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ITERATED FUNCTIONS


Given an Endomorphism ''f'' on a Set ''X''
:f: X o X
a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that
:\ f^n(x) = x
where f^n is the ''n''th Iterate of ''f''. The smallest positive integer ''n'' satisfying the above is called the prime period or '''least period''' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''' Periodic Function ''' with period ''n''.

If ''f'' is a Diffeomorphism of a Differentiable Manifold , so that the Derivative (f^n)^\prime is defined, then one says that a periodic point is hyperbolic if