Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.