Riemann map and holomorphic dynamics

1986 ◽  
Vol 85 (3) ◽  
pp. 439-455 ◽  
Author(s):  
Feliks Przytycki
Keyword(s):  
Holomorphic Dynamics ◽  
Riemann Map

Frontiers in Complex Dynamics

2014 ◽  
Author(s):  
Araceli Bonifant ◽  
Misha Lyubich ◽  
Scott Sutherland
Keyword(s):  
Historical Perspective ◽  
Complex Dynamics ◽  
Short Paper ◽  
Combinatorial Group Theory ◽  
Entropy Theory ◽  
Holomorphic Dynamics ◽  
Eightieth Birthday ◽  
Several Variables ◽  
Fields Medal ◽  
Combinatorial Group

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.

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 A Comparison of Classical Holomorphic Dynamics and Holomorphic Semigroup Dynamics-II

2020 ◽  
Vol 37 (1-2) ◽  
pp. 86-95
Author(s):  
Bishnu Hari Subedi ◽  
Ajaya Singh
Keyword(s):  
Group Dynamics ◽  
Dynamical Behavior ◽  
Holomorphic Dynamics ◽  
Semi Group ◽  
Holomorphic Semigroup

In this paper, we prove that the escaping set of a transcendental semi group is S-forward invari-ant. We also prove that if a holomorphic semi group is a belian, then the Fatou, Julia, and escaping sets are S-completely invariant. We also investigate certain cases and conditions that the holomorphic semi group dynamics exhibits the similar dynamical behavior just like a classical holomorphic dynamics. Frequently, we also examine certain amount of connections and contrasts between classical holomorphic dynamics and holomorphic semi group dynamics.

scholarly journals Green’s function and anti-holomorphic dynamics on a torus

2016 ◽  
Vol 144 (7) ◽  
pp. 2911-2922 ◽  
Author(s):  
Walter Bergweiler ◽  
Alexandre Eremenko
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Holomorphic Dynamics

Stability and bifurcations in projective holomorphic dynamics

2018 ◽  
Vol 115 ◽  
pp. 37-71
Author(s):  
François Berteloot ◽  
Fabrizio Bianchi
Keyword(s):  
Holomorphic Dynamics ◽  
Stability And Bifurcations

INVERSE TOPOLOGICAL PRESSURE WITH APPLICATIONS TO HOLOMORPHIC DYNAMICS OF SEVERAL COMPLEX VARIABLES

2004 ◽  
Vol 06 (04) ◽  
pp. 653-679 ◽  
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Technical Condition ◽  
Topological Pressure ◽  
Holomorphic Dynamics ◽  
Critical Set ◽  
Axiom A ◽  
Type Definition ◽  
Basic Set ◽  
Unique Zero ◽  
Stable Dimension ◽  
The One

In this paper, we introduce a few notions of inverse topological pressure [Formula: see text], defined in terms of backward orbits (prehistories) instead of forward orbits. This inverse topological pressure has some properties similar to the regular (forward) pressure but, in general, if the map is not a homeomorphism, they do not coincide. In fact, there are several ways to define inverse topological pressure; for instance, we show that the Bowen type definition coincides with the one using spanning sets. Then we consider the case of a holomorphic map [Formula: see text] which is Axiom A and such that its critical set does not intersect a particular basic set of saddle type Λ. We will prove that, under a technical condition, the Hausdorff dimension of the intersection between the local stable manifold and the basic set is equal to ts, i.e. [Formula: see text], for all points x belonging to Λ. Here ts represents the unique zero of the function t→P-(tϕs), with P- denoting the inverse topological pressure and [Formula: see text], y∈Λ. In general, [Formula: see text] will be estimated above by ts and below by [Formula: see text], where [Formula: see text] is the unique zero of the map t→P_(tϕs). As a corollary we obtain that, if the stable dimension is non-zero, then Λ must be a non-Jordan curve, and also, if f|Λ happens to be a homeomorphism (like in the examples from [13]), then the stable dimension cannot be zero.

scholarly journals One hundred years of complex dynamics

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150453 ◽  
Author(s):  
Mary Rees
Keyword(s):  
Complex Dynamics ◽  
Rational Functions ◽  
Holomorphic Dynamics ◽  
Area Of Study

Holomorphic dynamics has been a key area of study for 100 years. In this survey, we focus on the dynamics of rational functions.

Sign in / Sign up

Export Citation Format

Share Document