Cubic critical portraits and polynomials with wandering gaps

Abstract

AbstractThurston introduced $\sigma _d$-invariant laminations (where $\sigma _d(z)$ coincides with $z^d:\mathbb S ^1\to \mathbb S ^1$, $d\ge 2$) and defined wandering $k$-gons as sets ${\mathbf {T}}\subset \mathbb S ^1$ such that $\sigma _d^n({\mathbf {T}})$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\sigma _d^n({\mathbf {T}})$ in the plane are pairwise disjoint. He proved that $\sigma _2$ has no wandering $k$-gons. Call a lamination with wandering $k$-gons a 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. Here we use a new approach to construct cubic WT-laminations with the above properties so that any wandering branch point of $J$ has a dense orbit in each subarc of $J$ (we call such orbits condense), and show that critical portraits corresponding to such laminations are dense in the space ${\mathcal A}_3$of all cubic critical portraits.