On the Hyperbolic Hausdorff Dimension of the Boundary of a Basin of Attraction for a Holomorphic Map and of Quasirepellers
Volume 54 / 2006
Abstract
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.
