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Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps

Published online by Cambridge University Press:  20 November 2018

Sze-Man Ngai
Sze-Man Ngai*
Affiliation:
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China and Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, U.S.A. email: smngai@georgiasouthern.edu
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Abstract

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We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Keywords

28A8035P2035J0543A0547A75spectral dimensionfractalLaplacianself-similar measureiterated function system with overlapssecond-order self-similar identities
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 3 , 01 June 2011 , pp. 648 - 688
Copyright
Copyright © Canadian Mathematical Society 2011

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