Hausdorff Dimension of Exponential Parameter Rays and Their Endpoints

We investigate the set $I$ of parameters $\kappa$ for which the singular value of $z\mapsto e^z+\kappa$ converges to $\mathrm{\infty }$. The set $I$ consists of uncountably many parameter rays, plus landing points of some of these rays. 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; our result shows that this also holds in parameter space.