Let me write a short scheme of proof that I learned in the course of Ergodic Theory in Warwick University (notes are available on-line. You will find several different versions of this proof).

Let us call $X=SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ and consider the geodesic flow $g_t:=\begin{pmatrix}e^{\frac{t}{2}} & 0\\ 0 & e^{-\frac{t}{2}}\end{pmatrix}$ on $X.$

**Theorem:** The geodesic flow $g_t:X\to X$ is ergodic.

Let us define the matrices:
$$h_t:=\begin{pmatrix} 1 & t \\ 0 & 1\end{pmatrix} \mbox{ and } h^-_t:=\begin{pmatrix} 1 & 0 \\ t & 1\end{pmatrix}.$$

**Lemma 1.** $g_t h_s g_{-t}=h_{se^t}$ and $g_t h^-_s g_{-t}=h^-_{se^{-t}}.$

Proof: Direct.

**Lemma 2.** If $\mu$-as $\lim_{T\to\infty}\frac{1}{T}\int_0^T f(g_t x)dt$ is constant in $x\in X$ (i.e. it depends on $f,$ but not on $x$) and for all $f$ continuous on $X,$ then the flow $g_t:X\to X$ is ergodic with respect to $\mu.$

Proof: This a classic exercise in Ergodic theory.

**Lemma 3.** Almost all the matrices $\gamma\in SL(2,\mathbb{R})$ can be write in the form $\gamma=h_{s_1}g_t h_{s_2}^-.$

Proof: Direct.

**Corollary 4.** Given almost all points $x,x'\in X,$ we can chose $\gamma\in SL(2,\mathbb{R})$ such that $x'=h_{s_1}g_t h_{s_2}^-x.$

Proof: Direct.

**Lemma 5.** If $y=g_s x,$ then $\lim_{T\to\infty}\frac{1}{T}\int_0^T f(g_t x)dt=\lim_{T\to\infty}\frac{1}{T}\int_0^T f(g_t y)dt.$

Proof: Direct.

**Lemma 6.** If $y=h_s x,$ then $\lim_{T\to\infty}\frac{1}{T}\int_0^T f(g_t x)dt=\lim_{T\to\infty}\frac{1}{T}\int_0^T f(g_t y)dt.$

Proof. Use $g_t$ invariance of the measure $\mu$ and Lemma 1.

**Lemma 7.** If $y=h_s^- x,$ then $\lim_{T\to\infty}\frac{1}{T}\int_0^T f(g_t x)dt=\lim_{T\to\infty}\frac{1}{T}\int_0^T f(g_t y)dt.$

Proof. Use $g_t$ invariance of the measure $\mu$ and Lemma 1.

**Proof of Theorem**: Use Lemma 2 to characterise ergodicity. For $\mu$-a.e. $x,x',$ we can write (because of Corollary 4) $y_1=h_{s_2}^-x,$ $y_2=g_t y_1$ and $x'=h_{s_1}y_2.$ Then apply Lemmas 5,6 and 7 to conclude the theorem. $\square$

**Remark**: The measure $\mu$ is explicit in the Corollary 4 (and omitted in our statement of the Theorem), corresponds to the Liouville measure.

Geodesic flows on closed Riemannian manifolds of negative curvature. This paper, in Russian, is available here: mathnet.ru/php/… $\endgroup$3more comments