Fractals in Noncommutative Geometry

Abstract

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 R^n, 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.

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

References in corpus (4)

• Hopf algebras, cyclic cohomology and the transverse index theory
• Dimensions and singular traces for spectral triples, with applications to fractals
• Singular Traces and Compact Operators. I
• Singular traces, dimensions, and Novikov-Shubin invariants

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• A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers
• Spectral triple and Sinai - Ruelle - Bowen measures