Provinf Orbits are eventually fixed or eventually prime-2-periodic

Question

Please I need help with this question:

Let $S = \{a,b,c,d\}$ be a finite set and suppose that $f \colon S \to S$ has the property that: $$f(a) = f(b) = f(c) = d$$ Prove that each of the orbits of the dynamical system $(S, f)$ is either eventually fixed or eventually prime-2-periodic.

Answer 1

Indeed:

• if $f(d)=d$, then all orbits are eventually fixed (all will go to $d$);

• if $f(d)\ne d$, then all orbits have eventually period $2$ (draw a graph).