Koch, Sarah Department of Mathematics, Harvard University, Cambridge, Massachusetts.
- Quadratic polynomials
- Links to Primary Literature
- Additional Readings
Complex dynamics is a field of mathematics in which one studies dynamical systems on mathematical spaces, where these spaces are defined over the complex numbers C. More generally, a dynamical system is a pair of objects (X, f), where X is a space (for example, X might be the real line R or the complex plane C), and f is a transformation of the space X to itself. That is, f is a map (written f : X → X) which takes a point x ∊ X as input, and returns the point f(x) ∊ X as output [the mathematical symbols “x ∊ X” mean that the point x is an element of the set X, and similarly for “f(x) ∊ X”]. In this setting, one can iterate the map f on the space X and study the behavior of subsequent iterates. In this article, we will consider a specific type of complex dynamical system: the space X will be the complex plane C, and the transformation f : C → C will be a complex polynomial; that is, f(z) = adzd + ad−1zd−1 + ⋅⋅⋅ + a1z + a0, where the coefficients ai are all complex numbers, ad ≠ 0, and z is a complex variable.
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