[proofplan] The geodesic flow on a compact hyperbolic surface is the model Anosov flow. The Hopf argument proves ergodicity of the Liouville measure using absolute continuity of the stable and unstable foliations. Strong mixing follows from the classical mixing theorem for compact hyperbolic geodesic flows, equivalently from the Howe-Moore decay theorem for the diagonal flow on $\Gamma\backslash \mathrm{PSL}_2(\mathbb{R})$. [/proofplan] [step:Identify the flow with a homogeneous diagonal flow] Write \begin{align*} \Sigma=\Gamma\backslash\mathbb{H} \end{align*} for a cocompact lattice $\Gamma<\mathrm{PSL}_2(\mathbb{R})$. The unit tangent bundle $T^1\Sigma$ is naturally identified with \begin{align*} \Gamma\backslash\mathrm{PSL}_2(\mathbb{R}). \end{align*} Under this identification, the geodesic flow is right translation by the diagonal subgroup \begin{align*} a_t= \begin{pmatrix} e^{t/2}&0\\ 0&e^{-t/2} \end{pmatrix}. \end{align*} Liouville probability measure corresponds to the normalized Haar measure on $\Gamma\backslash\mathrm{PSL}_2(\mathbb{R})$. [/step] [step:Use the Hopf argument for ergodicity] The stable and unstable foliations of the geodesic flow on a compact hyperbolic surface are absolutely continuous, and the flow is Anosov. Hopf's argument says that any $\phi_t$-invariant integrable function is constant along stable leaves and along unstable leaves almost everywhere. Local product structure and absolute continuity of the two foliations then force the function to be constant almost everywhere on $T^1\Sigma$. Therefore every invariant measurable set has $\nu$-measure $0$ or $1$, so the geodesic flow is ergodic. [/step] [step:Use decay of matrix coefficients for strong mixing] The Howe-Moore theorem for $\mathrm{PSL}_2(\mathbb{R})$, applied to the unitary representation of $\mathrm{PSL}_2(\mathbb{R})$ on \begin{align*} L^2_0(\Gamma\backslash\mathrm{PSL}_2(\mathbb{R})), \end{align*} gives decay of diagonal matrix coefficients: \begin{align*} \langle F\circ a_t,G\rangle_{L^2}\longrightarrow0 \end{align*} for all $F,G\in L^2_0(\Gamma\backslash\mathrm{PSL}_2(\mathbb{R}))$ as $t\to+\infty$. Taking \begin{align*} F=\mathbb{1}_A-\nu(A), \qquad G=\mathbb{1}_B-\nu(B) \end{align*} gives \begin{align*} \nu(\phi_{-t}A\cap B)-\nu(A)\nu(B)\to0. \end{align*} This is strong mixing. [/step]