Show that if $T_A : \mathbb{T}^n \rightarrow \mathbb{T}^n$ is an endomorphism of the torus and $x \in \mathbb{Q}^n \backslash \mathbb{Z}^n$, then there exists an $m \in \mathbb{N}$ such $T_A^m (x)$ is periodic for $T_A$. The matrix $A$ that induced $T_A$ has entries in $\mathbb{Z}$.
This is a question posed in a book about Dynamical Systems Theory. And I know how to solve the problem for the case in which $T_A$ is and automorphism, but in this case this is all I have:
If there exists an $m$-periodic $x$, then $x$ is also $mk$-periodic, with $k \in \mathbb{N}$. This implies the two following equations:
$[A^m x] = [x] \Leftrightarrow A^m x - x = p_1 \in \mathbb{Z}^n$
$[A^{mk} x] = [x] \Leftrightarrow A^{mk} x - x = p_2 \in \mathbb{Z}^n$
Subtracting the first equation to the second yields:
$A^{mk} x - A^m x = (A^{mk} - A^m)x = p_2 - p_1 = p \in \mathbb{Z}^n$
But how can I find $m$ such that the result above is true?