Density of orbits in laminations and the spaceof critical portraits

Alexander Blokh; Clinton Curry; Lex Oversteegen

doi:10.3934/dcds.2012.32.2027

Article Contents

2012, Volume 32, Issue 6: 2027-2039. Doi: 10.3934/dcds.2012.32.2027

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Density of orbits in laminations and the space of critical portraits

1.
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170
2.
Department of Mathematics, Huntingdon College, Montgomery, AL 36106-2114

Received: March 31, 2011

Revised: October 31, 2011

Published: May 2012

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Abstract

Thurston introduced $\sigma_d$-invariant laminations (where $\sigma_d(z)$ coincides with $z^d:\mathbb{S}\to \mathbb{S}$, $d\ge 2$). He defined wandering $k$-gons as sets $T\subset \mathbb{S}$ such that $\sigma_d^n(T)$ consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets $\sigma_d^n(T)$ in the plane are pairwise disjoint. Thurston proved that $\sigma_2$ has no wandering $k$-gons and posed the problem of their existence for $\sigma_d$, $d\ge 3$.
Call a lamination with wandering $k$-gons a WT-lamination. Denote the set of cubic critical portraits by $\mathcal{A}_3$. A critical portrait, compatible with a WT-lamination, is called a WT-critical portrait; let $\mathcal{WT}_3$ 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 condense orbits of vertices of order three in their dendritic Julia sets, are dense and locally uncountable in $\mathcal{A}_3$ ($D\subset X$ is condense in $X$ if $D$ intersects every subcontinuum of $X$). Here we show that $\mathcal{WT}_3$ is a dense first category subset of $\mathcal{A}_3$, that critical portraits, whose laminations have a condense orbit in the topological Julia set, form a residual subset of $\mathcal{A}_3$, and that the existence of a condense orbit in the Julia set $J$ implies that $J$ is locally connected.

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Mathematics Subject Classification: Primary: 37F20; Secondary: 37B45, 37F10, 37F50.

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