Complex dynamics is the study of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s defined by
iteration
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
of functions on
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
spaces. Complex analytic dynamics is the study of the dynamics of specifically
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s.
Techniques
*General
**
Montel's theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic ...
**
Poincaré metric
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry ...
**
Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapp ...
**
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphi ...
**
Carathéodory's theorem (conformal mapping) In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that the conformal mapping sending the unit disk to ...
**
Böttcher's equation Böttcher's equation, named after Lucjan Böttcher, is the functional equation
::F(h(z)) = (F(z))^n
where
* is a given analytic function with a superattracting fixed point of order at , (that is, h(z)=a+c(z-a)^n+O((z-a)^) ~, in a neighbour ...
*
Combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
** Hubbard trees
** Spider algorithm
** Tuning
**
Laminations
**
Devil's Staircase algorithm (Cantor function)
**
Orbit portraits
**
Yoccoz puzzles
Parts
* Holomorphic dynamics (dynamics of
holomorphic functions)
Surveys in Dynamical systems available on-line at Dynamical Systems Homepage of Institute for Mathematical Sciences SUNY at Stony Brook
/ref>
** in one complex variable
** in several complex variables
* Conformal dynamics unites holomorphic dynamics in one complex variable with differentiable dynamics in one real variable.
See also
*Arithmetic dynamics Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is ...
* Chaos theory
* Complex analysis
*Complex quadratic polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.
Properties
Quadratic polynomials have the following properties, regardless of the form:
*It is a unicritical polynomial, i.e. it has on ...
*Fatou set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wit ...
*Infinite compositions of analytic functions
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the ...
*Julia set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
*Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
*Symbolic dynamics In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (e ...
Notes
References
*Alan F. Beardon,
Iteration of Rational Functions: complex analytic dynamical systems
', Springer, 2000,
*Araceli Bonifant, Misha Lyubich, Scott Sutherland (editors),
', Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financia ...
, 2014.
*Daniel S. Alexander,
A History of Complex Dynamics: From Schröder to Fatou and Julia
', Aspect of Mathematics, 1994,
*Lennart Carleson
Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 2006 fo ...
, Theodore W. Gamelin,
Complex Dynamics
', Springer, 1993,
*John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
,
Dynamics in One Complex Variable
' (Third edition), Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financia ...
, 2006
*Shunsuke Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda,
Holomorphic Dynamics
', Cambridge University Press, 2000,
External LinksA Primer on the Elementary Theory of Infinite Compositions of Complex Functions
Emergence
{{mathanalysis-stub