Periodic solution of a difference equation

Remarks

Example: for $x_{n+2} = x_{n+1} / x_n$ on $S = (0, \infty)$ with $x_0 = a$ and $x_1 = b$, the solution is $a, b, b/a, 1/a, 1/b, a/b, a, b, \dots$, so every solution is periodic with period $6$.

Non-example: for $x_{n+1} = 2 x_n$ on $\R$ with $x_0 = 1$, the solution $x_n = 2^n$ is strictly increasing, so $x_{n+p} \ne x_n$ for all $p \ge 1$ and it is not periodic.

For $k = 1$ and $F = f$, the solution starting at $x_0$ is periodic with period $p$ if and only if $f^p(x_0) = x_0$, that is, $x_0$ is a periodic point of $f$. A periodic solution of period $1$ is an equilibrium. For a non-autonomous equation, one asks for periodic solutions when the equation itself is periodic in $n$.