proofs of ergodicity of Sl(2, Z) action on R^2 without using duality

Question

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.

Answer 1

What you call "duality" (that for two commuting actions under reasonable conditions ergodic properties of either action on the space of orbits of the other one are the same) goes back to the work of Furstenberg in the 60s. This idea is especially useful when dealing with the so-called geometric flows: the horocycle and the geodesic ones. However, usually one goes in the direction opposite to the one you mention, namely first studies the action on the space of $\Gamma$-orbits and then deduces the corresponding properties of geometric flows.

You are referring to the situation with the horocycle flow: its ergodicity on the quotient of the hyperbolic plane $\mathbf H^2$ by a discrete subgroup $\Gamma$ is equivalent to ergodicity of the action of $\Gamma$ on the space of horocycles in $\mathbf H^2$ (the latter one being isomorphic to the linear action of $\Gamma$ on $\mathbb R^2\setminus\{0\}$). In the case of the geodesic flow the space of its orbits, i.e., the space of geodesics in the hyperbolic plane, is isomorphic to the square of the boundary circle with removed diagonal $\partial\mathbf H^2\times\partial\mathbf H^2\setminus \text{diag}$. In particular, invariant measures of the geodesic flow are in natural one-to-one correspondence with the so-called geodesic currents, i.e., invariant Radon measures on $\partial\mathbf H^2\times\partial\mathbf H^2\setminus \text{diag}$.

For geodesic flows the case of "big" surfaces and manifolds (with infinite volume) was being considered from the very beginning of the theory in the 30s (and in the constant curvature case finalized in the so-called Hopf-Tsuji-Sullivan theorem in the late 70s), whereas for the horocycle flow it wasn't really considered (and settled) until the late 90s. Generally speaking, proving ergodicity of the horocycle flow on a general surface consists of two components: (1) establishing ergodicity of the boundary action, (2) proving ergodicty of the so-called Busemann cocycle (which determines the extension from the boundary circle to the space of horocycles). For more details see the works of Babillot - Ledrappier MR1699356, Kaimanovich MR1738739, MR1919405, MR2731695, Pollicott MR1739598, Coudene MR1836430 , Solomyak MR1860491, Ledrappier MR1871151, Hamenstadt MR1926280, Ledrappier - Pollicott MR1953296, Babillot MR2087786, Ledrappier - Sarig MR2226490, Sarig MR2827866.

Answer 2

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.