For what values does my dynamical system produce periodic orbits?

Question

For what values of $a$ does the function, $f$, contain periodic orbits, where $f$ is given by: $$f(x)=a+x \mod 1.$$ It seems for any rational number $a$ you get periodic orbits although I don't know how to prove that. Does anyone know if you get periodic orbits if $a$ is irrational? I don't really know anything about dynamical systems so help in approaching this problem would be appreciated.

Answer 1

$f^{n}(x) = n a + x \mod 1$, where $f^{n}$ is $f$ iterated $n$ times. This is $x$ if and only if $n a$ is an integer, so $a$ is an integer divided by $n$, i.e. a rational number.