Hausdorff dimension of biaccessible angles for quadratic polynomials, preprint

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 biac-cessible external angles of quadratic polynomials, both in the dynamical and param-eter space. We 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 z2, as well as limit points thereof. We also show that the Hausdorff dimension of biaccessible dynamical angles de-pends in a Hölder sense on the parameter angle, and that this dimension, up to a factor log 2, equals the core entropy, i.e., the topological entropy of the dynamics of the Hubbard tree. 1