Complex dynamics

Article By:

Koch, Sarah Department of Mathematics, Harvard University, Cambridge, Massachusetts.

Last reviewed:2014

DOI:https://doi.org/10.1036/1097-8542.YB140338

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) = a_dz^d + a_d−1z^d−1 + ⋅⋅⋅ + a₁z + a₀, where the coefficients a_i are all complex numbers, a_d ≠ 0, and z is a complex variable.

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