叠层中轨道的密度和临界象的空间
Density of orbits in laminations and the space of critical portraits
摘要：
Thurston引入了$\si_d$ -不变叠层（其中$\si_d(z)$与$z^d:\ucirc\to \ucirc$，$d\ge 2$一致）.他定义\emph{wandering $k$-gons}为集$\T\subset \ucirc$，使$\si_d^n(\T)$组成的$k\ge 3$不同点的所有$n\ge 0$和凸包的所有集$\si_d^n(\T)$在平面上是成对不相交。Thurston证明了$\si_2$没有游荡的$k$ -环，并对$\si_d$，$d\ge 3$提出了它们的存在性问题.将$k$ -游离的层压称为\emph{WT-lamination}。用$\A_3$表示一组立方临界肖像。与WT-层压兼容的关键肖像被称为\emph{WT-critical portrait} ;让$\WT_3$成为所有这些的集合。作者最近证明了三次WT-叠层的存在和三次WT-临界肖像，定义多项式的\emph{condense}轨道的阶数为3的树突朱莉娅集，是密集的和局部不可数的$\A_3$（$D\subset X$是\emph{condense in $X$}，如果$D$相交于$X$的每个子连续统）。在这里，我们表明，$\WT_3$是一个密集的第一类子集的$\A_3$。我们还证明了：（a）临界画像，其叠层在拓扑Julia集中有一个凝聚轨道，形成$\A_3$的一个剩余子集;（b）Julia集$J$中凝聚轨道的存在意味着$J$是局部连通的.
英文摘要：
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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