Show simple item record

dc.contributor.authorPearson, John Clifforden_US
dc.date.accessioned2008-09-17T19:28:06Z
dc.date.available2008-09-17T19:28:06Z
dc.date.issued2008-05-13en_US
dc.identifier.urihttp://hdl.handle.net/1853/24657
dc.description.abstractAn analogue of the Riemannian structure of a manifold is created for an ultrametric Cantor set using the techniques of Noncommutative Geometry. In particular, a spectral triple is created that can recover much of the fractal geometry of the original Cantor set. It is shown that this spectral triple can recover the metric, the upper box dimension, and in certain cases the Hausdorff measure. The analogy with Riemannian geometry is then taken further and an analogue of the Laplace-Beltrami operator is created for an ultrametric Cantor set. The Laplacian then allows to create an analogue of Brownian motion generated by this Laplacian. All these tools are then applied to the triadic Cantor set. Other examples of ultrametric Cantor sets are then presented: attractors of self-similar iterated function systems, attractors of cookie cutter systems, and the transversal of an aperiodic, repetitive Delone set of finite type. In particular, the example of the transversal of the Fibonacci tiling is studied.en_US
dc.publisherGeorgia Institute of Technologyen_US
dc.subjectFractal geometryen_US
dc.subjectNoncommutative geometryen_US
dc.subjectCantor seten_US
dc.subject.lcshRiemannian manifolds
dc.subject.lcshNoncommutative differential geometry
dc.subject.lcshGeometry
dc.subject.lcshCantor sets
dc.titleThe noncommutative geometry of ultrametric cantor setsen_US
dc.typeDissertationen_US
dc.description.degreePh.D.en_US
dc.contributor.departmentMathematicsen_US
dc.description.advisorCommittee Chair: Bellissard, Jean; Committee Member: Baker, Matt; Committee Member: Bakhtin, Yuri; Committee Member: Garoufalidis, Stavros; Committee Member: Putnam, Ianen_US


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record