Let me add some detail first.
An Anosov automorphism on $R^2$ is a mapping from the unit square $S$ onto $S$ of the form
$\begin{bmatrix}x \\y\end{bmatrix}\rightarrow\begin{bmatrix}a && b \\c && d \end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}mod \hspace{1mm}1$
in which (i) $a, b, c, and \hspace{1mm} d$ are integers, (ii) the determinant of the matrix is $\pm$1, and (iii) the eigenvalues of the matrix do not have magnitude $1$.
It is easy to show that Arnold's cat map is an Anosov automorphism, and that it is chaotic.
To define "chaotic" in this context,
A mapping $T$ of $S$ onto itself is said to be chaotic if:
(i) $S$ contains a dense set of periodic points of the mapping $T$
(ii) There is a point in $S$ whose iterates under $T$ are dense in $S$.
That said, it is said that all Anosov automorphisms are chaotic mappings. Based on the definition of chaotic, how can one prove that statement?
Any feedback will be appreciated.