Cubic Critical Portraits and Polynomials with Wandering Gaps
arXiv:1003.4467
Abstract
Thurston introduced $\si_d$-invariant laminations (where $\si_d(z)$ coincides with $z^d:\ucirc\to \ucirc$, ) and defined \emph{wandering -gons} as sets $\T\subset \ucirc$ such that $\si_d^n(\T)$ consists of distinct points for all 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 -gons. Call a lamination with wandering -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 , 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 has a dense orbit in each subarc of (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.
31 pages, 4 figures; this is the last, third version of the paper which is to appear in Ergodic Theory and Dynamical Systems