I want to prove that the hyperbolic toral automorphism $L_{A}$ on the $n$ dimensional torus $T^{n}$ is topologically transitive.
The $n$ dimensional torus is defined by $T^{n} := \mathbb{R}^{n} / \mathbb{Z}^{n}$.
Let $A$ be an $n \times n$ matrix with integers entries. Then, we define the hyperbolic toral automorphism $L_{A} : T^{n} \rightarrow T^{n}, x \mapsto L_{A}(x) := A \cdot x$ such that $\det(A) = \pm 1$ and $A$ does not have eigenvalues equal to absolute value $1$.
My attempt:
So let $U, V \subseteq T^{n}$ be open sets. We want to find a $k > 0$ such that $L_{A}^{k}(U) \cap V \neq \emptyset$.
What I try to show is that $L_{A}^{k}(U) \subseteq V$ for some $k > 0$. I.e. that for some $L^{k}_{A}(p) \in L_{A}^{k}(U)$ implies that $L^{k}_{A}(p) \in V$. Which amounts to showing for some $p \in U$ and some $k > 0$ that $L^{k}_{A}(p) \in V$.
Select points $[r] \in U$ and $[s] \in V$ which are homoclinic to $[0]$.
Essentially, I am trying to generalize the $2$ dimensional case which is given below. I am stuck on the part where we say, "Choose an open interval $I_{u}$ ...". Since we are now in $n$-dimensions, do we still choose an open interval or do we now choose an open ball?
Also below is mentioned that $L_{A}^{n}(I_{u})$ and $L_{A}^{-n}(I_{s})$ are parallel to $W^{s}[0]$ and $W^{u}[0]$, respectively. Is this still true for dimension $n$? In short, does the below proof for the $2$ torus generalize to the $n$-torus?
As stated, your attempt at proving $L_A^k(U) \cap L_A^k(V) \ne \emptyset$ by proving $L_A^k(U) \subset L_A^k(V)$ cannot work.
Here is a counterexample, namely an example of two open subsets $U,V \subset T^n$ such that $L_A^k(U) \not\subset L_A^k(V)$ for all $k \ge 1$. In fact, as you'll see, you can take $U,V$ to be any two open balls of the same small radius.
Let $\lambda$ be an eigenvalue of $A$ with $|\lambda| > 1$, let $W \subset \mathbb R^n$ be an $A$-invariant subspace with the property $|Aw| = \lambda|w|$ for each $w \in W$, let $\widetilde{\mathcal F}$ be the folation of $\mathbb R^n$ by affine subspaces of the form $W + \vec p$, and let $\mathcal F$ be the projected foliation of $T^n$.
Now let $U,V \subset T^n$ be open balls of the same tiny radius $r$ and the same diameter $2r$. Some leaf $F$ of $\mathcal F$ passes through the center of $U$, and so the component of $F \cap U$ containing the center of $U$ is an open subdisc of dimension equal to the dimension of $V$, such that the diameter of that subdisc equals the diameter of $U$ itself. One can now see, using that subdisc, that the diameter of $L_A^k(U)$ is $\ge 2r|\lambda|$ for all $k \ge 1$, and therefore $L_A^k(U)$ cannot be a subset of $V$.