Smirnov, Stanislav K. (1996) Spectral analysis of Julia sets. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-09152006-144938
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.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Subject Keywords:||Julia set; multifractal; thermodynamic formalism|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||10 May 1996|
|Non-Caltech Author Email:||Stanislav.Smirnov (AT) math.unige.ch|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||15 Sep 2006|
|Last Modified:||26 Dec 2012 03:00|
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