I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an integer column vector $v \in \mathbb{Z}^n$ which is not equal to the zero vector, I want to know about the asymptotics of the sequence of vector norms $| M^j v|$ for positive integers $j$, with particular emphasis on boundedness and polynomial growth.

One can deduce some things about this using the Jordan decomposition for the action of $M$ on $\mathbb{C}^n$. For example, here is a nice, precise characterization of boundedness. The following are equivalent:

- $| M^j v|$ is bounded
- The vector $v$, regarded as a complex vector, is contained in the direct sum of the eigenspaces corresponding to eigenvalues of $M$ that are roots of unity.

What one uses to deduce $1 \implies 2$ is a discreteness argument: $\mathbb{Z}^n$ has only finitely many elements with a given bound, and hence boundedness of $|M^j v|$ implies the existence of $j$ such that $M^j v = v$.

I would like a similarly precise characterization of polynomial growth. The best I know at the moment is that the following are equivalent:

- $| M^j v|$ is bounded above by a polynomial function of $j$, for positive integers $j$.
- $v$ is contained in the direct sum of generalized eigenspaces corresponding to eigenvalues $\lambda \in \mathbb{C}$ such that $\lambda \le 1$, i.e. to those $\lambda$ that lie on or inside the unit circle of $\mathbb{C}$.

But statement 4 is somewhat loose, in that it allows some seeming possibilities that turn out to be impossible using a discreteness argument. For example, it is impossible that $v$ be contained in the direct sum of the generalized eigenspaces corresponding to eigenvalues that lie inside the unit circle, for in that case one can show that the sequence $M^j v$ converges to the zero vector, an impossibility for $M \in GL_n(\mathbb{Z})$ and $v \in \mathbb{Z}^n$.

What stronger statements are there which characterize polynomial growth of $|M^j v|$? For example, given $M \in GL_n(\mathbb{Z})$ and $v \in \mathbb{Z}^n$, are Statements 3 and 4 above equivalent to the following nice, strong statement?

- $v$ is contained in the direct sum of generalized eigenspaces corresponding to eigenvalues of $M$ that are roots of unity.