scholarly journals Preperiodic portraits for unicritical polynomials

2016 ◽  
Vol 144 (7) ◽  
pp. 2885-2899 ◽  
Author(s):  
John R. Doyle
Keyword(s):  
Unicritical Polynomials

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

scholarly journals Orbit counting in polarized dynamical systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Wade Hindes
Keyword(s):  
Dynamical Systems ◽  
Finitely Generated ◽  
Projective Varieties ◽  
Infinite Sets ◽  
Unicritical Polynomials

<p style='text-indent:20px;'>We extend recent orbit counts for finitely generated semigroups acting on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{P}^N $\end{document}</tex-math></inline-formula> to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.</p>

scholarly journals A question for iterated Galois groups in arithmetic dynamics

2020 ◽  
pp. 1-17
Author(s):  
Andrew Bridy ◽  
John R. Doyle ◽  
Dragos Ghioca ◽  
Liang-Chung Hsia ◽  
Thomas J. Tucker
Keyword(s):  
Dynamical System ◽  
Galois Groups ◽  
General Question ◽  
Arithmetic Dynamics ◽  
Polynomial Maps ◽  
Special Cases ◽  
Unicritical Polynomials

Abstract We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

scholarly journals Galois groups of iterates of some unicritical polynomials

2017 ◽  
Vol 181 (1) ◽  
pp. 57-73 ◽  
Author(s):  
Michael R. Bush ◽  
Wade Hindes ◽  
Nicole R. Looper
Keyword(s):  
Galois Groups ◽  
Unicritical Polynomials

scholarly journals Multicorns are not path connected

2014 ◽  
Author(s):  
John Hamal Hubbard ◽  
Dierk Schleicher
Keyword(s):  
Julia Set ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Unicritical Polynomials ◽  
Path Connected ◽  
Special Case ◽  
Connectedness Locus

This chapter proves that the tricorn is not locally connected and not even pathwise connected, confirming an observation of John Milnor from 1992. The tricorn is the connectedness locus in the space of antiholomorphic quadratic polynomials z ↦ ̄z² + c. The chapter extends this discussion more generally for antiholomorphic unicritical polynomials of degrees d ≥ 2 and their connectedness loci, known as multicorns. The multicorn M*subscript d is the connectedness locus in the space of antiholomorphic unicritical polynomials psubscript c(z) = ̄zsubscript d + c of degree d, i.e., the set of parameters for which the Julia set is connected. The special case d = 2 is the tricorn, which is the formal antiholomorphic analog to the Mandelbrot set.

scholarly journals The dynamical André–Oort conjecture: Unicritical polynomials

2017 ◽  
Vol 166 (1) ◽  
pp. 1-25 ◽  
Author(s):  
D. Ghioca ◽  
H. Krieger ◽  
K. D. Nguyen ◽  
H. Ye
Keyword(s):  
Unicritical Polynomials

scholarly journals Finite index theorems for iterated Galois groups of unicritical polynomials

2020 ◽  
Vol 374 (1) ◽  
pp. 733-752 ◽  
Author(s):  
Andrew Bridy ◽  
John R. Doyle ◽  
Dragos Ghioca ◽  
Liang-Chung Hsia ◽  
Thomas J. Tucker
Keyword(s):  
Finite Index ◽  
Galois Groups ◽  
Index Theorems ◽  
Unicritical Polynomials

scholarly journals Combinatorial rigidity for unicritical polynomials

2009 ◽  
Vol 170 (2) ◽  
pp. 783-797 ◽  
Author(s):  
Artur Avila ◽  
Jeremy Kahn ◽  
Mikhail Lyubich ◽  
Weixiao Shen
Keyword(s):  
Unicritical Polynomials ◽  
Combinatorial Rigidity
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