  We prove that the boundary of a component $U$ of the basin of an attracting
periodic cycle (of period greater than 1) for an exponential map on the complex
plane has Hausdorff dimension greater than 1 and less than 2. Moreover, the set
of points in the boundary of $U$ which do not escape to infinity has Hausdorff
dimension (in fact: hyperbolic dimension) greater than 1, while the set of
points in the boundary of $U$ which escape to infinity has Hausdorff dimension
1.
