English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We prove that the hyperbolic Hausdorff dimension of FrΩ, 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. Urbanski, 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 Holder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Holder potentials on Julia sets.

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