Complex dynamics

# Complex dynamics

Discussion

Encyclopedia
Complex dynamics is the study of dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s defined by iteration
Iterated function
In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...

of functions on complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic function
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, categories that are similar in some ways, but different in others...

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## Techniques

• General
• Montel's theorem
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 Paul Montel, and give conditions under which a family of holomorphic functions is normal....

• Poincaré metric
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 or Riemann surfaces.There are three equivalent...

• Schwarz lemma
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 stronger theorems, such as the Riemann mapping theorem, which it helps to prove...

• Riemann mapping theorem
• Carathéodory's theorem (conformal mapping)
Carathéodory's theorem (conformal mapping)
In mathematical complex analysis, Carathéodory's theorem, proved by , states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann map...

• Combinatorial
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

• Hubbard trees
• The Spider algorithm
• Tuning
• Laminations
Lamination (topology)
In topology, a branch of mathematics, a lamination is a :* "A topological space partitioned into subsets"* decoration of a manifold in which some subset of the manifold is partitioned into sheets of some lower dimension, and the sheets are locally parallel.Lamination of surface is a partition of...

• Devil's Staircase algorithm
Cantor function
In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Devil's staircase.-Definition:See figure...

• Orbit portrait
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 :...

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## Parts

• Holomorphic dynamics ( dynamics of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

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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.

• Complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

• John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...

A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.-Forms:When the quadratic polynomial has only one variable , one can distinguish its 4 main forms:...

• Fatou set
• Julia set
Julia set
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...

• Mandelbrot set
Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...

• Chaos Theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...