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

Feliks Przytycki. "On the Hyperbolic Hausdorff Dimension of the Boundary of a Basin of Attraction for a Holomorphic Map and of Quasirepellers." Bulletin of the Polish Academy of Sciences. Mathematics 54.1 (2006): 41-52. <http://eudml.org/doc/280343>.

@article{FeliksPrzytycki2006,
abstract = { e 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. 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 ∂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 Hölder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Hölder potentials on Julia sets. },
author = {Feliks Przytycki},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {boundary of basin of attraction; Gibbs mesure; Hausdorff dimension; hyperbolic dimension; coding tree; iteration of holomorphic function; central limit theorem},
language = {eng},
number = {1},
pages = {41-52},
title = {On the Hyperbolic Hausdorff Dimension of the Boundary of a Basin of Attraction for a Holomorphic Map and of Quasirepellers},
url = {http://eudml.org/doc/280343},
volume = {54},
year = {2006},
}

TY - JOUR
AU - Feliks Przytycki
TI - On the Hyperbolic Hausdorff Dimension of the Boundary of a Basin of Attraction for a Holomorphic Map and of Quasirepellers
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2006
VL - 54
IS - 1
SP - 41
EP - 52
AB - e 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. 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 ∂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 Hölder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Hölder potentials on Julia sets.
LA - eng
KW - boundary of basin of attraction; Gibbs mesure; Hausdorff dimension; hyperbolic dimension; coding tree; iteration of holomorphic function; central limit theorem
UR - http://eudml.org/doc/280343
ER -