Let $A = SL(n,\mathbb{Z})$ be a matrix and $\mathbb{T}^{n} = \mathbb{R}^{n}/\mathbb{Z}^{n}$ be the $n$-dimensional torus. If we assume that none of the eigenvalues of $A$ are roots of unity, then $A$ induces an automorphism of $\mathbb{T}^{n}$, called hyperbolic toral automorphism. I have two questions:
I had learned a while ago that the periodic points of a hyperbolic toral automorphism are dense when $n = 2$. Does this hold when the dimension $n$ is greater than 2?
Since we can keep applying $A$ iteratively, $A$ gives a dynamical system on $\mathbb{T}^{n}$. Is there a dynamical system way to tell whether the characteristic polynomial of $A$ is irreducible?
The first question has a positive answer.
I call $T_A$ the automorphism defined by $T_A(\overline{x}) = \overline{Ax}$, where for all $x \in \mathbb{R}^n$, the equivalence class of $x$ in $\mathbb{T}^n$ is denoted by $\overline{x}$.
Given any integer $N \ge 1$, the linear map $x \mapsto Ax$ induces a bijection from $N^{-1}\mathbb{Z}^n$ to itself, so $T_A$ induces a permutation on the finite set $N^{-1}\mathbb{Z}^n/\mathbb{Z}^n$. Therefore, all points of $N^{-1}\mathbb{Z}^n/\mathbb{Z}^n$ are periodic. Thus periodic points are dense in $\mathbb{T}^n$.
For the second question, I presume that you mean irreducible in $\mathbb{Z}[X]$ or equivalently in $\mathbb{Q}[X]$ since $\chi_A$ is monic? And your ask whether this property can be viewed when looking only at a generic orbit?