JOURNALOF MODERN DYNAMICS doi: 10.3934/jmd.2016.10.113 VOLUME 10, 2016, 113–134 HOROCYCLE FLOWS FOR LAMINATIONS BY HYPERBOLIC RIEMANN SURFACES AND HEDLUND’S THEOREM MATILDE MARTíNEZ, SHIGENORI MATSUMOTO AND ALBERTO VERJOVSKY (Communicated by Federico Rodriguez Hertz) To Étienne Ghys, on the occasion of his 60th birthday ABSTRACT. We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle (Mˆ ,T 1F ) of a compact minimal lamination (M,F ) by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if F has a leaf which is not simply connected, the horocyle flow is topologically transitive. 1. INTRODUCTION The geodesic and horocycle flows over compact hyperbolic surfaces have been studied in great detail since the pioneering work in the 1930’s by E. Hopf and G. Hedlund. Such flows are particular instances of flows on homogeneous spaces induced by one-parameter subgroups, namely, if G is a Lie group, K a closed subgroup and N a one-parameter subgroup of G, then N acts on the homogeneous space K \G by right multiplication on left cosets. One very important case is when G SL(n,R), K SL(n,Z) and N is a unipo- Æ Æ tent one parameter subgroup of SL(n,R), i.e., all elements of N consist of ma- trices having all eigenvalues equal to one. In this case SL(n,Z)\SL(n,R) is the space of unimodular lattices. By a theorem by Marina Ratner (see [28]), which gives a positive answer to the Raghunathan conjecture, the closure of the or- bit under the unipotent flow of a point x SL(n,Z)\SL(n,R) is the orbit of x 2 under the action of a closed subgroup H(x). This particular case already has very important applications to number theory; for instance, it was used by G. Margulis and Dani in [10] and Margulis in [18] to give a positive answer to the Oppenheim conjecture. Received April 15, 2015; revised February 23, 2016. 2010 Mathematics Subject Classification: Primary: 37C85, 37D40, 57R30. Key words and phrases: Hyperbolic surfaces, horocycle and geodesic flows, hyperbolic lami- nations, minimality. MM: Research partially supported by ANII FCE-2007-106 and CSIC, Proyecto de Grupo de Investigación 618 (Sistemas Dinámicos). SM: Research partially supported by Grant-in-Aid for Scientific Research (C) No. 25400096. AV: Research partially supported by CONACyT proyecto U1 55084, and PAPIIT (Universidad Nacional Autónoma de México) # IN103914. INTHEPUBLICDOMAINAFTER 2044 113 ©2016 AIMSCIENCES 114 MATILDE MARTÍNEZ,SHIGENORI MATSUMOTO AND ALBERTO VERJOVSKY When n 2 and ¡ is a discrete subgroup of SL(2,R) such that M ¡\SL(2,R) Æ Æ is of finite Haar volume, and N is any unipotent one-parameter subgroup acting on M, Hedlund proved that any orbit of the flow is either a periodic orbit or dense. When ¡ is cocompact the flow induced by N has every orbit dense, so it is a minimal flow. The horocycle flow on a compact hyperbolic surface is a homogeneous flow of the previous type and most of the important dynamic, geometric and ergodic features of the general case are already present in this 3-dimensional case. On the other hand, the study of Riemann surface laminations has recently played an important role in holomorphic dynamics (see [14] and [17]), polyg- onal tilings of the Euclidean or hyperbolic plane (see [5], [25]), moduli spaces of Riemann surfaces (see [23]), polygonal billiards (see [21] and [37]), etc. It is natural then to consider compact laminations by surfaces with a Riemannian metric of negative curvature and consider the positive and negative horocycle flows on the unit tangent bundle of the lamination. In the spirit of Raghunathan and Ratner, in this paper we study the closures of the horocycle orbits in this non-transitive PSL(2,R)-space, giving examples of orbit closures which are not algebraic in the sense of Ratner. Throughout this paper, (M,F ) is a compact metric space with a chosen met- ric and carrying a lamination by hyperbolic surfaces for which all leaves are dense. The unit tangent bundle of the lamination, called (Mˆ ,T 1F ), is the phase space of the geodesic flow and the horocycle flow. The space Mˆ is in a natural way a compact metric space since it fibers over M with fibre the circle. The joint action of these flows is an action of the affine group B. In Section2 we introduce all the basic definitions and notation, as well as preliminary results. Section3 is devoted to the study of the