Quasiconformal mappings and fractal geometry

Chrontsios Garitsis, Efstathios Konstantinos

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

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

Copyright and License Information

Copyright 2023 Efstathios Konstantinos Chrontsios Garitsis

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