Let $f : \mathbb{T}^2 \to \mathbb{T}^2$ be the "cat map" defined by $f(x,y)=(2x+y \mod 1,x+y\mod 1)$.
How can I prove that $f$ is exponentially mixing on $\mathbb{T}^2$ endowed with the Lebesgue measure? That is, there exists $\lambda <1$ and $C>1$ such that for all measurable sets $A, B$, $\left\vert\,\,|f^{-n}(A)\cap B|-|A||B|\,\,\right\vert \leqslant C \lambda^n$.
When one deals with ergodicity there are some handy characterizations for which one can use that $f$ acts like the matrix $\begin{bmatrix}2&1\\1&1\end{bmatrix}$. Is there a similar characterization for the notion of strong mixing?
Thanks.