Let $S$ be a closed surface endowed with a Riemannian metric of negative curvature and let $US$ be the unit tangent bundle. Let $\mu$ be the Liouville measure on $US$.

Let $f: US\rightarrow\mathbb{R}$ be a smooth function and $M=\int_{US}f(v)d\mu(v)/\int_{US}d\mu$ be its mean value.

For each $v\in US$, let $(v(t))_{t\in\mathbb{R}}\subset US$ denote the geodesic with initial velocity $v$. Ergodicity of geodesic flow and Birkhoff theorem imply that for almost all $v$ in $US$ we have $$ (\star)\quad\quad\quad\lim_{T\rightarrow+\infty}\frac{1}{T}\int_0^Tf(v(t))\, dt=M. $$

In general, one can not expect this to hold for every $v\in US$. However, for some reason I want an answer to the following questions.

**Questions**

Does there exist $x\in\Sigma$ such that $(\star)$ holds for every $v\in U_xS$ ?

If yes, is there a $x$ such that the convergence in $(\star)$ is uniform on $U_xS$?