Definitions
For the purpose of this post, a (topological) dynamical system is a compact metric space $X$ equipped with a homemomorphism $T:X\to X$.
We say that a subset $S$ of $\mathbb Z$ is relatively dense in $\mathbb Z$ if there is a positive integer $N$ such that for all $a\in \mathbb Z$ the set $\{a+1, a+2, \ldots, a+N\}$ has non-empty intersection with $S$.
Let $x$ be a point in a dynamical system $(X, T)$.
$\bullet$ The orbit of $x$ is defined as $O_x=\{T^nx:\ n\in\mathbb Z\}$.
$\bullet$ We say that a point $x\in X$ is almost periodic if for all neighborhoods $U$ of $x$ in $X$, the set $\{n\in \mathbb Z:\ T^nx\in U\}$ is relatively dense in $\mathbb Z$.
$\bullet$ We say $x$ periodic if the orbit of $x$ is finite.
Clearly, any periodic point is almost periodic.
Question 1
Assuming $(X, T)$ is a dynamical system with $|X|=|\mathbb N|$, is it necessary that every almost periodic point is also periodic?
I do not know the answer to the above question. In fact, I do not know any "good" examples of a countable dynamical system. If you are aware of good examples then please feel free to share.
Question 2
Assuming $(X, T)$ is a dynamical system with $|X|=|\mathbb N|$, is it necessary that $X$ has a periodic point?
The answer to this question is in the affirmative.
This is because we know that there is a $T$-invariant probability measure $\mu$ on $X$. Since $X$ is countable, there is a point $x$ in $X$ such that $\mu(x)>0$. Now the orbit of $x$ must be finite, for otherwise, by the $T$-invariance of $\mu$, we would have that $\mu(X)=\infty$.
Can we give an argument with does not go via measure theory and is purely topological in nature?
Best Answer
Here is the answer rewritten to avoid the transfinite induction.
Given a topological space $X$, let $X'$ denote the set of non-isolated points of $X$.
Definition. Given a homeomorphism $T: X\to X$, a point $x\in X$ is called recurrent (with respect to $(X,T)$) if for each neighborhood $U$ of $x$ there exists $n\ge 1$ such that $T^n(x)\in U$.
This condition is weaker than almost periodic.
Lemma 1. Let $X$ be a countable compact metrizable space, $T: X\to X$ a homeomorphism. Then every recurrent point $x\in X$ is periodic.
Proof. Consider the collection ${\mathcal I}_x$ of all compact $T$-invariant subsets of $X$ containing $x$. Let $C_x$ denote the intersection of all members of ${\mathcal I}_x$. Clearly, $C_x\in {\mathcal I}_x$. I claim that $x$ is an isolated point of $C_x$. Indeed, since $C_x$ is countable and compact metrizable, it has some isolated points, $C'_x\ne C_x$. If $x\in C'_x$ then $C'_x\in {\mathcal I}_x$ and $C'_x$ is strictly smaller than $C_x$, which is a contradiction. Hence, $x$ is isolated in $C_x$. The point $x$ is still recurrent with respect to $(C_x,T)$. Since $x$ is isolated in $C_x$, $\{x\}$ is a neighborhood of $x$ in $C_x$. Hence, by recurrence, there exists $n\ge 1$ such that $T^n(x)\in \{x\}$, i.e. $T^n(x)=x$, i.e. $x$ is $T$-periodic. qed
This answers Question 1. To answer Question 2, I will prove a stronger result:
Lemma 2. Let $X$ be a nonempty compact metrizable topological space, $T: X\to X$ is a homeomorphism. Then $X$ contains recurrent points. Equivalently, $X$ contains a $T$-invariant compact nonempty subset $X_0$ such that every $T$-orbit in $X_0$ is dense in $X_0$.
Proof. Consider the poset ${\mathcal I}$ of all nonempty $T$-invariant compact subsets of $X$ (with the partial order given by inclusion). Clearly, the intersection of members of each totally ordered (nonempty) subset in ${\mathcal I}$ belongs to ${\mathcal I}$. Hence, by Zorn's Lemma, ${\mathcal I}$ contains a minimal element $X_0$. By the minimality, every $T$-orbit in $X_0$ is dense (otherwise, take the closure of a non-dense $T$-orbit in $X_0$). qed
Combining the two lemmata, we see that if $X$ is countable, compact, metrizable, nonempty, then for each homeomorphism $T: X\to X$, there exists a $T$-periodic point.