I am new to Dynamical Systems and recently I found the book of Brin and Stuck and decided to begin with this. I have a problem to understand one example it mentions about hyperbolic toral automorphisms.
To begin with, first we take the matrix $ A= \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} $ , which has eigenvalues $ \lambda = \frac{3+\sqrt{5}}{2} $ and $\frac{1}{\lambda} $ and corresponding eigenvectors $ v_{\lambda}=(\frac{1+\sqrt{5}}{2},1)$ and $ v_{\frac{1}{\lambda}}=(\frac{1-\sqrt{5}}{2},1) $ . The matrix has integer entries, so it induces a map from the torus $ \mathbb{T}^{2}= {\mathbb{R}^2}/{\mathbb{Z}^2} $ to the torus $ \mathbb{T}^2$ written explicitly as $$ A \begin{pmatrix}x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} (2x_1 + x_2) \mod 1 \\ (x_1 + x_2) \mod 1 \end{pmatrix} $$ Ok,my problem is to understand what the book says in the next lines. Precisely it states : ''the lines in $\mathbb{R}^2$ parallel to the eigenvector $v_{\lambda}$ project to a family $W^u$ of parallel lines in $\mathbb{T}^2$. For $x\in \mathbb{T}^2$, the line $W^u (x)$ through $x$ is called the unstable manifold of $x$. The family $W^u$ partitions $\mathbb{T}^2$ and is called the unstable foliation of A. This foliation has the invariance property,meaning that $A(W^u(x))=W^u(A(x))$. Also $A$ expands each line in $W^u$ by a factor of $\lambda$. Similarly the stable foliation $W^s$ is obtained by projecting the lines of $\mathbb{R}^2$ parallel to $v_{1/{\lambda}}$. This foliation os also invariant under $A$ and $A$ contracts each stable manifold by a factor $1/{\lambda}$. Since the slopes of both eigenvectors are irrational, it follows that each stable and unstable manifold is dense in $\mathbb{T}^2 $ . Similarly any integer entried matrix $B \in \mathbb{Z}^{n\times n}$ induces a group endomorphism of the torus $\mathbb{T}^n$. This map is invertible iff $|\det(B)|=1$. If $B$ is invertible and the eigenvalues do not lie in the unit circle, then $B : \mathbb{T}^n \to \mathbb{T}^n $ has expanding and contracting subspaces of complementary dimensions and is calles a hyperbolic toral automorphism. One can show that all eigenvalues of a two-dimensional hyperbolic toral automorphism are irrational, hence the stable and unstable manifolds are dense in $\mathbb{T}^2$.''
My first question is this : ok,I understand why all the lines parallel to $v_{\lambda}$ are being projected to parallel lines of $\mathbb{T}^2$ (Each line parallel to $v_{\lambda}$ has the form $\{(x, \frac{\sqrt{5}-1}{2}x+b ) | x\in \mathbb{R} \}$ and this is mapped to $ \{ (2x + \frac{\sqrt{5}-1}{2}x+b \mod 1, x+\frac{\sqrt{5}-1}{2}x+b \mod 1) | x\in \mathbb{R} \}$). But I do not see why each manifold has the invariance property. Is this a matter of writing down and doing the calculations or is there a difficult idea behind it?
The second question is about the density argument. How can we prove that each line (in fact it is the CLASS of the line on the quotient ${\mathbb{R}^2}/{\mathbb{Z}^2}$, right? ) $W^u(x)$ is dense in $\mathbb{T}^2$ ?
Any hint will be much appreciated. Thank you a lot! :)
