University of Illinois Urbana-Champaign

Quasiconformal mappings and fractal geometry

Chrontsios Garitsis, Efstathios Konstantinos

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CHRONTSIOSGARITSIS-DISSERTATION-2023.pdf

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https://hdl.handle.net/2142/121393

Description

Title
Quasiconformal mappings and fractal geometry
Author(s)
Chrontsios Garitsis, Efstathios Konstantinos
Issue Date
2023-05-04
Director of Research (if dissertation) or Advisor (if thesis)
  • Hinkkanen, Aimo
  • Tyson, Jeremy
Doctoral Committee Chair(s)
Erdogan, Burak
Committee Member(s)
Hildebrand, AJ
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Quasiconformal, fractal, geometry, analysis
Abstract
This thesis discusses three different projects regarding fractals and quasiconformal mappings. The first project involves certain fractal Fourier series motivated by exponential sums studied in areas of number theory. We showed with Hildebrand that if for a function $f:\N\rightarrow \C$ the sum $|\sum_{n\leq x}f(n)e^{2\pi i n t}|$ grows at a specific uniform rate, then the Fourier series $\sum_{n=1}^\infty \frac{f(n)}{n}e^{2\pi i n t}$ is H\"older continuous, which provides upper bounds on the box-counting dimension of fractal sets associated with the series. The second project investigates the quasiconformal distortion of certain dimension notions and spectra. More specifically, we showed with Tyson that a quasiconformal map between domains in Euclidean spaces cannot change the dimensions of sets uncontrollably. This was previously known for the Hausdorff and box-counting dimensions, but our result applies on the Assouad dimension and spectrum, with the latter being the first result of its kind. The third project is motivated by the geometric definition of quasiconformality on the complex plane and provides interesting properties for certain quadrilaterals. Namely, we showed with Hinkkanen that every quadrilateral $Q$ of modulus in some interval $[1/K,K]$, $K>1$, needs to contain a disk of radius comparable to the maximum of the internal distances of $Q$, where the comparability constant only depends on $K$.
Graduation Semester
2023-08
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/121393
Copyright and License Information
Copyright 2023 Efstathios Konstantinos Chrontsios Garitsis

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