As the question suggests, I am trying to find an ergodic transformation on $[0,1]$ (equipped with the Borel algebra and the Lebesgue measure) that leaves $\mathbb{Q}\cap[0,1]$ invariant. Any thought is appreciated.
2026-03-27 23:37:51.1774654671
Is the set of rational numbers between $0$ and $1$ invariant under some ergodic transformation
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Consider $$f(x)=\begin{cases}x&\text{if }x\in\Bbb Q+\sqrt 2\Bbb Z\\x+\sqrt 2\bmod 1&\text{otherwise}\end{cases} $$