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Dimension properties of the boundaries of exponential basins

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.