2002/02/28 by Daniele Guido, Tommaso Isola · 8 citations
Mathematics · #Advanced Operator Algebra Research #Mathematical Dynamics and Fractals #advanced mathematical theories #math.OA #msc:28A80 #msc:46Lxx
paper · pdf · doi:10.1016/s0022-1236(03)00230-1
published as Journ. Funct. Analysis, 203 (2003), 362-400 · latex, 36 pages, no figures, to appear on Journ. Funct. Analysis
arxiv created 2002/11/12 · openalex publication_date 2003/09/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a spectral triple (A,D,H), the functionals on A of the form a -> tauomega(a|D|^(-t)) are studied, where tauomega is a singular trace, and omega is a generalised limit. When tauomega 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 t's for which there exists a singular trace tauomega giving rise to a non-trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The corresponding functionals 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.