2001/02/27 by Daniele Guido, Tommaso Isola
Mathematics · #math.OA #msc:46Lxx #msc:28A80
published as in the Proceedings of the Conference "Mathematical Physics in Mathematics and Physics", Siena 2000, R. Longo Ed., Fields Institute Communications, Vol. 30, AMS, Providence, RI, 2001 · 15 pages, LaTeX with fic-l.cls at ftp://ftp.ams.org/pub/author-info/packages/fic/amslatex/fic-l.cls To appear in the proceedings of the conference "Mathematical Physics in Mathematics and Physics", Siena 2000
arxiv created 2001/02/27 · arxiv updated 2009/11/30
To any spectral triple (A,D,H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that |D|-d has non trivial logarithmic Dixmier trace. Moreover, when d is finite non-zero, there always exists a singular trace which is finite nonzero on |D|-d, giving rise to a noncommutative integration on A. Such results are applied to fractals in R, using Connes' spectral triple, and to limit fractals in Rn, a class which generalises self-similar fractals, using a new spectral triple. The noncommutative dimension or measure can be computed in some cases. They are shown to coincide with the (classical) Hausdorff dimension and measure in the case of self-similar fractals.