Density of orbits in laminations and the space of critical portraits

Abstract

Thurston introduced $\si_d$-invariant laminations (where $\si_d(z)$ coincides with $z^d:\ucirc\to \ucirc$, $d\ge 2$). He 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. Thurston proved that $\si_2$ has no wandering $k$-gons and posed the problem of their existence for $\si_d$,\, $d\ge 3$. Call a lamination with wandering $k$-gons a \emph{WT-lamination}. Denote the set of cubic critical portraits by $\A_3$. A critical portrait, compatible with a WT-lamination, is called a \emph{WT-critical portrait}; let $\WT_3$ 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 \emph{condense} orbits of vertices of order three in their dendritic Julia sets, are dense and locally uncountable in $\A_3$ ($D\subset X$ is \emph{condense in $X$} if $D$ intersects every subcontinuum of $X$). Here we show that $\WT_3$ is a dense first category subset of $\A_3$. We also show that (a) critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of $\A_3$, (b) the existence of a condense orbit in the Julia set $J$ implies that $J$ is locally connected.