BULLETIN OF THE POLISH ACADEMY OF SCIENCES MATHEMATICS Vol. 54, No. 1, 2006 DYNAMICAL SYSTEMS AND ERGODIC THEORY On the Hyperbolic Hausdorff Dimension of the Boundary of a Basin of Attraction for a Holomorphic Map and of Quasirepellers by Feliks PRZYTYCKI Presented by Andrzej LASOTA Summary. We prove that the hyperbolic Hausdorff dimension of Fr Ω, the boundary of the simply connected immediate basin of attraction Ω 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. Urba´ nski, A. Zdunik and the author in the late 80-ties. To prove that the dimension is larger than 1, we use expanding repellers in ∂Ω constructed in [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 H¨older potential on the shift space, under a non-cohomology assumption. We alsoconsider Gibbs measures for H¨older potentials on Julia sets. 0. Introduction. This paper completes [Z1], [Z2] and [P2]. In particular we prove the following Theorem A. Let f : C → C be a rational map of degree d ≥ 2 of the Riemann sphere and Ω be the simply connected immediate basin of attrac- tion to a periodic attracting orbit. Then , provided f is not a finite Blaschke product in some holomorphic coordinates , or a quotient of a Blaschke prod- 2000 Mathematics Subject Classification : Primary 37F35; Secondary 37F15, 37D25. Key words and phrases : boundary of basin of attraction, Gibbs mesure, Hausdorff dimension, hyperbolic dimension, coding tree, iteration of holomorphic function, central limit theorem. Supported by Polish KBN grant 2P03A 03425. [41]