Title:
The noncommutative geometry of ultrametric cantor sets

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dc.contributor.advisor Bellissard, Jean
dc.contributor.author Pearson, John Clifford en_US
dc.contributor.committeeMember Baker, Matt
dc.contributor.committeeMember Bakhtin, Yuri
dc.contributor.committeeMember Garoufalidis, Stavros
dc.contributor.committeeMember Putnam, Ian
dc.contributor.department Mathematics en_US
dc.date.accessioned 2008-09-17T19:28:06Z
dc.date.available 2008-09-17T19:28:06Z
dc.date.issued 2008-05-13 en_US
dc.description.abstract An 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.description.degree Ph.D. en_US
dc.identifier.uri http://hdl.handle.net/1853/24657
dc.publisher Georgia Institute of Technology en_US
dc.subject Fractal geometry en_US
dc.subject Noncommutative geometry en_US
dc.subject Cantor set en_US
dc.subject.lcsh Riemannian manifolds
dc.subject.lcsh Noncommutative differential geometry
dc.subject.lcsh Geometry
dc.subject.lcsh Cantor sets
dc.title The noncommutative geometry of ultrametric cantor sets en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Bellissard, Jean
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
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relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
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