Decomposing Multitwists

TL;DR

This paper develops a method to decompose certain bi-Lipschitz maps on the sphere that spiral around Cantor sets, using Dehn twists and a bi-Lipschitz path, advancing understanding of the decomposition problem in geometric topology.

Contribution

It constructs a decomposition for bi-Lipschitz maps spiraling around Cantor sets with Assouad dimension less than one, linking uniform disconnectedness to pants decompositions.

Findings

Decomposition applies to maps spiraling around Cantor sets with small Assouad dimension.

Shows equivalence between uniform disconnectedness of the set and bounded hyperbolic length in pants decomposition.

Provides a new approach to the decomposition problem in the class of bi-Lipschitz maps.

Abstract

The Decomposition Problem in the class $LIP(\mathbb{S}^2)$ is to decompose any bi-Lipschitz map $f:\mathbb{S}^2 \to \mathbb{S}^2$ as a composition of finitely many maps of arbitrarily small isometric distortion. In this paper, we construct a decomposition for certain bi-Lipschitz maps which spiral around every point of a Cantor set $X$ of Assouad dimension strictly smaller than one. These maps are constructed by considering a collection of Dehn twists on the Riemann surface $\mathbb{S}^2 \setminus X$. The decomposition is then obtained via a bi-Lipschitz path which simultaneously unwinds these Dehn twists. As part of our construction, we also show that $X \subset \mathbb{S}^2$ is uniformly disconnected if and only if the Riemann surface $\mathbb{S}^2 \setminus X$ has a pants decomposition whose cuffs have hyperbolic length uniformly bounded above, which may be of independent interest.