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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!

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    $\begingroup$ If you reverse time in the underlying geodesic flow, isn't the new negative horocycle flow the old positive horocycle flow? $\endgroup$
    – Anthony Quas
    Commented Feb 6, 2015 at 2:46
  • $\begingroup$ 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? $\endgroup$
    – Frank_A
    Commented Feb 6, 2015 at 12:06

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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.

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