Complex Dynamics
Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by Iterated function, iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is iterated. In geometric terms, that amounts to iterating a mapping from some algebraic variety to itself. The related theory of arithmetic dynamics studies iteration over the rational numbers or the p-adic numbers instead of the complex numbers. Dynamics in complex dimension 1 A simple example that shows some of the main issues in complex dynamics is the mapping f(z)=z^2 from the complex numbers C to itself. It is helpful to view this as a map from the complex projective line \mathbf^1 to itself, by adding a point \infty to the complex numbers. (\mathbf^1 has the advantage of being compact space, compact.) The basic question is: given a point z in \mathbf^1, how does its ''orbit'' (or ''forward orbit'') :z,\; f(z)=z^2,\; f(f(z))=z^4, f(f(f(z ...
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Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ...
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Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X eith ...
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Classification Of Fatou Components
In mathematics, Fatou components are connected component (analysis), components of the Fatou set. They were named after Pierre Fatou. Rational case If f is a rational function :f = \frac defined in the extended complex plane, and if it is a nonlinear function (degree > 1) : d(f) = \max(\deg(P),\, \deg(Q))\geq 2, then for a periodic connected component (analysis), component U of the Fatou set, exactly one of the following holds: # U contains an attracting periodic point # U is parabolic # U is a Siegel disc: a simply connected Fatou component on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle. # U is a Herman ring: a double connected Fatou component (an Annulus (mathematics), annulus) on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle. File:Julia-set_N_z3-1.png, Julia set (white) and Fatou set (dark red/green/blue) fo ...
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Julia Set (Rev Formula 02)
In complex dynamics, 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 with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is " chaotic". The Julia set of a function    is commonly denoted \operatorname(f), and the Fatou set is denoted \operatorname(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Formal definition Let f(z) be a non-constant meromorphic function from the Riemann sphere onto itself. Such functions f(z) are ...
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Parabolic Julia Set For Internal Angle 1 Over 3
Parabolic usually refers to something in a shape of a parabola, but may also refer to a parable. Parabolic may refer to: *In mathematics: **In elementary mathematics, especially elementary geometry: **Parabolic coordinates **Parabolic cylindrical coordinates ** parabolic Möbius transformation ** Parabolic geometry (other) ** Parabolic spiral ** Parabolic line **In advanced mathematics: *** Parabolic cylinder function ***Parabolic induction ***Parabolic Lie algebra ***Parabolic partial differential equation A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, for example, engineering science, quantum mechanics and financial ma ...

Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. So ...
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Mandelbrot Set
The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting at z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Images of the Mandelbrot set exhibit an infinitely complicated Boundary (topology), boundary that reveals progressively ever-finer Recursion, recursive detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a ''fractal curve''. The "style" of this recursive detail depends on the region of the set boundary being ...
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Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ...
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Fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine geometry, affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they Scaling (geometry), scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). ...
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Julia Set
In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under iterated function, repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small Perturbation theory, perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaos theory, chaotic". The Julia set of a function    is commonly denoted \operatorname(f), and the Fatou set is denoted \operatorname(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Form ...
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Degree Of A Continuous Mapping
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations. The degree of a map between general manifolds was first defined by Brouwer, who showed that the degree is homotopy invariant and used it to prove the Brouwer fixed point theorem. Less general forms of the concept existed before Brouwer, such as the winding number and the Kronecker characteristic (or Kronecker integral). In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number. Definitions of the degree From ''S''''n'' to ''S''''n'' The simplest and most important ca ...
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Gaston Julia
Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French mathematician who devised the formula for the Julia set. His works were popularized by Benoit Mandelbrot; the Julia and Mandelbrot fractals are closely related. He founded, independently with Pierre Fatou, the modern theory of holomorphic dynamics. Military service Julia was born in the Algerian town of Sidi Bel Abbes, at the time governed by the French. During his youth, he had an interest in mathematics and music. His studies were interrupted at the age of 21, when France became involved in World War I and Julia was conscripted to serve with the army. During an attack he suffered a severe injury, losing his nose. His many operations to remedy the situation were all unsuccessful, and for the rest of his life he resigned himself to wearing a leather strap around the area where his nose had been. Career in mathematics Julia gained attention for his mathematical work at the age of 25, in 1918, when his 199-pag ...
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