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Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

✍ Scribed by Mariusz Urbański; Anna Zdunik

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
433 KB
Volume
171
Category
Article
ISSN
0001-8708

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✦ Synopsis

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of

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