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 $U\_xS$?



On the Birkhoff ergodic theorem for geodesic flows
The answer is NO. There is no chance for such a thing. 


The reason is the following. The asymptotic behaviour of the ergodic average depends on the asymptotic geometric behaviour of the ... geodesic you are looking at! And the hyperbolicity of the geodesic flow means that starting from everywhere (for example from $x$) you can get every possible ergodic behaviour.


In your problem, consider a nonconstant continuous map $f$. 
Assume for example that there exists another probability measure $\nu$ invariant under the geodesic flow, such that $\int\_{US} f\,d\nu\neq \int\_{US} f\,d\mathcal{L}$, where $\mathcal{L}$ is the normalized Liouvillle measure. 
Without loss of generality, you can assume $\nu$ to be ergodic. 


Choose a generic vector $v\in US$ for $\nu$. Then the ergodic average of $f$ along the geodesic $(g^t v)\_{t\ge 0}$ defined by $v$ converges to $\int f d\nu\neq M$.


But now, by hyperbolicity, given any $x\in US$ there exists (a lot of) $w\in U\_xS$ which is in the (weak) stable manifold of $v$. (If you are not convinced, draw a picture in the universal cover, lifting $x, v$ and find $w$...) In particular, the geodesics of $v$ and $w$ are asymptotic, and the ergodic averages converge to the same limit. 


It is the (negative) answer to your question. 


Bests


Barbara


