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Multifractal formalisms for the local spectral and walk dimensions

Published online by Cambridge University Press:  17 June 2002

B. M. HAMBLY ,
JUN KIGAMI  and
TAKASHI KUMAGAI
B. M. HAMBLY
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford, OX1 3LB. e-mail: hambly@maths.ox.ac.uk
JUN KIGAMI
Affiliation:
Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan. e-mail: kigami@i.kyoto-u.ac.jp
TAKASHI KUMAGAI
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. e-mail: kumagai@kurims.kyoto-u.ac.jp
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Abstract

We introduce the concepts of local spectral and walk dimension for fractals. For a class of finitely ramified fractals we show that, if the Laplace operator on the fractal is defined with respect to a multifractal measure, then both the local spectral and walk dimensions will have associated non-trivial multifractal spectra. The multifractal spectra for both dimensions can be calculated and are shown to be transformations of the original underlying multifractal spectrum for the measure, but with respect to the effective resistance metric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 132 , Issue 3 , May 2002 , pp. 555 - 571
Copyright
2002 Cambridge Philosophical Society

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