Unicritical Laminations

TL;DR

This paper extends Thurston's invariant lamination theory from quadratic to unicritical polynomials of any degree, establishing structural properties, a minor lamination set, and verifying the Fatou conjecture within this context.

Contribution

It generalizes Thurston's methods to unicritical laminations of arbitrary degree and introduces the Unicritical Minor Lamination set, verifying the Fatou conjecture for these laminations.

Findings

The set of minors forms a Unicritical Minor Lamination $ ext{UML}_d$.

Structural properties like the Central Strip Lemma are extended to degree $d$.

The Fatou conjecture is verified for 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 $\mathrm{UML}_d$.