Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple (A,H,D), the functionals on A of the form a↦τω(a|D|−α) are studied, where τω is a singular trace, and ω is a generalised limit. When τω is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|−1, and that the set of α's for which there exists a singular trace τω giving rise to a non trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The functionals corresponding to points in the traceability interval are called Hausdorff–Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff–Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit ω.

[1]  C. Tricot.  Curves and Fractal Dimension , 1994 .

[2]  Hideki Kosaki,et al.  Generalized s-numbers of τ-measurable operators , 1986 .

[3]  Michel L. Lapidus,et al.  Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann-Zeta-Function , 1997 .

[4]  T. Isola,et al.  Singular Traces on Semifinite von Neumann Algebras , 1995 .

[5]  Noncommutative Riemann Integration and Novikov–Shubin Invariants for Open Manifolds , 1998, math/9802015.

[6]  Joseph C. Várilly,et al.  Elements of Noncommutative Geometry , 2000 .

[7]  A remark on trace properties of K-cycles , 1995, funct-an/9506003.

[8]  Alain Connes,et al.  Noncommutative geometry , 1988 .

[9]  Kenneth S. Washington.  The Talk of the Conference. , 1971 .

[10]  S. Albeverio,et al.  Singular Traces and Compact Operators , 1993, funct-an/9308001.

[11]  Michel L. Lapidus,et al.  The Riemann Zeta-Function and the One-Dimensional Weyl-Berry Conjecture for Fractal Drums , 1993 .

[12]  Spectral flow and Dixmier traces , 2002, math/0205076.

[13]  Fractals in Noncommutative Geometry , 2001, math/0102209.

[14]  C. Bennett,et al.  Interpolation of operators , 1987 .

[15]  R. Longo.  Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects , 2001 .

[16]  D. Gatzouras,et al.  Lacunarity of self-similar and stochastically self-similar sets , 1999 .

[17]  Traces on irregular ideals , 1989 .

[18]  T. Fack,et al.  On von Neumann spectral triples , 2000, math/0012233.

[19]  T. Figiel,et al.  Commutator structure of operator ideals , 2004 .

[20]  R. Mauldin,et al.  Random recursive constructions: asymptotic geometric and topological properties , 1986 .

[21]  J. Kahane,et al.  Ensembles parfaits et séries trigonométriques , 1963 .

[22]  E. M. Semenov,et al.  Symmetric Functionals and Singular Traces , 1998 .

[23]  M. Lapidus,et al.  Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions , 2005 .

[24]  J. Kigami,et al.  Self-Similarity of Volume Measures for Laplacians¶on P. C. F. Self-Similar Fractals , 2001 .