B-action mentioned above, and it contains the main results of the present paper. In Theorem9 we prove that this action is topologically transitive, and in Theorem8 that it has a unique minimal set M Mˆ . This set may or may not be the whole of Mˆ . A family of ½ laminations for which M is a proper minimal set is described in Proposition3; it is the family of hyperbolic surface laminations that come from an action of the affine group. Example 13 shows a lamination which does not come from an affine action and for which M is also proper. Two sufficient conditions for the B-action to be minimal appear in Proposition4 and in Theorem 14. The last one is the existence of a transverse holonomy-invariant measure for F . Section4 gives an assortment of examples of minimal sets for the horocy- cle flow. Some of these minimal sets coincide with the minimal set M for the affine action, and some do not. We study the dynamics of the geodesic and the horocycle flows, separately, and the main result of this section is Theorem24, which states that if M Mˆ and F has a non simply connected leaf, then the Æ horocycle flow is topologically transitive. Finally, we ask the following question: does the minimality of the B-action imply the minimality of the horocycle flow? There is strong evidence that the answer might be positive, see [1], [2], and [19]. JOURNALOF MODERN DYNAMICS VOLUME 10, 2016, 113–134 HOROCYCLE FLOWS FOR LAMINATIONS BY HYPERBOLIC RIEMANN SURFACES 115 2. LAMINATIONS BY HYPERBOLIC SURFACES For the definition of a compact lamination or foliated space in general, we refer the reader to Chapter 11 of [8]. In this paper, we only concern ourselves with compact laminations by hyperbolic surfaces. 2.1. Laminations by hyperbolic surfaces and their unit tangent bundles. A compact lamination by hyperbolic surfaces (M,F ) (or simply F ) consists of a compact metrizable space M together with a family {(U®,'®)} such that {U } is an open covering of M, ² ® ' : U D T is a homeomorphism, where D is a bounded open ² ® ® ! ® £ ® ® disk in the Poincaré upper half plane H and T® is a topological space, and 1 for (y,t) ' (U U ), ' '¡ (y,t) (g (t)y,h (t)), where g (t) is in ² 2 ¯ ® \ ¯ ® ± ¯ Æ ®¯ ®¯ ®¯ PSL(2,R). Notice that since '® is a homeomorphism, the map h®¯ is a homeomorphism from an open subset of T¯ to an open subset of T®, and g®¯(t) is continuous in t. In the sequel M will always be compact. Each U® is called a foliated chart, a set 1 1 of the form '¡ ({y} T ) being its transversal. The sets of the form '¡ (D ® £ ® ® ® £ {t}), called plaques, glue together to form maximal 2-dimensional connected manifolds called leaves. The lamination F is the partition of M into leaves. The leaves have structures of hyperbolic surfaces with local charts the restriction of p ' ’s to plaques, where p : D T D is the projection onto the first 1 ± ® 1 ® £ ® ! ® factor. A lamination is said to be minimal if all the leaves are dense. A holonomy-invariant measure is a family of Radon measures in all transver- sals which are invariant under the maps h®¯ when restricted to their domains and ranges. For each x M, let us denote by T 1F the set of the unit tangent vectors of 2 x the leaf through x, and define the laminated unit tangent bundle Mˆ of (M,F ) ˆ S 1 ˆ 1 by M x M Tx F . Points of M are denoted by (x,v), where x M and v Tx F . Æ 2 ˆ 1 2 1 2 Denote the natural projection by ¦ : M M. Thus ¦¡ (x) Tx F . There are 1 1 ! Æ homeomorphisms î : ¦¡ (U®) T D® T® defined by the leafwise differ- ! £ 1 1 entials of Á . The locally defined transition function à á : T D T ® ® ± ¯ ¯ £ ¯ ! T 1D T is given by ® £ ® 1 î á (w,t) (g®¯(t) (w),h®¯(t)), ± ¯ Æ ¤ 1 1 where g®¯(t) is the differential of g®¯(t). Plaques î¡ (T D® {t}) all together ¤ £ define a lamination which we shall denote by T 1F . 1 1 1 When we identify T H with PSL(2,R) as usual, then T D® and T D¯ are open subsets of PSL(2,R), and the transition function takes the form 1 à á (g,t) (g (t)g, h (t)), ® ± ¯ Æ ®¯ ®¯ where g PSL(2,R) and g (t)g is the group multiplication. Since the transition 2 ®¯ functions commute with the locally defined right translation of PSL(2,R) on the first factor, we obtain a right action of the universal covering group of PSL(2,R) JOURNALOF MODERN DYNAMICS VOLUME 10, 2016, 113–134 116 MATILDE MARTÍNEZ,SHIGENORI MATSUMOTO AND ALBERTO VERJOVSKY on Mˆ .