Article Details

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

Poonam Sharma*, in Journal of Advances and Scholarly Researches in Allied Education | Multidisciplinary Academic Research


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.