Unicritical laminations
Volume 258 / 2022
Abstract
Thurston introduced invariant (quadratic) laminations in his 1984 preprint as a vehicle for understanding the connected Julia sets and the parameter space of quadratic polynomials. Important ingredients of his analysis of the angle doubling map $\sigma _2$ on the unit circle $\mathbb {S}^1$ were the Central Strip Lemma, non-existence of wandering polygons, the transitivity of the first return map on vertices of periodic polygons, and the non-crossing of minors of quadratic invariant laminations. We use Thurston’s methods to prove similar results for unicritical laminations of arbitrary degree $d$ and to show that the set of so-called minors of unicritical laminations themselves form a Unicritical Minor Lamination ${\rm UML}_d$. In the end we verify the Fatou conjecture for the unicritical laminations and extend the Lavaurs algorithm onto ${\rm UML}_d$.
