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

Given a spectral triple , the functionals on 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 , 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.