Let $S$ be a compact connected orientable surface of variable negative curvature, and let $M=T^1S$ be the unit tangent bundle of $S$. Then, we know from the paper of Brian Marcus (*) that the negative horocycle flow $\{h^-_t\}_{t\in\mathbb R}$ on $M$ is uniquely ergodic with respect to some Borel probability measure $\mu_-$.

**Questions:** Is the positive horocycle flow $\{h^+_t\}_{t\in\mathbb R}$ on $M$ also uniquely ergodic with respect to some Borel probability measure $\mu_+$? If yes, are the measures $\mu_-$ and $\mu_+$ equal or related$\;\!$?

(*) B. Marcus, Unique ergodicity of the horocycle flow: Variable negative curvature case, Israel J. Math., 1975

Thanks!