Periodic points of complex quadratic mappings

This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.

These periodic points play a role in the theories of Fatou and Julia sets.

Let

be the complex quadric mapping, where ${\displaystyle z}$ and ${\displaystyle c}$ are complex-valued.

Notationally, ${\displaystyle \ f_{c}^{(k)}(z)}$ is the ${\displaystyle \ k}$ -fold composition of ${\displaystyle f_{c}\,}$ with itself—that is, the value after the k-th iteration of function ${\displaystyle f_{c}.\,}$ Thus

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