Editing Complex dynamics (section)

==Dynamics in complex dimension 1==
{{Main|Julia set}}
A simple example that shows some of the main issues in complex dynamics is the mapping <math>f(z)=z^2</math> from the complex numbers '''C''' to itself. It is helpful to view this as a map from the [[complex projective line]] <math>\mathbf{CP}^1</math> to itself, by adding a point <math>\infty</math> to the complex numbers. (<math>\mathbf{CP}^1</math> has the advantage of being [[compact space|compact]].) The basic question is: given a point <math>z</math> in <math>\mathbf{CP}^1</math>, how does its ''orbit'' (or ''forward orbit'')
:<math>z,\; f(z)=z^2,\; f(f(z))=z^4, f(f(f(z)))=z^8,\; \ldots </math>
behave, qualitatively? The answer is: if the [[absolute value#Complex numbers|absolute value]] |''z''| is less than 1, then the orbit converges to 0, in fact more than [[exponential decay|exponentially]] fast. If |''z''| is greater than 1, then the orbit converges to the point <math>\infty</math> in <math>\mathbf{CP}^1</math>, again more than exponentially fast. (Here 0 and <math>\infty</math> are ''superattracting'' [[fixed point (mathematics)|fixed point]]s 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 <math>|z|=1</math>, 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 set|dense]] in the circle, and in fact [[equidistributed sequence|uniformly distributed]] on the circle. There are also infinitely many [[periodic point]]s on the circle, meaning points with <math>f^r(z)=z</math> for some positive integer ''r''. (Here <math>f^r(z)</math> means the result of applying ''f'' to ''z'' ''r'' times, <math>f(f(\cdots(f(z))\cdots))</math>.) 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 <math>f^r(z)=z</math>, the derivative of <math>f^r</math> at ''z'' has absolute value greater than 1.)

[[Pierre Fatou]] and [[Gaston Julia]] showed in the late 1910s that much of this story extends to any complex algebraic map from <math>\mathbf{CP}^1</math> to itself of [[degree of a continuous mapping|degree]] greater than 1. (Such a mapping may be given by a polynomial <math>f(z)</math> with complex coefficients, or more generally by a rational function.) Namely, there is always a compact subset of <math>\mathbf{CP}^1</math>, the '''[[Julia set]]''', on which the dynamics of ''f'' is chaotic. For the mapping <math>f(z)=z^2</math>, the Julia set is the unit circle. For other polynomial mappings, the Julia set is often highly irregular, for example a [[fractal]] in the sense that its [[Hausdorff dimension]] is not an integer. This occurs even for mappings as simple as <math>f(z)=z^2+c</math> for a constant <math>c\in\mathbf{C}</math>. The [[Mandelbrot set]] is the set of complex numbers ''c'' such that the Julia set of <math>f(z)=z^2+c</math> is [[connected space|connected]].
[[File:Parabolic Julia set for internal angle 1 over 3.png|thumb|The Julia set of the polynomial <math>f(z)=z^2+az</math> with <math>a\doteq -0.5+0.866i</math>.]]
[[File:Julia set (Rev formula 02).jpg|thumb|The Julia set of the polynomial <math>f(z)=z^2+c</math> with <math>c\doteq 0.383-0.0745i</math>. This is a [[Cantor set]].]]

There is a rather complete [[classification of Fatou components|classification of the possible dynamics]] of a rational function <math>f\colon\mathbf{CP}^1\to \mathbf{CP}^1</math> in the '''Fatou set''', the complement of the Julia set, where the dynamics is "tame". Namely, [[Dennis Sullivan]] showed that each [[connected component (topology)|connected component]] ''U'' of the Fatou set is pre-periodic, meaning that there are natural numbers <math>a<b</math> such that <math>f^a(U)=f^b(U)</math>. Therefore, to analyze the dynamics on a component ''U'', one can assume after replacing ''f'' by an iterate that <math>f(U)=U</math>. Then either (1) ''U'' contains an attracting fixed point for ''f''; (2) ''U'' is ''parabolic'' in the sense that all points in ''U'' approach a fixed point in the boundary of ''U''; (3) ''U'' is a [[Siegel disk]], meaning that the action of ''f'' on ''U'' is conjugate to an irrational rotation of the open unit disk; or (4) ''U'' is a [[Herman ring]], meaning that the action of ''f'' on ''U'' is conjugate to an irrational rotation of an open [[annulus (mathematics)|annulus]].<ref>Milnor (2006), section 13.</ref> (Note that the "backward orbit" of a point ''z'' in ''U'', the set of points in <math>\mathbf{CP}^1</math> that map to ''z'' under some iterate of ''f'', need not be contained in ''U''.)