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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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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