The noncommutative geometry of ultrametric cantor sets

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.

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Georgia Institute of Technology

Year dc:date.issued

2008

Author dc:creator

• Pearson, John Clifford

Advisor dc:contributor.advisor

• Bellissard, Jean

Committee members dc:contributor.committeemember

• Baker, Matt
• Bakhtin, Yuri
• Garoufalidis, Stavros
• Putnam, Ian

• Fractal geometry
• Noncommutative geometry
• Cantor set

Handle dc:identifier.uri

http://hdl.handle.net/1853/24657

OAI identifier oai:identifier

oai:repository.gatech.edu:1853/24657