×

Hausdorff dimension of biaccessible angles for quadratic polynomials. (English) Zbl 1375.37129

The biaccessibility dimension is defined as the Hausdorff dimension of those external angles for which the corresponding rays land together with some other ray. The authors discuss a purely combinatorial characterization of biaccessible (both dynamic and parameter) angles, and detailed estimates of the Hausdorff dimension of the set of biaccessible angles are also given. This paper is self-contained and provides all necessary aspects which are here used. The authors begin this paper by giving a short review of rays and their landing properties. The authors consider only the case of quadratic polynomials \(p_{c}=z^{2}+c\).
The following two sets are investigated
(i)
\(\mathrm{Biac}_{\vartheta}\), the set of combinatorially biaccessible \(\varphi \in \mathbb{S}^{1}\) with respect to \(\vartheta\)
(ii)
\(\mathrm{Biac}\), the set of combinatorially biaccessible parameter angles \(\vartheta \in \mathbb{S}^{1}\), where \(\vartheta\) is the parameter angle.

The Hausdorff dimension of biaccessible angles \(\dim_{H}(\mathrm{Biac}_{\vartheta})\) and the Hausdorff dimension of biaccessible parameter angles \(\dim_{H}(\mathrm{Biac})\) are discussed. The authors describe also the dimension of renormalizable angles and kneading sequences. Biaccessible angles in the Julia set and the Mandelbrot are also topologically and combinatorially investigated. The authors give two pairs of bounds for the Hausdorff dimension of biaccessible itinearies and kneading sequences. They use an embedding of the Hubbard tree in their computations.

MSC:

37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37B10 Symbolic dynamics
37E25 Dynamical systems involving maps of trees and graphs
37E45 Rotation numbers and vectors
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
30C10 Polynomials and rational functions of one complex variable
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI arXiv