Iterating Linear Functions

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.