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I have the following two questions that I found somewhere and I would like to find their answers: I am using regular terminology:

How to prove the following claims:

  1. if $\omega(x)$ is a finite set then it has only one period.
  2. If $\omega(x)$ is a countable set such that $X$ is the set where $f:X\to X$ which we are making the iterations over it is a metric and complete space (i.e Baire theorem follows) then $\omega(x)$ contains at least one period.

Just for those who do not know what is a period we have: $f^p(x)=x$ for some $p \in \mathbb{Z}$.

Appreciate for your advice or answers, references etc...

for those unacquianted $\omega$ limit is the set: $\omega(x):=\{y\in X : \exists n_k\to \infty f^{(n_k)}(x)\to y \}$.

MathematicalPhysicist
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  • Is an irrational rotation on the unit circle a counter-example? Every orbit is dense, thus the limit set the full circle, but no periodic orbit exists. – Lutz Lehmann May 21 '21 at 15:27
  • @LutzLehmann is this a counterexample to the second of first question? – MathematicalPhysicist May 21 '21 at 17:46
  • The second. As to the first, depending on how you define period, every multiple of a period is again a period. Did you want to express in the first question that the orbit is irreducible? – Lutz Lehmann May 21 '21 at 18:01
  • @LutzLehmann yes every multiple of a period is again a period, – MathematicalPhysicist May 22 '21 at 05:16
  • @LutzLehmann as for the second the lecturer wasn't sure this claim is correct. – MathematicalPhysicist May 22 '21 at 06:25
  • Irrational rotation means the map $x\mapsto x+\alpha\mod 1$, $\alpha\notin\Bbb Q$, which can be transformed to a map on the unit circle by associating $x$ with the angle $2\pi x$. Under slight perturbations of $\alpha$ it becomes rational, so the nearby maps have finite periods. But the irrational rotation itself has dense orbits from any point. – Lutz Lehmann May 22 '21 at 07:10
  • So the second question is blatant wrong? – MathematicalPhysicist May 22 '21 at 07:24
  • As stated, yes. There might be the idea of something like the chaotic mode of the logistic map behind it, where the generic orbit is dense, but unstable cyclic orbits for all periods exist. – Lutz Lehmann May 22 '21 at 07:29

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