I know this might be much to read since I reference paragraphs from the book. I hope that the question is not too cluttered.
The $n$-torus is defined to be the set of all equivalence classes under the equivalence relation that any two points in $\mathbb{R}^{n}$ are equivalent if and only if their coordinates differ by an integer. And the hyperbolic toral automorphism is defined by
Definition. Let $L_{A}(x) = A \cdot x$, where $A$ is an $n \times n$ matrix such that
(1) All entries of $A$ are integers; (2) $\det (A) = \pm 1$; (3) $A$ is hyperbolic, i.e. none of its eigenvalues have absolute value $1$.
I am trying to prove the following statement
Proposition. If $[p] \in T^{n}$, then the stable set $W^{s}[p]$ at $[p]$ and the unstable set $W^{u}[p]$ are dense in $T^{n}$.
However, I do not understand what the stable and unstable sets actually are for the $n$-torus. I believe that the reason I cannot even begin with attempting a proof is that the book never defined the stable and unstable sets for dimension $n$, but only for $2$. In the book, for the $2$-torus, the author says the following
After which he presents the proof for the $2$-dimensional case that the stable and unstable sets are dense given by
EDIT: This is the definition that Devaney gives for the stable $W^{s}$ and unstable $W^{u}$ subspaces. I believe that $L$ is a linear map from $\mathbb{R}^{3}$ into $\mathbb{R}^{3}$.
