Rational Maps with Fatou Components of Arbitrary Connectivity Number

2007 ◽  
Vol 7 (2) ◽  
pp. 415-427 ◽  
Author(s):  
Marcus Stiemer
Keyword(s):  
Rational Maps ◽  
Fatou Components ◽  
Arbitrary Connectivity

Rational maps whose Fatou components are Jordan domains

1996 ◽  
Vol 16 (6) ◽  
pp. 1323-1343 ◽  
Author(s):  
Kevin M. Pilgrim
Keyword(s):  
Jordan Curve ◽  
Critical Points ◽  
Julia Set ◽  
Rational Maps ◽  
Rational Map ◽  
Fatou Component ◽  
Jordan Curves ◽  
Fatou Components

AbstractWe prove: If f(z) is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of f is a Jordan curve. If f(z) is a hyperbolic critically finite rational map all of whose postcritical points are periodic, then there exists a cycle of Fatou components whose boundaries are Jordan curves. We give examples of critically finite hyperbolic rational maps f with a Fatou component ω satisfying f(ω) = ω and f|∂ω not topologically conjugate to the dynamics of any polynomial on its Julia set.

scholarly journals On Connectivity of Fatou Components concerning a Family of Rational Maps

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Junyang Gao ◽  
Gang Liu
Keyword(s):  
Rational Maps ◽  
Rational Map ◽  
Fatou Component ◽  
Fatou Components

I. N. Baker established the existence of Fatou component with any given finite connectivity by the method of quasi-conformal surgery. M. Shishikura suggested giving an explicit rational map which has a Fatou component with finite connectivity greater than 2. In this paper, considering a family of rational mapsRz,tthat A. F. Beardon proposed, we prove thatRz,thas Fatou components with connectivities 3 and 5 for anyt∈0,1/12. Furthermore, there existst∈0,1/12such thatRz,thas Fatou components with connectivity nine.

On the dynamics of generalized McMullen maps

2013 ◽  
Vol 34 (6) ◽  
pp. 2093-2112 ◽  
Author(s):  
YINGQING XIAO ◽  
WEIYUAN QIU ◽  
YONGCHENG YIN
Keyword(s):  
Critical Points ◽  
Julia Sets ◽  
Dynamical Behavior ◽  
Parameter Family ◽  
Rational Maps ◽  
Topological Description ◽  
Image Position ◽  
Fatou Components ◽  
Two Parameter

AbstractIn this paper, we study the dynamics of the two-parameter family of rational maps $$\begin{eqnarray*}{F}_{a, b} (z)= {z}^{n} + \frac{a}{{z}^{n} } + b.\end{eqnarray*}$$ We give the topological description of Julia sets and Fatou components of ${F}_{a, b} $ according to the dynamical behavior of the orbits of its free critical points.

scholarly journals Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 237
Author(s):  
Rostislav Grigorchuk ◽  
Supun Samarakoon
Keyword(s):  
Operator Algebras ◽  
Probabilistic Approach ◽  
Schur Complement ◽  
Automata Theory ◽  
Random Schrödinger Operators ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Von Neumann ◽  
Intermediate Growth ◽  
Quasi Crystals

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

scholarly journals An Algebraic Brascamp–Lieb Inequality

2021 ◽  
Author(s):  
Jennifer Duncan
Keyword(s):  
General Class ◽  
Recent Progress ◽  
Algebraic Groups ◽  
Hölder’S Inequality ◽  
Duke Math ◽  
Rational Maps ◽  
Affine Invariant ◽  
Linear Maps ◽  
Nonlinear Maps ◽  
Funct Anal

AbstractThe Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

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