# 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!

• If you reverse time in the underlying geodesic flow, isn't the new negative horocycle flow the old positive horocycle flow? – Anthony Quas Feb 6 '15 at 2:46
• I have the impression that this applies in the case where the horocycle flow is uniformly parametrised along the horocycle foliation. My interest resides in the general case, as in the paper of Marcus, where the parametrisation can be different from the uniform one. Is it also true in that case? – Frank_A Feb 6 '15 at 12:06

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).