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I want to check the ergodicity of irrational rotation map without using Fourier series. $T:\frac{\mathbb{R}}{\mathbb{Z}} \to \frac{\mathbb{R}}{\mathbb{Z}}$ which $ x \mapsto x+q$ , $\mod \ 1$ where $q \in \mathbb{Q}^c$.

Let $\mathcal{B}$ be the Borel-$\sigma$- algebra and $\mu(B) = \mu(a+B)$ , $B \in \mathcal{B}$ where $a+B=\{a+b | b \in B \}$

I wanna check the ergodicity by using the fact that for every $A,B$ in the Borel sigma algebra we have :

$\lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} \mu(T^{-i}A \cap B)=\mu(A)\cdot\mu(B)$

or

$\forall B$ in the Borel sigma algebra where $\mu(B) > 0$ : $\mu(\bigcup T^{-i}B)=1$

Any help ?

Kernel
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Reza Yaghmaeian
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  • Does this answer your question? Ergodicity of irrational rotation – Kernel May 11 '21 at 07:26
  • No I want to use the above theorem to prove this. I mean : $\lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n-1} \mu(T^{-i}A \cap B)=\mu(A)\cdot\mu(B)$

    or

    $\forall B$ in the Borel sigma algebra where $\mu(B) > 0$ : $\mu(\bigcup T^{-i}B)=1$

     – Reza Yaghmaeian May 11 '21 at 08:03
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    It's usually not possible or very difficult to prove a system is ergodic directly from the definition you want to use. The proof in the linked answer is the most standard proof of this fact that doesn't use Fourier analysis – Adam May 14 '21 at 01:09

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