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

Abstract

Given a spectral triple ( A, H,D) , the functionals on A of the form a↦ τ ω ( a| D| − α ) are studied, where τ ω is a singular trace, and ω is a generalised limit. When τ ω 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 τ ω 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.