scholarly journals Critical portraits for postcritically finite polynomials

2009 ◽  
Vol 203 (2) ◽  
pp. 107-163 ◽  
Author(s):  
Alfredo Poirier
2013 ◽  
Vol 1 ◽  
Author(s):  
MATTHEW BAKER ◽  
LAURA DE MARCO

AbstractWe study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on ${ \mathbb{P} }^{1} $ containing a Zariski-dense subset of postcritically finite maps.


2017 ◽  
Vol 11 (1) ◽  
pp. 57-98 ◽  
Author(s):  
Thomas Gauthier ◽  
◽  
Gabriel Vigny

2018 ◽  
Vol 39 (10) ◽  
pp. 2855-2880
Author(s):  
KHUDOYOR MAMAYUSUPOV

We obtain a unique, canonical one-to-one correspondence between the space of marked postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of entire maps that take the form $p(z)\exp (q(z))$ for $p(z)$, $q(z)$ polynomials and $\exp (z)$, the complex exponential function. This bijection preserves the dynamics and embedding of Julia sets and is induced by a surgery tool developed by Haïssinsky.


2012 ◽  
Vol 33 (3) ◽  
pp. 713-738 ◽  
Author(s):  
ALEXANDER BLOKH ◽  
CLINTON CURRY ◽  
LEX OVERSTEEGEN

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.


Author(s):  
William P. Thurston

This chapter studies the topological entropy h of postcritically finite one-dimensional maps and, in particular, the relations between dynamics and arithmetics of eʰ, presenting some constructions for maps with given entropy and characterizing what values of entropy can occur for postcritically finite maps. In particular, the chapter proves: h is the topological entropy of a postcritically finite interval map if and only if h = log λ‎, where λ‎ ≥ 1 is a weak Perron number, i.e., it is an algebraic integer, and λ‎ ≥ ∣λ‎superscript Greek Small Letter Sigma∣ for every Galois conjugate λ‎superscript Greek Small Letter Sigma ∈ C. Unfortunately, the author of this chapter has died before completing this work, hence this chapter contains both the original manuscript as well as a number of notes which clarify many of the points mentioned therein.


2019 ◽  
Vol 16 ◽  
pp. 975-982
Author(s):  
N. V. Abrosimov ◽  
M. V. Chanchieva ◽  
A. V. Tetenov

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