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Concept
Complex dynamics
Summary
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. The related theory of arithmetic dynamics studies iteration over the rational numbers or the p-adic numbers instead of the complex numbers.
Julia set
A simple example that shows some of the main issues in complex dynamics is the mapping from the complex numbers C to itself. It is helpful to view this as a map from the complex projective line to itself, by adding a point to the complex numbers. ( has the advantage of being compact.) The basic question is: given a point in , how does its orbit (or forward orbit)
behave, qualitatively? The answer is: if the absolute value |z| is less than 1, then the orbit converges to 0, in fact more than exponentially fast. If |z| is greater than 1, then the orbit converges to the point in , again more than exponentially fast. (Here 0 and are superattracting fixed points of f, meaning that the derivative of f is zero at those points. An attracting fixed point means one where the derivative of f has absolute value less than 1.)
On the other hand, suppose that , meaning that z is on the unit circle in C. At these points, the dynamics of f is chaotic, in various ways. For example, for almost all points z on the circle in terms of measure theory, the forward orbit of z is dense in the circle, and in fact uniformly distributed on the circle. There are also infinitely many periodic points on the circle, meaning points with for some positive integer r. (Here means the result of applying f to z r times, .) Even at periodic points z on the circle, the dynamics of f can be considered chaotic, since points near z diverge exponentially fast from z upon iterating f. (The periodic points of f on the unit circle are repelling: if , the derivative of at z has absolute value greater than 1.
Official source
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