Recurrent point

Remarks

Equivalent form: $x$ is recurrent if and only if $x \in \omega(x)$, where $\omega(x)$ is the $\omega$-limit set of $x$, the set of limits of sequences $f^{n_k}(x)$ with $n_k \to \infty$. The same condition $x \in \omega(x)$ defines recurrence on a general topological space. Every point on a periodic orbit is recurrent: if $f^p(x) = x$, take $n_k = kp$. Every point of a minimal set is recurrent. Example: for $f(x) = x + \alpha \bmod 1$ on the circle with $\alpha$ irrational, Dirichlet's approximation theorem gives infinitely many $n$ with $n\alpha$ within $1/n$ of an integer, so $f^{n}(x) \to x$ along these $n$. Non-example: for $f(x) = x/2$ on $\R$, the point $x = 1$ has $f^n(1) = 2^{-n} \to 0$, so no subsequence tends to $1$.