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Geodesic paths and horocycle flow on abelian covers. (English) Zbl 0967.37020
Dani, S. G. (ed.), Lie groups and ergodic theory. Proceedings of the international colloquium, Mumbai, India, January 4-12, 1996. New Delhi: Narosa Publishing House. Stud. Math., Tata Inst. Fundam. Res. 14, 1-32 (1998).
Let \(\hat{M}\) be the homology cover of a hyperbolic surface \(M= \mathbb{H}^2/\Gamma\) with deck transformation group \(\Gamma/ \hat{\Gamma} \cong H_1(M,\mathbb{Z}) \cong \mathbb{Z}^d\). The authors construct for every \(\nu\in \mathbb{R}^d\) a Gibbs measure for the geodesic flow on the unit tangent bundle \(T^1 M\) of \(M\) whose lift to \(T^1 \hat{M}\) is invariant and ergodic under the horocycle flow. The conditionals of this measure on transversal \(\hat{A}\) to the strong stable foliation transform under the action of \(\Gamma/ \widetilde{\Gamma}\) via \(a\hat{\mu}_{\hat{A}}= e^{\langle v|a\rangle} \hat{\mu}_{a\hat{A}}\). The measure is uniquely characterized by this property.
The proofs are inspired by earlier work of M. Pollicott and R. Sharp [Invent. Math. 117, 275-302 (1994; Zbl 0804.58009)] and use symbolic dynamics. The authors give also counting results for the number of orbits in a given homology class of a prescribed length which significantly improve the results of Pollicott and Sharp. Most of the results are valid for general abelian covers of compact invariant basic hyperbolic sets for smooth flows.
For the entire collection see [Zbl 0927.00025].

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)