Dixmier traces and coarse multifractal analysis

Kenneth Falconer; Tony Samuel

doi:10.1017/S0143385709001102

Dixmier traces and coarse multifractal analysis

Kenneth Falconer, Tony Samuel

Mathematics

Research output: Contribution to journal › Article › peer-review

12 Citations (Scopus)

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 noncommutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.

Original language	English
Pages (from-to)	369-381
Journal	Ergodic Theory and Dynamical Systems
Volume	31
Issue number	2
DOIs	https://doi.org/10.1017/S0143385709001102
Publication status	Accepted/In press - 2 Apr 2011

Keywords

Spectral metrics
Subshifts of finite type
Gibbs measures
Equlibrium states

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10.1017/S0143385709001102

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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 noncommutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.",

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AU - Falconer, Kenneth

AU - Samuel, Tony

PY - 2011/4/2

Y1 - 2011/4/2

N2 - 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 noncommutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.

AB - 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 noncommutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.

KW - Spectral metrics

KW - Subshifts of finite type

KW - Gibbs measures

KW - Equlibrium states

UR - https://arxiv.org/abs/0905.3052

U2 - 10.1017/S0143385709001102

DO - 10.1017/S0143385709001102

M3 - Article

SN - 0143-3857

VL - 31

SP - 369

EP - 381

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 2

ER -

Dixmier traces and coarse multifractal analysis

Abstract

Keywords

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