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Abstract
We investigate different measures defined geometrically or dynamically on polynomial Julia sets and their scaling properties. Our main concern is the relationship between harmonic and Hausdorff measures.
We prove that the fine structure of harmonic measure at the more exposed points of an arbitrary polynomial Julia set is regular, and dimension spectra or pressure for the corresponding (negative) values of parameter are real-analytic. However, there is a precisely described class of polynomials where a set of preperiodic critical points can generate a unique very exposed tip, which manifests in the phase transition for some kinds of spectra.
For parabolic and subhyperbolic polynomials, and also semihyperbolic quadratics we analyze the spectra for the positive values of parameter, establishing the extent of their regularity.
Results are proved through spectral analysis of the transfer (Perron-Frobenius-Ruelle) operator.