Density of orbits in laminations and the space of critical portraits

Abstract

Thurston introduced $\sigma_d$-invariant laminations (where $\sigma_d(z)$ coincides with $z^d:\mathbb{S}\to \mathbb{S}$, $d\ge 2$). He defined wandering $k$-gons as sets $T\subset \mathbb{S}$ such that $\sigma_d^n(T)$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\sigma_d^n(T)$ in the plane are pairwise disjoint. Thurston proved that $\sigma_2$ has no wandering $k$-gons and posed the problem of their existence for $\sigma_d$, $d\ge 3$. Call a lamination with wandering $k$-gons a WT-lamination. Denote the set of cubic critical portraits by $\mathcal{A}_3$. A critical portrait, compatible with a WT-lamination, is called a WT-critical portrait; let $\mathcal{WT}_3$ be the set of all of them. It was recently shown by the authors that cubic WT-laminations exist and cubic WT-critical portraits, defining polynomials with condense orbits of vertices of order three in their dendritic Julia sets, are dense and locally uncountable in $\mathcal{A}_3$ ($D\subset X$ is condense in $X$ if $D$ intersects every subcontinuum of $X$). Here we show that $\mathcal{WT}_3$ is a dense first category subset of $\mathcal{A}_3$, that critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of $\mathcal{A}_3$, and that the existence of a condense orbit in the Julia set $J$ implies that $J$ is locally connected.