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 i ...
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 holomorphi ...
**
Poincaré metric
**
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)
**
Böttcher's equation
*
Combinatorial
** Hubbard trees
** Spider algorithm
** Tuning
**
Laminations
**
Devil's Staircase algorithm (Cantor function)
**
Orbit portrait
In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps.
In simple words one can say that it is :
* a list of external angles for which rays lan ...
s
**
Yoccoz puzzles
Parts
* Holomorphic dynamics (dynamics of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s)
Surveys in Dynamical systems available on-line at Dynamical Systems Homepage of Institute for Mathematical Sciences SUNY at Stony Brook
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** 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
*Chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to hav ...
*Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
* Complex quadratic polynomial
* Fatou set
*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
*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
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 financial ...
, 2014.
*Daniel S. Alexander,
A History of Complex Dynamics: From Schröder to Fatou and Julia
', Aspect of Mathematics, 1994,
* Lennart Carleson, Theodore W. Gamelin,
Complex Dynamics
', Springer, 1993,
* John Milnor,
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 financial ...
, 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
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