scholarly journals Density of orbits in laminations and the space of critical portraits

2012 ◽  
Vol 32 (6) ◽  
pp. 2027-2039 ◽  
Author(s):  
Alexander Blokh ◽  
◽  
Clinton Curry ◽  
Lex Oversteegen ◽  
Keyword(s):  
Critical Portraits

scholarly journals Cubic critical portraits and polynomials with wandering gaps

2012 ◽  
Vol 33 (3) ◽  
pp. 713-738 ◽  
Author(s):  
ALEXANDER BLOKH ◽  
CLINTON CURRY ◽  
LEX OVERSTEEGEN
Keyword(s):  
Branch Point ◽  
Pairwise Disjoint ◽  
Julia Sets ◽  
Convex Hulls ◽  
Dense Orbit ◽  
New Approach ◽  
Quotient Spaces ◽  
Cubic Polynomials ◽  
Induced Maps ◽  
Critical Portraits

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.

scholarly journals The landing of parameter rays at non-recurrent critical portraits

2018 ◽  
Vol 61 (12) ◽  
pp. 2267-2282
Author(s):  
Yan Gao ◽  
Jinsong Zeng
Keyword(s):  
Critical Portraits
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