Role of Geometric Properties of Connes' Spectral Triple | Original Article

ABSTRACT:

In this paper examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of R with no isolated points, we develop a noncommutative coarse multifractal formalism. Speci μ cally, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satis μ es a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1; +)-summable spectral triples for a one-sided topologically exact subshift ofnite type ( μ NA ;). These spectral triples are constructed using equilibrium measures obtained from the Perron Frobenius-Ruelle operator, whose potential function is non-arithemetic and Holder continuous.