Unicritical Laminations

Abstract

Thurston introduced \emph{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 \emph{unicritical} laminations of arbitrary degree $d$ and to show that the set of so-called \emph{minors} of unicritical laminations themselves form a \emph{Unicritical Minor Lamination} $\mathrm{UML}_d$. In the end we verify the \emph{Fatou conjecture} for the unicritical laminations and extend the \emph{Lavaurs algorithm} onto $\mathrm{UML}_d$.