Dimensions and singular traces for spectral triples, with applications to fractals
By D. Guido, T. Isola. 2002
87 citations recorded by the source corpus.
Given a spectral triple (A,H,D), the functionals on A of the form a↦τω(a|D|−α) are studied, where τω is a singular trace, and ω is a generalised limit. When τω 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 α's for which there exists a singular trace τω 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 ω.
Subfields: Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/3369348ef9d60a82aec98e241474bf7ced2577e5
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Citations in the corpus
- Dixmier traces and some applications in non-commutative geometry (2006, background): “However, a priori it is not clear (see [37]) whether there exists (for example on the ideal L (1,∞)(H )) a rearrangement-invariant singular functional which is not necessarily symmetric, or whether there exists a trace on L (1,∞)(H ) which does not coincide with a Dixmier trace (see § 5 below).”
- Dynamical Systems on Spectral Metric Spaces (2010, methodology): “Then several works appeared, like the important contributions of Buyalo [10, 11] or the ones by Guido & Isola [34, 35, 36] in the analysis of metric spaces and fractals.”