Let $A$ be an $n \times n$ matrix (let's say hyperbolic, but this might be irrelevant). Consider the action of $A$ on $(\mathbb{R} \backslash \mathbb{Z})^n$. Does this action always have periodic orbits with minimal period being an even number $k\geq2$?
It is easy to exhibit such orbits explicitly in specific cases. E.g. if $n=2$ and $c \neq 0 \ mod \ a+d $, taking the orbit of $(\frac{1}{a+d},0)$. However I've been unable to find a proof in general.
EDIT: the general answer is no, however the question still holds under the assumption that $A$ is a hyperbolic matrix (no eigenvalues of absolute value $1$).
Not in general, no. If $A$ has odd order - that is, if $A^k=I_n$ for some odd number $k$ - then all orbits are periodic with periods dividing $k$, hence all odd. For example, $A=\begin{pmatrix}0&-1\\1&-1\end{pmatrix}$ satisfies $A^3=I_2$, so every orbit has size $3$ (except the orbit of the fixed point $(0,0)$).