Density of orbits in laminations and the space of critical portraits

Thurston introduced ${\text{si}}_d$-invariant laminations (where ${\text{si}}_d(z)$ coincides with $z^d:\text{ucirc}\to \text{ucirc}$, $d\ge 2$). He defined \emph{wandering $k$-gons} as sets $\text{T}\subset \text{ucirc}$ such that ${\text{si}}_d^n(\text{T})$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets ${\text{si}}_d^n(\text{T})$ in the plane are pairwise disjoint. Thurston proved that ${\text{si}}_2$ has no wandering $k$-gons and posed the problem of their existence for ${\text{si}}_d$,, $d\ge 3$. Call a lamination with wandering $k$-gons a \emph{WT-lamination}. Denote the set of cubic critical portraits by ${\text{A}}_3$. A critical portrait, compatible with a WT-lamination, is called a \emph{WT-critical portrait}; let ${\text{WT}}_3$ 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 ${\text{A}}_3$ ($D\subset X$ is \emph{condense in $X$} if $D$ intersects every subcontinuum of $X$). Here we show that ${\text{WT}}_3$ is a dense first category subset of ${\text{A}}_3$. We also show that (a) critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of ${\text{A}}_3$, (b) the existence of a condense orbit in the Julia set $J$ implies that $J$ is locally connected.