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Complex dynamics. Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by 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.
Dynamical system. The Lorenz attractor arises in the study of the Lorenz oscillator, a 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, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock ...
Definition. A topological space is called countably generated if for every subset , is closed in whenever for each countable subspace of the set is closed in .Equivalently, is countably generated if and only if the closure of any equals the union of closures of all countable subsets of .
This definition can be extended to real x and y, y ≠ 0, by the formula = ⌊ ⌋. Then it follows from the definition of floor function that this extended operation satisfies many natural properties. Notably, x mod y is always between 0 and y, i.e., if y is positive,
As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. For instance, a value of c belongs to the Mandelbrot set if and only if the corresponding Julia set is connected.
Glossary of mathematical symbols. A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various ...
External ray. An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. [1] Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External rays are used in complex analysis, particularly in complex dynamics and geometric function theory .
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, [1] algebra, [2] geometry, [1] and analysis, [3 ...