# Iterating Linear Functions

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Sequences of iterates of nonlinear functions can behave in complicated ways as can be seen from a variety of Demonstrations (see Related Links).

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Contributed by: Chris Boucher (March 2011)

Open content licensed under CC BY-NC-SA

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## Details

Given a function and an initial value , the sequence of iterates of is the sequence defined recursively by . If , then . If is smaller than one in absolute value, then clearly , which is the solution to the equation . If >1, then the sequence of iterates diverges to infinity or minus infinity depending on the sign of ; that is, depending on which side of the fixed point the sequence of iterates starts. If , then the sequence of iterates alternates between values to the left and right of the fixed point whose distance from the fixed point grows without bound.

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