Periodic Point - Iterated Functions

Iterated Functions

Given an endomorphism f on a set X

a point x in X is called periodic point if there exists an n so that

where is the nth 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 with period n.

If f is a diffeomorphism of a differentiable manifold, so that the derivative is defined, then one says that a periodic point is hyperbolic if

and that it is attractive if

and it is repelling if

If the dimension of the stable manifold of a periodic point or fixed point is zero, the point is called a source; if the dimension of its unstable manifold is zero, it is called a sink; and if both the stable and unstable manifold have nonzero dimension, it is called a saddle or saddle point.