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

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).
