Spectral metric spaces for Gibbs measures

Abstract

We construct spectral metric spaces for Gibbs measures on a one-sided topologically exact subshift of finite type. That is, for a given Gibbs measure we construct a spectral triple and show that Connes' corresponding pseudo-metric is a metric and that its metric topology agrees with the weak-*-topology on the state space over the set of continuous functions defined on the subshift. Moreover, we show that each Gibbs measure can be fully recovered from the noncommutative integration theory and that the noncommutative volume constant of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of the shift invariant Gibbs measure.

25 pages

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• A classification of aperiodic order via spectral metrics and Jarník sets
• The sectional curvature of the infinite dimensional manifold of Hölder equilibrium prababilities
• Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples
• Multiresolution analysis for Markov Interval Maps