Skip to main content Accessibility help

Unique ergodicity of the horocycle flow on Riemannnian foliations

  • F. ALCALDE CUESTA (a1), F. DAL’BO (a2), M. MARTÍNEZ (a3) and A. VERJOVSKY (a4)


A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations.



Hide All
[1] Ahlfors, L. and Bers, L.. Riemann’s mapping theorem for variable metrics. Ann. of Math. (2) 72 (1960), 385404.
[2] Alcalde Cuesta, F.. Groupoïde d’homotopie d’un feuilletage Riemannien et réalisation symplectique de certaines variétés de Poisson. Publ. Mat. 33(3) (1989), 395410.
[3] Alcalde Cuesta, F. and Dal’Bo, F.. Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces. Expo. Math. 33(4) (2015), 431451.
[4] Alcalde Cuesta, F., Dal’Bo, F., Martínez, M. and Verjovsky, A.. Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology. Discrete Contin. Dyn. Syst. 36(9) (2016), 46194635. Corrigendum to ‘Minimality of the horocycle flow on laminations by hyperbolic surfaces with non-trivial topology’. Discrete Contin. Dyn. Syst. 39(8) (2017), 4585–4586.
[5] Alcalde Cuesta, F. and Hector, G.. Feuilletages en surfaces, cycles évanouissants et variétés de Poisson. Monatsh. Math. 124(3) (1997), 191213.
[6] Bakhtin, Y. and Martínez, M.. A characterization of harmonic measures on laminations by hyperbolic Riemann surfaces. Ann. Inst. Henri Poincaré Probab. Stat. 44(6) (2008), 10781089.
[7] Bonatti, C., Eskin, A. and Wilkinson, A.. Projective cocycles over $\text{SL}(2,\mathbb{R})$ actions: measures invariant under the upper triangular group. Preprint, 2016, arXiv:1709.02521.
[8] Bonatti, C. and Gómez-Mont, X.. Sur le comportement statistique des feuilles de certains feuilletages holomorphes. Essays on Geometry and Related Topics (Monographs of L’Enseignement Mathématique, 38) . Vols 1, 2. European Mathematical Society, Geneva, 2001, pp. 1541.
[9] Candel, A.. Uniformization of surface laminations. Ann. Sci. Éc. Norm. Supèr. (4) 26(4) (1993), 489516.
[10] Connell, C. and Martínez, M.. Harmonic and invariant measures on foliated spaces. Trans. Amer. Math. Soc. 369(7) (2017), 49314951.
[11] Coudène, Y.. A short proof of the unique ergodicity of horocyclic flows. Ergodic Theory (Contemporary Mathematics, 485) . American Mathematical Society, Providence, RI, 2009, pp. 8589.
[12] Dal’Bo, F.. Geodesic and horocyclic trajectories. Universitext (EDP Sciences, Les Ulis) . Springer, London, 2011. Translated from the 2007 French original.
[13] Deroin, B. and Kleptsyn, V.. Random conformal dynamical systems. Geom. Funct. Anal. 17(4) (2007), 10431105.
[14] Furstenberg, H.. The unique ergodicity of the horocycle flow. Recent Advances in Topological Dynamics (Proc. Conf., Yale University, New Haven, CN, 1972; in honor of Gustav Arnold Hedlund) (Lecture Notes in Mathematics, 318) . Springer, Berlin, 1973, pp. 95115.
[15] Furstenberg, H.. Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573601.
[16] Gallego, E., Gualandri, L., Hector, G. and Reventós, A.. Groupoïdes Riemanniens. Publ. Mat. 33(3) (1989), 417422.
[17] Ghys, É. and Sergiescu, V.. Stabilité et conjugaison différentiable pour certains feuilletages. Topology 19(2) (1980), 179197.
[18] Hedlund, G. A.. Fuchsian groups and transitive horocycles. Duke Math. J. 2(3) (1936), 530542.
[19] Martínez, M., Matsumoto, S. and Verjovsky, A.. Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund’s theorem. J. Mod. Dyn. 10 (2016), 113134.
[20] Matsumoto, S.. The unique ergodicity of equicontinuous laminations. Hokkaido Math. J. 39(3) (2010), 389403.
[21] Matsumoto, S.. Weak form of equidistribution theorem for harmonic measures of foliations by hyperbolic surfaces. Proc. Amer. Math. Soc. 144(3) (2016), 12891297.
[22] Matsumoto, S.. Remarks on the horocycle flows for foliations by hyperbolic surfaces. Proc. Amer. Math. Soc. 145(1) (2017), 355362.
[23] Molino, P.. Géométrie globale des feuilletages Riemanniens. Nederl. Akad. Wetensch. Indag. Math. 44(1) (1982), 4576.
[24] Molino, P.. Riemannian Foliations (Progress in Mathematics, 73) . Birkhäuser Boston, Boston, MA, 1988. Translated from the French by Grant Cairns, with appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu.
[25] Ratner, Marina. Invariant measures and orbit closures for unipotent actions on homogeneous spaces. Geom. Funct. Anal. 4(2) (1994), 236257.
[26] Reinhart, B. L.. Foliated manifolds with bundle-like metrics. Ann. of Math. (2) 69 (1959), 119132.
[27] Samelson, H.. Notes on Lie Algebras, 2nd edn. Springer, New York, 1990.
[28] Verjovsky, A.. A uniformization theorem for holomorphic foliations. The Lefschetz Centennial Conference, Part III (Mexico City, 1984) (Contemporary Mathematics, 58) . American Mathematical Society, Providence, RI, 1987, pp. 233253.
[29] Winkelnkemper, H. E.. The graph of a foliation. Ann. Global Anal. Geom. 1(3) (1983), 5175.
[30] Zimmer, R. J.. Ergodic Theory and Semisimple Groups (Monographs in Mathematics, 81) . Birkhäuser, Basel, 1984.


MSC classification

Unique ergodicity of the horocycle flow on Riemannnian foliations

  • F. ALCALDE CUESTA (a1), F. DAL’BO (a2), M. MARTÍNEZ (a3) and A. VERJOVSKY (a4)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed