A discrete dynamical system (I, f) can have infinity many fixed points: one can, for example, to take f as the function f(x) = x.

On the contrary, the discrete dynamical system ([1, 1], f) whit f(x) =x^2 has a unique fixed point. Periodic points with a finite orbit are not necessarily periodic points. For example, given the discrete dynamical system (R, f) with f(x) = |x 1|, the orbit of the point 2 is the set {2, 1, 0}.

Let (X, f) be a discrete dynamical system. A point x X is called eventually fixed if there is a natural number N such that f^n+1(x) = f^n(x) whenever n N. The point x is called eventually periodic of period k if there is a natural number N such that f^n+k(x) = f^n(x) whenever n N.

Question: Prove that a point x is a periodic point of period k of the discrete dynamical system (X, f ) if and only if x is a fixed point of f k and it is not a fixed point off^m for all m < k

Hint: Notice that periodic of period K is equivalent to be a fixed point of the iterate k.