Title:
Riemannian geometry of compact metric spaces
Riemannian geometry of compact metric spaces
Author(s)
Palmer, Ian Christian
Advisor(s)
Bellissard, Jean
Editor(s)
Collections
Supplementary to
Permanent Link
Abstract
A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the
space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.
Sponsor
Date Issued
2010-05-21
Extent
Resource Type
Text
Resource Subtype
Dissertation