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Suppose $\Gamma\subset SL_2(\mathbb{Z})$ is a non-amenable subgroup, especially, $\Gamma=SL_2(\mathbb{Z})$. Consider the natural action of $\Gamma$ on $S^1\times S^1=T^2$.

How to check that this action is strongly ergodic? ($T^2$ is equipped with the Haar measure).

Recall that an action $\Gamma\curvearrowright (X,\mu)$ is called strongly ergodic if for every sequence of measurable sets $A_n\subset X$ such that $\lim\limits_{n\to\infty}\mu(A_n\Delta\gamma A_n)=0,\forall \gamma\in \Gamma$, we have that $\lim\limits_{n\to\infty}\mu(A_n)(1-\mu(A_n))=0$.

Note that since $T^2$ is considered as the Pontryagin dual of $\mathbb{Z}^2$, we know that for any $f\in T^2$, such that $f(n,m)=z_1^nz_2^m, (z_2,z_2)\in T^2\forall (n,m)\in\mathbb{Z}^2$, then, for $$\gamma=\begin{bmatrix}a&b\\c&d\end{bmatrix},$$ $\gamma f$ satisfies $(\gamma f)(n,m)=(z_1^az_2^c)^n(z_1^bz_2^d)^m$. But how to check the strong ergodicity condition? Since I do not know how to calculate $\mu(A_n\Delta\gamma A_n)$ in practice. Maybe we need to do some qualitative analysis instead of quantitive analysis, but how to proceed?

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up vote 1 down vote accepted

It is a result in Klaus Schmidt's paper Asymptotically invariant sequences and an action of $SL(2,\mathbb{Z})$ on the 2-sphere.

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