Squeezing functions and Cantor Sets

TL;DR

This paper investigates the geometric properties of Cantor set complements using squeezing functions, demonstrating how some resemble the unit disk while others do not, and analyzing Julia set complements with degenerate squeezing functions.

Contribution

It constructs large Cantor sets with complements that either do or do not resemble the unit disk in terms of squeezing functions, and studies Julia set complements with degenerate squeezing functions.

Findings

Some Cantor set complements can be arbitrarily close to the unit disk.

Other Cantor set complements significantly differ from the unit disk.

Julia set complements have degenerate squeezing functions despite high Hausdorff dimension.

Abstract

We construct "large" Cantor sets whose complements resemble the unit disk arbitrarily well from the point of view of the squeezing function, and we construct "large" Cantor sets whose complements do not resemble the unit disk from the point of view of the squeezing function. Finally we show that complements of Cantor sets arising as Julia sets of quadratic polynomials have degenerate squeezing functions, despite of having Hausdorff dimension arbitrarily close to two.