# CONFLUENTES MATHEMATICI

On the half-trajectories of horocyclic flow on geometrically infinite hyperbolic surfaces
Confluentes Mathematici, Volume 14 (2022) no. 2, pp. 139-147.

We study the density of half-horocycles or half-orbits of the horocyclic flow on the unit tangent bundle of geometrically infinite hyperbolic surfaces. In [10] Schapira proved that under some assumptions, both half-horocycles ${\left({h}^{s}v\right)}_{s\ge 0}$ and ${\left({h}^{s}v\right)}_{s\le 0}$ are simultaneously dense or not in the nonwandering set of the horocyclic flow. We construct a counterexample, when the assumptions are not satisfied, on a surface of first kind, answering a question of Schapira [10].

Revised:
Accepted:
Published online:
DOI: 10.5802/cml.89
Classification: 51M10, 51M09, 37D40, 20B07, 37C10
Keywords: Geodesic flow, horocyclic flow, geometrically infinite surfaces

1 Université Dan Dicko Dankoulodo de Maradi, Niger
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Adamou Saidou. On the half-trajectories of horocyclic flow on geometrically infinite hyperbolic surfaces. Confluentes Mathematici, Volume 14 (2022) no. 2, pp. 139-147. doi : 10.5802/cml.89. https://cml.centre-mersenne.org/articles/10.5802/cml.89/

[1] A. Beardon. The geometry of discrete groups, Springer-Verlag, Berlin-Heidelberg-New York, 1983. | Zbl

[2] P. Buser. Geometry and spectra of compact Riemanian surfaces, Birkhäuser 1992. | Zbl

[3] F. Dal’bo. Topologie du feuilletage fortement stable, Ann. Inst. Fourier 50(3):981–993, 2000. | DOI | MR | Zbl

[4] F. Dalbo. Geodesic and Horocyclic Trajectories, Universitext, Springer-Verlag, London, 2011 | DOI

[5] D. Borthwick. Spectral Theory of Infinite-Area Hyperbolic Surfaces, Progress in Mathematics Volume 256, Birkhäuser, 2007. | DOI | Zbl

[6] P. Eberlein. Geodesic flows on negatively curved manifolds I, Ann. Math. (2) 95:492–510, 1972. | DOI | MR | Zbl

[7] A. Hatcher, P. Lochak and L. Schneps. On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521:1–24, 2000. | DOI | Zbl

[8] G.A. Hedlund. Fuchsian groups and transitive horocycles, Duke Math. J. 2:530–542, 1936. | DOI | MR | Zbl

[9] B. Schapira. Density and equidistribution of half-horocycles on geometrically finite hyperbolic surface. J. Lond. Math. Soc. 84(3):785–806, 2011. | DOI | MR | Zbl

[10] B Schapira. Density of half-horocycles on geometrically infinite hyperbolic surfaces. Erg. Th. Dyn. Sys. 33(4):1162–1177, 2013. | DOI | MR | Zbl

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