Ergodic Theory and Dynamical Systems

External rays and the real slice of the Mandelbrot set

a1 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA (e-mail:


This paper investigates the set of angles of the parameter rays which land on the real slice [-2,1/4] of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as bi-accessible points of quadratic Julia sets are given.

(Received November 22 2001)
(Revised September 17 2002)

p1 Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794-3651, USA