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

• Autorzy: Feliks Przytycki
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37F35: Conformal densities and Hausdorff dimension
• 37F15: Expanding maps; hyperbolicity; structural stability
• 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2006
• Tom: 54
• Numer: 1
• Strony: 41-52

Daty

wydano

2006

Twórcy

autor

Feliks Przytycki

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

Identyfikatory

DOI

10.4064/ba54-1-4