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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.

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$f^{n}(x) = n a + x \mod 1$, 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.

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  • $\begingroup$ 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? $\endgroup$ – Peanutlex Feb 5 at 20:48
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    $\begingroup$ It doesn't mean that, but it's true. $\endgroup$ – Robert Israel Feb 5 at 21:15
  • $\begingroup$ How could we show that it's true if you don't mind me asking? $\endgroup$ – Peanutlex Feb 5 at 21:26
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    $\begingroup$ E.g. it follows from the equidistribution theorem. $\endgroup$ – Robert Israel Feb 5 at 21:48
  • $\begingroup$ 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? $\endgroup$ – Peanutlex Feb 6 at 9:55

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