# Finding Counterexamples for Poincaré's Recurrence Theorem (Dropped Hypothesis)

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.

• Fix the conclusion in the statement of the theorem: it's not that almost every point is "Borel".
– KCd
Jan 20, 2021 at 18:58
• 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.
– KCd
Jan 20, 2021 at 19:06
• @KCd I see, thanks! The point being "Borel" was a bad typo haha. Jan 20, 2021 at 21:25
• @user56202 You should update the question in light of KCd's comment, or add an answer. Or delete the question. Jan 29, 2021 at 6:10