invariant measure of uniquely ergodic horocycle flow 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!
 A: The result of Marcus can be formulated in terms of strong (un)stable foliations of the geodesic flow in negative curvature. 
The  result, due to Bowen-Marcus in the compact case, to Roblin as soon as the geodesic flow (in neg. curvature) admits a finite measure of maximal entropy, says that there exists a unique (up to multiplicative constants) transverse measure to the strong (un)stable foliation, which  is invariant under the holonomy of the foliation. 
As said by A Quas, changing the stable to the unstable foliation (by sending a vector $v$ to its opposite $-v$) does not change anything, you get the same transverse invariant measure (by uniqueness for example). 
Now, in dimension 2, consider any parametrization of the 1-dimensional strong (un)stable manifolds as orbits of a flow,  then there is a unique (up to multiplicative constant) invariant measure, obtained by integrating along the orbits of the flow thanks to the parametrization, and then integrating transversally thanks to the transverse measure. 
