Dynamical systems, iteration and orbits

Let A and B be sets and let f be a functionwith domain A and codomain B. That is, for each x ∈ A there corresponds f(x) ∈ B. We may indicate this situation symbolically by writing f : A → B. In the case when the domain and codomain are equal to a set S say, and f : S→ S, we say that f is a transformation on S. In such a case, we call the pair (S, f) a dynamical system. Thus, a dynamical system consists of a set together with a transformation on the set. Given a dynamical system (S, f), S may be called the phase space and the elements of S may be called states. If f → A → B and g : B → C, the composition of g with f is the function g o f : A → C given by (g o f) (x) = g(f(x) for all x ∈ A.