Density of orbits in laminations and the space of critical portraits

Details Density of orbits in laminations and the space of critical portraits Density of orbits in laminations and the space of critical portraits

2011-04-20

arxiv.org/abs/1104.4130v2

Mathematics

Dynamical Systems

13 pages; accepted for publication in Discrete and Continuous Dynamical Systems

Scientific paper

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.

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