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Semirigidity of horocycle flows over compact surfaces of variable negative curvature. (English) Zbl 0633.58024

The geodesic flow \(g_ t\) on the unit tangent bundle U(S) of a compact surface S of negative curvature is Anosov. Thus there is a splitting of each tangent space of U(S) into a stable space, an unstable space and the space tangent to the geodesic vector field generated by g. The one dimensional stable and unstable spaces determine line fields which generate the stable and unstable horocycle foliations. One can find vector fields tangtent to these foliations but their lengths are not determined. If \(\mu\) is the g-invariant measure on U(S) of maximal entropy then there are (not necessarily smooth) reparametrizations of the corresponding horocycle flows \(h_ s\) and \(k_ s\) satisfying \(g_ th_ sg_{-t}=h_{e^ ts}\), \(g_ tk_ sg_{-t}=k_{e^{-t}s}\) which leave \(\mu\) invariant. A corollary to an important theorem of M. Ratner [Ann. Math., II. Ser. 115, 597-614 (1982; Zbl 0506.58030)] states the following: If S has constant curvature -1, and \(\phi\) : U(S)\(\to U(S')\) is a measure-theoretic conjugacy of (h,\(\mu)\) with \((h',\mu ')\), then \(\phi =h'_{s_ 0}\theta\) a.e., where \(\theta\) is the lift of some isometry \(S\to S'\) of the unit tangent bundles. The main result in the paper under review is a generalization of this theorem to surfaces with variable negative curvature, vis., for \(\phi\) a measure- theoretic isomorphism conjugating (h,\(\mu)\) to \((h',\mu)\) there is some \(s_ 0\) such that \(\phi =h'_{s_ 0}\theta\) a.e., with \(\theta\) a homeomorphic conjugacy of g with \(g'\). Also, this homeomorphic conjugacy is a \(C^ 1\) diffeomorphism.
Reviewer: C.Chicone

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 0506.58030
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References:

[1] Bowen, Israel J. Math. 43 pp 43– (1977)
[2] DOI: 10.1090/S0273-0979-1980-14845-4 · Zbl 0461.58008 · doi:10.1090/S0273-0979-1980-14845-4
[3] Berger, Manifolds All of Whose Geodesies Are Closed (1978)
[4] Anosov, Proc. Steklov Inst. of Math. 90 pp none– (1967)
[5] Yano, Curvature and Betti Numbers (1953)
[6] Ratner, In Conference in Modern Analysis and Probability, Contemporary Math. 26 pp none– (none)
[7] DOI: 10.2307/2007030 · Zbl 0556.28020 · doi:10.2307/2007030
[8] Ratner, Ergod. Th. & Dynam. Sys. 2 pp 465– (1982)
[9] Ratner, Ann. Math. 115 pp 587– (1982)
[10] DOI: 10.2307/2373755 · Zbl 0257.58007 · doi:10.2307/2373755
[11] DOI: 10.1007/BF01075620 · Zbl 0245.58003 · doi:10.1007/BF01075620
[12] DOI: 10.2307/2374316 · Zbl 0533.58029 · doi:10.2307/2374316
[13] Liv?ic, Mat. Zametki 10 pp 555– (1971)
[14] DOI: 10.1007/BF02760791 · Zbl 0314.58013 · doi:10.1007/BF02760791
[15] Hirsch, J. Diff. Geom. 10 pp 225– (1975)
[16] DOI: 10.1016/0040-9383(80)90015-4 · Zbl 0465.58027 · doi:10.1016/0040-9383(80)90015-4
[17] Ballman, Manifolds of no npositive curvature (1984)
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