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Rotation $x \to x+a \pmod 1$ of the circle is Ergodic if and only if $a$ is irrational

I have a book, Ergodic problems of classical mechanics by Arnold/Avez, and in it they prove that rotation $Tx = x+a \pmod 1$ of the circle $M=\{x \pmod 1\}$ is Ergodic if and only if a is irrational. ...