For a map $f\colon X\to X$, a point $p\in X$ is called non-wandering, provided for every neighborhood $U$ of $p$ there exists an integer $n>0$ such that $f^n(U)\cap U\neq\emptyset$.

Sometimes, I also read the following definition:

For either maps or flows a point $p\in X$ is said to be a nonwandering point, if given any neighborhood $U$ of $p$ there exists a sequence of times $t_n\to\infty$ such that $f^{t_n}(p)\in U$ for all $n$. Hence, non-wandering points are points whose neighborhoods the map visits infinitely often.

Are these definitions equivalent? Does the first definition also imply that each neighborhood of $p$ is visited infinitely often?


The second definition is dubious; I would call such a point recurrent instead. They are not equivalent: for example, take $f:[0,1[\to[0,1[$, $f(x)=2x \pmod 1$. Then $x=1/2$ is a nonwandering point (in fact the nonwandering set is $[0,1[$ itself), but it's certainly not recurrent in the previous sense since $f^n(x)=0$ for all $n>0$.

And yes, in the first definition, you can ask $n$ to be arbitrary large, at least for metric space. Assuming it's not, notice that $$N(U)=\sup \{ n>0 \, : \, f^{-n}(U)\cap U\neq \emptyset \},$$ is decreasing as the neighborhood $U$ decreases. Thus, you have a countable basis $(U_k)_k$ of decreasing neighborhood of $p$, $N(U_k)$ is a decreasing sequence of positive integers, so is constant, equal to some $N$, for $k$ sufficiently large.

Now consider a point $x_k\in f^{-N}(U_k)\cap U_k$, which exists since it's a non-empty set. The sequence $(x_k)_k$ must converge to $p$, but so does $f^N(x_k)$ since these are elements of $U_k$. By continuity, $f^N(p)=p$ so $p$ is periodic, and clearly periodic points come back infinitely often (contradicting the assumption).


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