Spectral triples for the Sierpinski Gasket

Abstract

We construct a family of spectral triples for the Sierpinski Gasket $K$. For suitable values of the parameters, we determine the dimensional spectrum and recover the Hausdorff measure of $K$ in terms of the residue of the volume functional $a\to$ tr$(a|D{|}^{-s})$ at its abscissa of convergence $d_D$, which coincides with the Hausdorff dimension $d_H$ of the fractal. We determine the associated Connes' distance showing that it is bi-Lipschitz equivalent to the distance on $K$ induced by the Euclidean metric of the plane, and show that the pairing of the associated Fredholm module with (odd) $K$-theory is non-trivial. When the parameters belong to a suitable range, the abscissa of convergence ${\delta }_D$ of the energy functional $a\to$ tr$(|D{|}^{-s/2}|[D,a]{|}^2|D{|}^{-s/2})$ takes the value $d_E=\frac{\log (12/5)}{\log 2}$, which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on $K$.

48 pages, 9 figures. Final version, to appear in J.Funct.Anal

References in corpus (4)

• Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory
• Dynamical Systems on Spectral Metric Spaces
• Derivations and Dirichlet forms on fractals
• Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket

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• Asymptotic and exact expansions of heat traces
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• Finite energy coordinates and vector analysis on fractals
• Quantum entropy couples matter with geometry
• Examples of infinite direct sums of spectral triples
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• A Spectral Triple for a Solenoid Based on the Sierpinski Gasket
• A noncommutative Sierpinski Gasket
• Variations in noncommutative potential theory: finite energy states, potentials and multipliers
• Riesz transforms, Hodge-Dirac operators and functional calculus for multipliers I
• Spectral Triples for the Variants of the Sierpinski Gasket