Periodic points of complex quadratic mappings

This article describes periodic point

Periodic point

In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.- Iterated functions :...

s of some complex quadratic map

Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.-Forms:When the quadratic polynomial has only one variable , one can distinguish its 4 main forms:...

s. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic

Quadratic equation

In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

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 is a complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

. A periodic point

Periodic point

of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.

This theory is applied in relation with the theories of Fatou and Julia set

Julia set

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...

s.
Let

where

and

are complex-valued

Complex number