Methods and Applications of Analysis

Volume 25 (2018)

Number 3

In Memory of Professor John N. Mather: Part 1 of 3

Guest Editors: Sen Hu, University of Science and Technology, China; Stanisław Janeczko, Polish Academy of Sciences, Poland; Stephen S.-T. Yau, Tsinghua University, China; and Huaiqing Zuo, Tsinghua University, China.

Unwinding spirals I

Pages: 225 – 232

DOI: https://dx.doi.org/10.4310/MAA.2018.v25.n3.a3

Authors

Alexander Fish (School of Mathematics and Statistics, University of Sydney, NSW, Australia)

Laurentiu Paunescu (School of Mathematics and Statistics, University of Sydney, NSW, Australia)

Abstract

Inspired by the previous works by Freedman and He [FH], and Katznelson, Subhashis Nag, and Sullivan [KNS], we study the spiraling behaviour around a singularity of bi-Lipschitz homeomorphisms in $\mathbb{R}^2$. In particular, we show that there is no bi-Lipschitz homeomorphism of $\mathbb{R}^2$ that maps a spiral with a sub-exponential decay of winding radii to an unwound arc. This result is sharp as shows an example of a logarithmic spiral.

Keywords

logarithmic spiral, unwinding spirals, planar geometry

2010 Mathematics Subject Classification

Primary 14H50. Secondary 51F99.

Full Text (PDF format)

Received 23 May 2018

Accepted 3 August 2018

Published 1 November 2019