Central Strips of Sibling Leaves in Laminations of the Unit Disk

TL;DR

This paper generalizes Thurston's Central Strip Lemma from quadratic laminations to all degrees, providing a key tool for understanding higher degree laminations and their associated Julia sets.

Contribution

It extends the Central Strip Lemma to laminations of all degrees and explores its applications to identity return polygons in higher degree settings.

Findings

Generalized Central Strip Lemma for degree d≥2

Applications to identity return polygons

Insights into higher degree Julia sets

Abstract

Quadratic laminations of the unit disk were introduced by Thurston as a vehicle for understanding the (connected) Julia sets of quadratic polynomials and the parameter space of quadratic polynomials. The "Central Strip Lemma" plays a key role in Thurston's classification of gaps in quadratic laminations, and in describing the corresponding parameter space. We generalize the notion of {\em Central Strip} to laminations of all degrees $d\ge2$ and prove a Central Strip Lemma for degree $d\ge2$. We conclude with applications of the Central Strip Lemma to {\em identity return polygons} that show it may play a role similar to Thurston's lemma for higher degree laminations.