Hausdorff dimension of exponential parameter rays and their endpoints

We investigate the set I of parameters κ for which the singular value of z ez + κ converges to ∞. The set I consists of uncountably many parameter rays, plus landing points of some of these rays (Forster et al 2008 Proc Am. Math. Soc. 136 at press (Preprint math.DS/0311427)). We show that the parameter rays have Hausdorff dimension 1, which implies (Qiu 1994 Acta Math. Sin. (N.S.) 10 362–8) that the ray endpoints in I alone have dimension 2. Analogous results were known for dynamical planes of exponential maps (Karpinska 1999 C. R. Acad. Sci. Paris Ser. I: Math. 328 1039–44; Schleicher and Zimmer 2003 J. Lond. Math. Soc. 67 380–400); our result shows that this also holds in parameter space.