Central Strips of Sibling Leaves in Laminations of the Unit Disk

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 $d\ge 2$ and prove a Central Strip Lemma for degree $d\ge 2$. 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.

31 pages and 19 figures