Given a spectral triple the functionals on A of the form a-->tau(omega)(a|D|(-alpha)) are studied, where tau(omega) is a-singular trace, and omega is a generalised limit. When tau(omega) is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|(-1), and that the set of alpha's for which there exists a singular trace tau(omega) giving rise to a non trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The functionals corresponding to points in the traceability interval are called Hausdorff-Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit omega. (C) 2003 Elsevier Inc. All rights reserved.

Guido, D., Isola, T. (2003). Dimensions and singular traces for spectral triples, with applications to fractals. JOURNAL OF FUNCTIONAL ANALYSIS, 203(2), 362-400 [10.1016/S0022-1236(03)00230-1].

Dimensions and singular traces for spectral triples, with applications to fractals

GUIDO, DANIELE;ISOLA, TOMMASO
2003-10-01

Abstract

Given a spectral triple the functionals on A of the form a-->tau(omega)(a|D|(-alpha)) are studied, where tau(omega) is a-singular trace, and omega is a generalised limit. When tau(omega) is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|(-1), and that the set of alpha's for which there exists a singular trace tau(omega) giving rise to a non trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The functionals corresponding to points in the traceability interval are called Hausdorff-Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit omega. (C) 2003 Elsevier Inc. All rights reserved.
ott-2003
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Fractals in the real line; Noncommutative Hausdorff dimensions; Noncommutative Hausdorff-Besicovitch; Singular traces; Spectral triples
Guido, D., Isola, T. (2003). Dimensions and singular traces for spectral triples, with applications to fractals. JOURNAL OF FUNCTIONAL ANALYSIS, 203(2), 362-400 [10.1016/S0022-1236(03)00230-1].
Guido, D; Isola, T
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/43494
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