Dirac operators and spectral triples for some fractal sets built on curves

Details Dirac operators and spectral triples for some fractal sets built on curves Dirac operators and spectral triples for some fractal sets built on curves

2006-10-06

arxiv.org/abs/math/0610222v2

Mathematics

Metric Geometry

48 pages, 4 figures. Elementary proofs omitted. To appear in Adv. Math

Scientific paper

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension $\log 3/ \log 2$.

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