TITLE: proofs of ergodicity of Sl(2, Z) action on R^2 without using duality
QUESTION [8 upvotes]: The group $
G=SL(2, R)$ acts linearly on $\mathbb R^2$. The Lebesgue measure of $\mathbb R^2$ is invariant and ergodic for $G$. There is a proof using duality theorem:
Let $U$ be the upper triangular unipotent subgroup of $G$ and let $\Gamma=SL(2, \mathbb Z)$. Then $U$ acts ergodically on $G/\Gamma$. Therefore duality tells us that $\Gamma$ acts ergodically on $G/U\cong \mathbb R^2\backslash 0$.
This method is good but it can not be extended to the group action on non homogeneous manifolds. Are there any other proofs? It is better that I can see different proofs to learn methods.

REPLY [3 votes]: I recommand the work of Thomas Roblin  "Ergodicité et équidistribution en courbure négative"  Mémoires de la SMF 95 (2003),  in addition to the references mentioned in the first answer. 
In particular, he does not need any compactness or finite volume assumption, he works with variable negative curvature, and even on CAT(-1) spaces $X$, not only on manifolds. 
In this framework ergodic properties of the horospherical foliation are related (by duality) to ergodic properties of the action of $\Gamma$ on the space of horospheres $\partial\tilde{X}\times \mathbb{R}$ (which identifies with $\mathbb{R}^2\setminus \{0 \}/\pm$ in the case of hyperbolic surfaces). 
These ergodic properties are strongly related to the mixing property of the geodesic flow. 
As mentioned in the beginning of the first answer, in the much earlier approach of Furstenberg, harmonic analysis is used on $\mathbb{R}^2$ to get unique ergodicity of the action of a cocompact fuchsian group $\Gamma$ on $\mathbb{R}^2\setminus\{0\}$, and the unique ergodicity of the horocyclic flow on $SL(2,\mathbb{R})/\Gamma$ is deduced by duality. 
The modern approach (as in Roblin, or Coudène, or others) through mixing is more dynamical.