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Dixmier traces and coarse multifractal analysis

Published online by Cambridge University Press:  02 February 2010

KENNETH FALCONER  and
TONY SAMUEL
KENNETH FALCONER
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews,Fife KY16 9SS, Scotland (email: kjf@st-and.ac.uk, as57@st-and.ac.uk)
TONY SAMUEL
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews,Fife KY16 9SS, Scotland (email: kjf@st-and.ac.uk, as57@st-and.ac.uk)
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Abstract

We show how multifractal properties of a measure supported by a fractal F⊆[0,1] may be expressed in terms of complementary intervals of F and thus in terms of spectral triples and the Dixmier trace of certain operators. For self-similar measures this leads to a non-commutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 369 - 381
Copyright
Copyright © Cambridge University Press 2010

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