A commutative noncommutative fractal geometry

Fractals;Symbolic dynamics;Ergodic theory

Samuel, Anthony ; Falconer, K. J. ; Falconer, K. J.

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Fractals;    Symbolic dynamics;    Ergodic theory;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/1710/AnthonySamuelPhDThesis.pdf?sequence=6&isAllowed=y

来源: DR-NTU

【 摘 要 】

In 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.