On the Hyperbolic Hausdorff Dimension of the Boundary of a Basin of Attraction for a Holomorphic Map and of Quasirepellers

Feliks Przytycki¹

¹ Institute of Mathematics Polish Academy of Sciences 00-956 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) no. 1, pp. 41-52.

Résumé

e prove that the hyperbolic Hausdorff dimension of $\mathop{\rm Fr} {\mit\Omega}$, the boundary of the simply connected immediate basin of attraction ${\mit\Omega}$ to an attracting periodic point of a rational mapping of the Riemann sphere, which is not a finite Blaschke product in some holomorphic coordinates, or a $2:1$ factor of a Blaschke product, is larger than 1. We prove a “local version” of this theorem, for a boundary repelling to the side of the domain. The results extend an analogous fact for polynomials proved by A. Zdunik and relies on the theory elaborated by M. Urbański, A. Zdunik and the author in the late 80-ties. To prove that the dimension is larger than 1, we use expanding repellers in $\partial{\mit\Omega}$ constructed in \cite{[P2]}.To reach our results, we deal with a quasi-repeller, i.e. the limit set for a geometric coding tree, and prove that the hyperbolic Hausdorff dimension of the limit set is larger than the Hausdorff dimension of the projection via the tree of any Gibbs measure for a Hölder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Hölder potentials on Julia sets.

Détail

DOI : 10.4064/ba54-1-4

Keywords: prove hyperbolic hausdorff dimension mathop mit omega boundary simply connected immediate basin attraction mit omega attracting periodic point rational mapping riemann sphere which finite blaschke product holomorphic coordinates factor blaschke product larger prove local version theorem boundary repelling side domain results extend analogous polynomials proved zdunik relies theory elaborated urba ski zdunik author late ties prove dimension larger expanding repellers partial mit omega constructed cite reach results quasi repeller limit set geometric coding tree prove hyperbolic hausdorff dimension limit set larger hausdorff dimension projection via tree gibbs measure lder potential shift space under non cohomology assumption consider gibbs measures lder potentials julia sets