Harmonic measure and expansion on the boundary of the connectedness locus

The paper develops a technique for proving properties that are typical in t he boundary of the connectedness locus with respect to the harmonic measure . A typical expansion condition along the critical orbit is proved. This co ndition implies a number of properties, including the Collet-Eckmann condit ion, Hausdorff dimension less than 2 for the Julia set, and the radial cont inuity in the parameter space of the Hausdorff dimensions of totally discon nected Julia sets.