Hausdorff dimension of exponential parameter rays and their endpoints, Nonlinearity 21

Abstract

Abstract. We investigate the set I of parameters κ for which the singular value of z ↦ → e z + κ converges to ∞. The set I consists of uncountably many parameter rays, plus landing points of some of these rays [FRS]. We show that the parameter rays have Hausdorff dimension 1, while the ray endpoints in I alone have dimension 2. Analogous results were known for dynamical planes of exponential maps [K, SZ]; our result shows that this also holds in parameter space. 1