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I have just learned the Poincaré Recurrence Theorem and I would like to understand why each hypothesis is necessary. To this end, I am dropping one hypothesis at a time, and seeing why the theorem fails. I have found counterexample for dropping some of the hypotheses, but not all. Could you please help me with the rest?

This is the version of the theorem I have

Theorem (Poincare Recurrence): Let $f:X \to X$ where

  1. $X$ is a compact metric space
  2. $f$ is an m.p.t. (measure preserving transformation)
  3. $f$ is continuous
  4. The $\sigma$-algebra on $X$ is Borel

Then almost every point of $X$ is recurrent.

I have counterexamples for dropping $1$ and $2$, but not for $3$ and $4$.

Counterexample for 1:

$f(x) = x+1$ on $\mathbb{R}$ with Lebesgue measure. For any $x$, $f^n(x) \to \infty$, so there are no recurrent points.

Counterexample for 2: $f(x) = \frac 12 x$ on $[0, 1]$ with Lebesgue measure. Then for every $x$, $f^n(x) \to 0$, so only $x = 0$ is recurrent.

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  • $\begingroup$ Fix the conclusion in the statement of the theorem: it's not that almost every point is "Borel". $\endgroup$
    – KCd
    Jan 20, 2021 at 18:58
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    $\begingroup$ You don't need 3 or 4 in order for the theorem to hold, and even 1 can be weakened. All you need for the theorem to hold is a measure space $(X,\mu)$ of finite measure and an m.p.t. $f \colon X \to X$. See Theorem 1 on the Wikipedia page for the Poincare recurrence theorem. $\endgroup$
    – KCd
    Jan 20, 2021 at 19:06
  • $\begingroup$ @KCd I see, thanks! The point being "Borel" was a bad typo haha. $\endgroup$
    – user56202
    Jan 20, 2021 at 21:25
  • $\begingroup$ @user56202 You should update the question in light of KCd's comment, or add an answer. Or delete the question. $\endgroup$ Jan 29, 2021 at 6:10

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