Complex dynamics studies iteration of holomorphic and rational maps, especially the division of the Riemann sphere into stable Fatou behavior and chaotic Julia behavior.
A family of holomorphic or meromorphic maps is normal when every sequence has a subsequence converging locally uniformly in the spherical metric, possibly to infinity.
A family of meromorphic functions on a domain that omits three fixed points of the Riemann sphere is normal. For holomorphic functions with values in the complex plane, omitting two fixed values suffices.
The Fatou set of a rational map is the largest open subset of the Riemann sphere on which its iterates form a normal family. Its complement is the Julia set.
A periodic Fatou component of a rational map is an immediate attracting basin, an immediate parabolic basin, a Siegel disc, or a Herman ring. Attracting and parabolic basins capture critical orbits, while the boundaries of rotation domains lie in the closure of the postcritical set.
Every immediate basin of an attracting fixed point of a rational map contains a critical point. Otherwise the map on the basin would be an unbranched covering; lifting it to the unit disc would give an automorphism fixing a point, contradicting strict contraction at the attracting fixed point.
A parabolic basin consists of points whose iterates converge to a parabolic cycle along a specified attracting direction. Each connected basin component lies in the Fatou set, and its boundary lies in the Julia set.
Every Fatou component of a rational map of degree at least two is eventually periodic. Thus the periodic-component classification accounts for the entire Fatou set.
The Julia set is the complement of the Fatou set. It is nonempty, closed, completely invariant, and perfect for every rational map of degree at least two.
If a closed set satisfies for a rational map of degree at least two, then either has at most two points and lies in the Fatou set, or . The complement omits every point of , so Montel theorem proves normality when .
The accumulation-point set of is closed and completely invariant. Applying the classification of completely invariant closed sets shows that it equals ; hence the Julia set is perfect.
For every , choose a th root of unity with , put , and setIts only critical points are and infinity, and both land on the repelling fixed point . The postcritically finite map has no possible Fatou component, so its Julia set is the whole sphere.
The degree- Chebyshev polynomial has Julia set the interval after the standard normalization. Its Fatou set is the connected complement of that interval in the Riemann sphere, so it has exactly one Fatou component.
Near a superattracting fixed point, a holomorphic map is conformally conjugate to its leading monomial.
For a degree- polynomial , the Böttcher coordinate near infinity satisfiesIts modulus extends naturally throughout the basin of infinity.
The escape-rate Green function isIt vanishes on the filled Julia set, is positive and harmonic on the basin of infinity, satisfies , and equals near infinity and by continuation throughout the basin.
The Julia set of a polynomial is connected exactly when every finite critical point belongs to the filled Julia set. In this case the Böttcher coordinate extends conformally over the entire basin of infinity; an escaping critical point is precisely an obstruction to that continuation.
For outside the Mandelbrot set, evaluate the dynamical Böttcher coordinate at the critical value:This parameter Böttcher map is a conformal isomorphism from the complement of the Mandelbrot set to the exterior unit disc, proving that the Mandelbrot set is full and connected.
The holomorphic fixed-point index isAt a simple fixed point of multiplier , it equals . The indices of all fixed points of a rational map, counted appropriately, sum to one.
If a quadratic rational map has three distinct fixed points with multipliers , then for . Otherwise the two corresponding indices already sum to one, contradicting the nonzero third index.
A parabolic cycle is a periodic orbit whose multiplier is a root of unity; an iterate then has multiplier one and attracting petals.
A hyperbolic component in a parameter space consists of maps with a specified attracting cycle. Its multiplier gives a holomorphic coordinate on the component in the quadratic family, and root-of-unity boundary multipliers give parabolic parameters.
For every prime , the polynomial has a nonzero root, giving a superattracting cycle of exact period . Moving in its hyperbolic component to a boundary point with cycle multiplier gives a parabolic cycle of exact period . Thus the set of parabolic periods in the quadratic family is infinite.
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Complex dynamics is a branch of mathematics that studies the behavior of dynamical systems in the context of complex numbers. It typically involves the iteration of complex functions, particularly polynomials and rational functions, and explores the patterns and structures that emerge from these iterations. Key concepts in complex dynamics include: 1. **Iteration**: Complex dynamics often focuses on iterating a function, meaning applying the function repeatedly.