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dc.contributor.advisorFalconer, K. J.
dc.contributor.advisorStratmann, Bernd
dc.contributor.authorSamuel, Anthony
dc.description.abstractIn this thesis 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. Specifically, 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 satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. 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 of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}}^{{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.en_US
dc.publisherUniversity of St Andrews
dc.rightsCreative Commons Attribution 3.0 Unported
dc.subjectFractal geometryen_US
dc.subjectMultifractal analysisen_US
dc.subjectSymbolic dynamicsen_US
dc.subjectErgodic theoryen_US
dc.subjectThermodynamic formalismen_US
dc.subjectRenewal theoryen_US
dc.subjectNoncommutative geometryen_US
dc.subjectSpectral triplesen_US
dc.subject.lcshSymbolic dynamicsen_US
dc.subject.lcshErgodic theoryen_US
dc.titleA commutative noncommutative fractal geometryen_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US

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Creative Commons Attribution 3.0 Unported
Except where otherwise noted within the work, this item's licence for re-use is described as Creative Commons Attribution 3.0 Unported