Doctoral Dissertations

Date of Award

5-2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Vyron Vellis

Committee Members

Vyron Vellis, Theodora Bourni, Matthew Badger, Joan Lind

Abstract

Rectifiability and parametrization of fractal sets are topics of substantial interest in geometric measure theory, including the celebrated theorem of Rademacher, Jones and Okikiolu's Analyst's Traveling Salesperson Theorem (ATST), and Remes's theorem on the parametrization of connected self-similar fractals. The first two of these, respectively, resolve fully questions related to tangents of Lipschitz curves and to Lipschitz rectifiability but leave wide open the analogous questions for H\"older curves, and the methods used by Remes have since been applied to related parametrization results for other classes of fractal sets.

First, we prove continuous and H\"older continuous parametrization results for connected fractals generated by infinite iterated function systems. In the latter portion, we investigate the tangent geometry of certain H\"older curves, proving that a certain class of randomly generated fractal sets are H\"older curves with the optimal exponent, and then proving that these curves display, at typical points, tangent geometry which is wildly more complicated than that of any self-similar curve. In our parametrization results we are able to construct these H\"older parametrizations of optimal exponent explicitly, by contrast to prior analogous parametrization results using Remes's method which must make some sacrifice here; either the parametrizations do not have optimal exponent or the proof is non-constructive. The investigation of tangent geometry to H\"older curves is a necessary first step toward obtaining a H\"older analogue of the ATST and thus in understanding H\"older rectifiability. Our results on the tangents to the randomly generated fractal sets demonstrate that tangents to H\"older curves can be extremely complicated, and indeed the geometry which can be attained is not captured by the tangents to self-similar sets.

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Analysis Commons

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