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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  that 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  that escape to infinity has Hausdorff dimension 1.

Dimension properties of the boundaries of exponential basins