A famous theorem of Birman and Series says that if $S$ is a compact hyperbolic surface, then the set of points that lie on simple geodesics is nowhere dense and has Hausdorff dimension one; in particular, it has measure zero. This is proved in


Birman, Joan S.; Series, Caroline,
Geodesics with bounded intersection number on surfaces are sparsely distributed.
Topology 24 (1985), no. 2, 217–225.


**Question**: Is this true for compact oriented surfaces equipped with metrics of variable negative curvature?



Birman-Series for variable negative curvature
Here is another argument which reduces the general result to the constant curvature case:


For any negatively curved metric $g$ on $S$, the set of geodesics of the universal cover $\tilde{S}$ is canonically identified with $\partial\_\infty \tilde{S}^{(2)}$, the set of pairs of distinct points in the Gromov boundary of $\tilde{S}$, in which $\overline{\Lambda}$ (the subset of geodesics that are embedded in $S$) is a closed subset. Note that the Gromov boundary and the set $\Lambda$ are quasi-isometric invariants, so they don't really depend on the choice of the metric $g$. What choosing the metric $g$ does is it puts a $\mathcal C^1$ structure on $\partial\_\infty \tilde{S}^{(2)}$.


Let $g\_0$ be another metric of constant curvature of minus one. Then Birman and Series' theorem tells you that $\overline{\Lambda} \subset \partial\_\infty \tilde{S}^{(2)}$ has Hausdorff dimension $0$ with respect to the $\mathcal C^1$ structure defined by $g\_0$. Now, one can prove that the two $\mathcal C^1$ structures on $\partial\_\infty \tilde{S}^{(2)}$ induced respectively by $g$ and $g\_0$ are Hölder regular with respect to each other, which implies that $\overline{\Lambda}$ also has Hausdorff dimension zero for the $\mathcal C^1$ structure defined by $g$.


