Document Type thesis Author Name Liu, Pan URN etd-042412-103445 Title On Quasi-Volume-Filling Surfaces Degree MS Department Mathematical Sciences Advisors Umberto Mosco, Advisor Keywords Fractal filling surface Quasi-filling surface Date of Presentation/Defense 2012-04-24 Availability unrestricted
The purpose of this paper is to construct a Quasi-volume-filling surface and study its properties. We start with the construction of a volume-filling surface, the PÃ³lya surface, based on PÃ³lya's curve, by rotating the PÃ³lya's curve in 3-dimensional space. Then we construct a Quasi-space-filling curve in 2-dimensions, the Quasi- PÃ³lya curve, which approximates the PÃ³lya's curve and fills a triangle up to a residual small surface of arbitrary size. We prove that the Quasi-PÃ³lya curve satisfies the open set condition, and there exists a unique invariant (self-similar) measure consistent with the normalized Hausdorff measure on it. Moreover, the energy form constructed on Quasi-PÃ³lya curve is proved to be a closed & regular form, and we prove that the Quasi-PÃ³lya curve is a variational fractal in the end. Next, we use the same idea, by rotating the Quasi-PÃ³lya curve in 3-dimensional space, to construct the Quasi-PÃ³lya surface, which is a Quasi-volume-filling surface and approximates to PÃ³lya surface in some sense.
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