# Article

# Article

- Mathematics
- Analysis (calculus)
- Complex dynamics

# 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*_{d}*z*^{d} + *a*_{d−1}*z*^{d−1} + ⋅⋅⋅ + *a*_{1}*z* + *a*_{0}, where the coefficients *a*_{i} are all complex numbers, *a*_{d} ≠ 0, and *z* is a complex variable.

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