
³ Mathematics Department and IMS, SUNY Stony Brook, Stony Brook, New York 11794

Journal of the American Mathematical Society, Tome 21 (2008) no. 2, pp. 305-363

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-07-00583-8

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