Hausdorff dimension of biaccessible angles for quadratic polynomials

TL;DR

This paper investigates the Hausdorff dimension of biaccessible external angles in quadratic polynomials, providing bounds and characterizations related to Julia sets and bifurcations.

Contribution

It offers explicit bounds and characterizations for the Hausdorff dimension of biaccessible angles in quadratic polynomials, linking them to Julia set structure and bifurcations.

Findings

Dimension equals 1 iff Julia set is an interval

Dimension equals 0 at finite bifurcations from z^2

Provides bounds for Hausdorff dimension in dynamical and parameter space

Abstract

A point $z$ in the Julia set of a polynomial $p$ is called biaccessible if two dynamic rays land at $z$; a point $z$ in the Mandelbrot set is called biaccessible if two parameter rays land at $z$. In both cases, we say that the external angles of these two rays are biaccessible as well. In this paper we give upper and lower bounds for the Hausdorff dimension of biaccessible external angles of quadratic polynomials, both in the dynamical and parameter space. In particular, explicitly describe those quadratic polynomials where this dimension equals 1 (if and only if the Julia set is an interval), and when it equals 0, namely, at finite direct bifurcations from the polynomial $z^2$, as well as limit points thereof.