## Horocycle flows on certain surfaces without conjugate points

HTML articles powered by AMS MathViewer

- by Patrick Eberlein PDF
- Trans. Amer. Math. Soc.
**233**(1977), 1-36 Request permission

## Abstract:

We study the topological but not ergodic properties of the horocycle flow $\{ {h_t}\}$ in the unit tangent bundle*SM*of a complete two dimensional Riemannian manifold

*M*without conjugate points that satisfies the “uniform Visibility” axiom. This axiom is implied by the curvature condition $K \leqslant c < 0$ but is weaker so that regions of positive curvature may occur. Compactness is not assumed. The method is to relate the horocycle flow to the geodesic flow for which there exist useful techniques of study. The nonwandering set ${\Omega _h} \subseteq SM$ for $\{ {h_t}\}$ is classified into four types depending upon the fundamental group of

*M*. The extremes that ${\Omega _h}$ be a minimal set for $\{ {h_t}\}$ and that ${\Omega _h}$ admit periodic orbits are related to the existence or nonexistence of compact “totally convex” sets in

*M*. Periodic points are dense in ${\Omega _h}$ if they exist at all. The only compact minimal sets in ${\Omega _h}$ are periodic orbits if

*M*is noncompact The flow $\{ {h_t}\}$ is minimal in

*SM*if and only if

*M*is compact. In general $\{ {h_t}\}$ is topologically transitive in ${\Omega _h}$ and the vectors in ${\Omega _h}$ with dense orbits are classified. If the fundamental group of

*M*is finitely generated and ${\Omega _h} = SM$ then $\{ {h_t}\}$ is topologically mixing in

*SM*.

## References

- R. L. Bishop and B. O’Neill,
*Manifolds of negative curvature*, Trans. Amer. Math. Soc.**145**(1969), 1–49. MR**251664**, DOI 10.1090/S0002-9947-1969-0251664-4 - Herbert Busemann,
*The geometry of geodesics*, Academic Press, Inc., New York, N.Y., 1955. MR**0075623** - Patrick Eberlein,
*The cut locus of noncompact finitely connected surfaces without conjugate points*, Comment. Math. Helv.**51**(1976), no. 1, 23–41. MR**431264**, DOI 10.1007/BF02568141
—, - Patrick Eberlein,
*Geodesic flow in certain manifolds without conjugate points*, Trans. Amer. Math. Soc.**167**(1972), 151–170. MR**295387**, DOI 10.1090/S0002-9947-1972-0295387-4 - Patrick Eberlein,
*Geodesic flows on negatively curved manifolds. I*, Ann. of Math. (2)**95**(1972), 492–510. MR**310926**, DOI 10.2307/1970869 - Patrick Eberlein,
*Geodesic flows on negatively curved manifolds. II*, Trans. Amer. Math. Soc.**178**(1973), 57–82. MR**314084**, DOI 10.1090/S0002-9947-1973-0314084-0 - Patrick Eberlein,
*Some properties of the fundamental group of a Fuchsian manifold*, Invent. Math.**19**(1973), 5–13. MR**400250**, DOI 10.1007/BF01418848
—, - P. Eberlein and B. O’Neill,
*Visibility manifolds*, Pacific J. Math.**46**(1973), 45–109. MR**336648**, DOI 10.2140/pjm.1973.46.45 - Harry Furstenberg,
*The unique ergodicity of the horocycle flow*, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., Vol. 318, Springer, Berlin, 1973, pp. 95–115. MR**0393339** - Anna Grant,
*Surfaces of negative curvature and permanent regional transitivity*, Duke Math. J.**5**(1939), no. 2, 207–229. MR**1546119**, DOI 10.1215/S0012-7094-39-00520-X - Robert Gulliver,
*On the variety of manifolds without conjugate points*, Trans. Amer. Math. Soc.**210**(1975), 185–201. MR**383294**, DOI 10.1090/S0002-9947-1975-0383294-0 - Gustav A. Hedlund,
*Fuchsian groups and mixtures*, Ann. of Math. (2)**40**(1939), no. 2, 370–383. MR**1503464**, DOI 10.2307/1968925 - Gustav A. Hedlund,
*Fuchsian groups and transitive horocycles*, Duke Math. J.**2**(1936), no. 3, 530–542. MR**1545946**, DOI 10.1215/S0012-7094-36-00246-6 - Ernst Heintze and Hans-Christoph Im Hof,
*Geometry of horospheres*, J. Differential Geometry**12**(1977), no. 4, 481–491 (1978). MR**512919** - Eberhard Hopf,
*Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung*, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl.**91**(1939), 261–304 (German). MR**1464** - Brian Marcus,
*Ergodic properties of horocycle flows for surfaces of negative curvature*, Ann. of Math. (2)**105**(1977), no. 1, 81–105. MR**458496**, DOI 10.2307/1971026 - Brian Marcus,
*Unique ergodicity of the horocycle flow: variable negative curvature case*, Israel J. Math.**21**(1975), no. 2-3, 133–144. MR**407902**, DOI 10.1007/BF02760791 - Peter J. Nicholls,
*Transitive horocycles for Fuchsian groups*, Duke Math. J.**42**(1975), 307–312. MR**365540**

*Geodesies and ends in certain surfaces without conjugate points*, Advances of Math. (to appear).

*When is a geodesic flow of Anosov type*? II, J. Differential Geometry

**8**(1973), 565-577. MR

**52**#1788.

## Additional Information

- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**233**(1977), 1-36 - MSC: Primary 58F15; Secondary 34C35, 53C20
- DOI: https://doi.org/10.1090/S0002-9947-1977-0516501-3
- MathSciNet review: 0516501