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Spectral Analysis of Julia Sets


Smirnov, Stanislav K. (1996) Spectral Analysis of Julia Sets. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/X37M-D376.


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
Major Option:Mathematics
Awards:Caltech Distinguished Alumni Award, 2015
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Makarov, Nikolai G.
Group:Caltech Distinguished Alumni Award
Thesis Committee:
  • Makarov, Nikolai G. (chair)
  • Kahn, Jeremy
  • Kechris, Alexander S.
  • Luxemburg, W. A. J.
Defense Date:10 May 1996
Non-Caltech Author Email:Stanislav.Smirnov (AT)
Record Number:CaltechETD:etd-09152006-144938
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3554
Deposited By: Imported from ETD-db
Deposited On:15 Sep 2006
Last Modified:22 Jul 2021 18:23

Thesis Files

PDF (Smirnov_sk_1996.pdf) - Final Version
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