Cubic Critical Portraits and Polynomials with Wandering Gaps

TL;DR

This paper constructs and analyzes cubic laminations with wandering gaps, demonstrating their abundance and complex dynamics, including dense orbits of wandering branch points, and shows these structures are densely represented among cubic critical portraits.

Contribution

The paper introduces a new method to construct cubic WT-laminations with condense wandering branch points and proves their critical portraits are uncountably dense in the space of all cubic critical portraits.

Findings

Existence of uncountably many cubic WT-laminations with condense wandering branch points.

Construction method for cubic WT-laminations with specified properties.

Density of critical portraits corresponding to such laminations in the space of cubic critical portraits.

Abstract

Thurston introduced $\si_d$-invariant laminations (where $\si_d(z)$ coincides with $z^d:\ucirc\to \ucirc$, $d\ge 2$) and defined \emph{wandering $k$-gons} as sets $\T\subset \ucirc$ such that $\si_d^n(\T)$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\si_d^n(\T)$ in the plane are pairwise disjoint. He proved that $\si_2$ has no wandering $k$-gons. Call a lamination with wandering $k$-gons a \emph{WT-lamination}. In a recent paper it was shown that uncountably many cubic WT-laminations, with pairwise non-conjugate induced maps on the corresponding quotient spaces $J$, are realizable as cubic polynomials on their (locally connected) Julia sets. In the present paper we use a new approach to construct cubic WT-laminations with all of the above properties and the extra property that the corresponding wandering branch point of $J$ has a dense orbit in each subarc of $J$ (we call such orbits \emph{condense}), and to show that critical portraits corresponding to such laminations are uncountably dense in the space $\A_3$ of all cubic critical portraits.