# For what values does my dynamical system produce periodic orbits?

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.

$$f^{n}(x) = n a + x \mod 1$$, 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.
• Does that mean that the set: $$\lim_{k\rightarrow\infty}\left\{x,f(x),f^2(x),...,f^k(x)\right\}$$ is the real number line, [0,1], if $a$ is irrational? – Peanutlex Feb 5 at 20:48
• But if: $$\lim_{k\rightarrow\infty}\left\{x_0,f(x_0),f^2(x_0),...,f^k(x_0)\right\}=[0,1],$$ then wouldn't that make the set $[0,1]$ a countable infinite set so it cannot be true? – Peanutlex Feb 6 at 9:55