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.
The proof that there is a dense set of periodic points should go something like this. Consider a point with rational components, both with denominator $q$ (I don't insist the rationals be in lowest terms). Then every point in the orbit of that point will also have rational components with denominator $q$. But there are only finitely many points in $S$ with rational components with denominator $q$, so the orbit must visit some point twice. Once it visits some point twice, it must keep visiting the same points over and over, periodically. So this proves every point with rational components is pre-periodic. But $T$ is one-one so pre-periodic points are periodic. As the points with rational components are dense, we're done.
EDIT: The proof that the map is topologically transitive (equivalent, in this setting, to having a point with dense orbit) seems to be harder. In Elaydi, Discrete Chaos, 2nd edition, it takes two pages, from mid-page 285 to mid-page 287.