Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathcal{F})$ of a compact minimal lamination $(M,\mathcal F)$ 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 $\mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.


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 an unipotent one parameter subgroup of SL(n, R), i.e., all elements of N consists of matrices 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 [24]), which gives a positive answer to the Raghunathan conjecture, the closure of the orbit under the unipotent flow of a point x ∈ SL(n, Z)\SL(n, R) is the orbit of x 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 [9] and Margulis in [16] to give a positive answer to the Oppenheim conjecture. When n = 2 and Γ is a discrete subgroup of SL(2, R) such that M := Γ\SL(2, R) is of finite Haar volume, and N is any unipotent oneparameter subgroup acting on M , Hedlund proved that any orbit of the flow is either a periodic orbit or dense. When Γ 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 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 [11] and [14]), polygonal tilings of the Euclidean or hyperbolic plane (see [3], [19]), moduli spaces of Riemann surfaces (see [17]), 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 this paper we give a condition that guarantees that both these flows are minimal if the lamination is minimal.

Preliminaries
Let S be a compact surface endowed with a Riemannian metric of constant curvature equal to -1. It is of the form Γ\h, where h is the hyperbolic plane and Γ is a cocompact discrete subgroup of P SL(2, R) containing only hyperbolic matrices. The compact unit tangent bundle T 1 S of S can be identified with Γ\P SL(2, R), and therefore has a P SL(2, R)-action induced by the right translation.
The geodesic flow on T 1 S corresponds to the action of the diagonal subgroup The following result was proved by Hedlund in [12]: This last condition says that the sets of the form ϕ −1 i (D i × {t}), called plaques, glue together to form d-dimensional manifolds that we call leaves. The sets E i are called foliated charts or flow boxes.
The leaf through x is denoted by L x . If d = 2 we say that L is a surface lamination.

Definition 2. A lamination is minimal if all its leaves are dense.
Consider a finite set τ 1 , . . . , τ M of transversals of L such that any leaf of L intersects at least one of the τ i 's.

is invariant under all holonomy transformations induced by paths connecting pairs of points in τ .
For the definition of holonomy transformations, see for example [5] or [7].
Consider a surface lamination L endowed with a C ∞ Riemannian metricthat is, a Riemannian metric on each leaf all whose derivatives vary continuously in the transverse direction. Each leaf L, being a Riemannian manifold, has its Laplace-Beltrami operator ∆ L . If f : L → R is twice differentiable in the leaf direction, and x ∈ L, we define where L x is the leaf passing through x and f | Lx is the restriction of f to L x . Harmonic measures were introduced by Lucy Garnett (see [10]), and they satisfy the following ergodic theorem: Theorem 2. (Garnett) Let µ be an ergodic harmonic probability measure on L. Let f be a real-valued function on L which is integrable with respect to µ. Then Let L be a surface lamination with a C ∞ Riemannian metric and L be a leaf of L. The restriction to L of the metric on L lifts to a Riemannian metric on the universal coverL of L. Let us fix x ∈L and call A(r) the area of the ball onL centered at x with radius r > 0.
Definition 5. We say that the leaf L is hyperbolic if This condition is of course independent of x, but it is also independent of the Riemannian metric chosen, since all metrics on the compact lamination L are quasi-isometric, and induce quasi-isometric metrics on leaves.
From this point onwards, L will denote a compact surface lamination whose leaves are hyperbolic. Each leaf has a metric of constant curvature -1 in each conformal class, and we can fix a conformal class that varies continuously in the transverse direction (for example, by taking any C ∞ Riemannian metric). According to a theorem due to Candel and Verjovsky (see [6] and [27]), metrics of constant curvature -1 thus defined, as well as their derivatives of all orders, have a continuous variation in the transverse direction.
Definition 6. For x ∈ L let T x L x denote the tangent plane of the leaf L x at the point x. Let Let {(E i , ϕ i )} be an atlas for L, as in Definition 1. For every i, let p i : D i × T i → D i be the projection onto the first factor. The restric- where we understand that the domain of p i • ϕ i is the plaque through x.
The family (Ẽ i ,φ i ) is an atlas that gives T 1 L the structure of a threedimensional lamination whose leaves are the unit tangent bundles of the leaves of L. We say that T 1 L is the unit tangent bundle of the lamination L. Naturally, there is a fibre bundle projection T 1 L → L whose fibers are the unit tangent spaces to the leaves.
The laminated geodesic flow is the flow g t on T 1 L that when restricted to the unit tangent bundle T 1 L of a leaf L coincides with the geodesic flow of L. The stable and unstable horocycle flows are the flows in T 1 L that when restricted to the unit tangent bundle T 1 L of a leaf L coincide with the stable and unstable horocycle flows of L, respectively. All these flows are continuous since they have a continuous dependence on derivatives of the metric.
Measures on T 1 L invariant under these flows are closely related to harmonic measures on L, see [2].
Notice that the stable horocycle flow of our compact lamination has no closed orbits: if x ∈ T 1 L were periodic, then its forward orbit under the geodesic flow g would accumulate on a fixed point of h + , and there are none.
The three-dimensional lamination T 1 L has a continuous right P SL(2, R)action whose orbits are the leaves. As in the case of surfaces, the geodesic and horocycle flows correspond to the action on T 1 L of the one-parameter subgroups D, H + and H − , respectively. A central role in this work will be played by the the affine group, seen as the subgroup If L has a C ∞ metric of negative but not necessarily constant curvature, the restriction of the geodesic flow to the unit tangent bundle of each leaf is still an Anosov flow. Its stable manifolds are called stable horocycles, and they can be continuously parametrized as flow lines of the so-called stable horocycle flow h + . Furthermore, we can do this in such a way that see [15].

Hedlund's theorem for compact minimal laminations
Theorem 3. Let L be a compact minimal hyperbolic surface lamination, endowed with a metric of negative curvature, and let T 1 L be its unit tangent bundle. If the action of the affine group on T 1 L is minimal, then the stable and unstable horocycle flow h + is minimal.
Let us observe that if we reverse the time of the geodesic flow, the unstable horocycles becomes stable ones, and conversely. It is therefore possible to state an analogous theorem for the unstable horocycle flow.
We will now give a sufficient condition for the affine group action to be minimal, and later on we will show that this is not always the case.
Let L be a compact, minimal, hyperbolic surface lamination endowed with a metric of constant curvature -1.
Let M ⊂ T 1 L be a minimal set for the action of the affine group. Since L is minimal, M projects onto L via the canonical projection π : T 1 L → L.
Let us take a point x ∈ L and call L x the leaf of L that passes through x. If M is not the whole of T 1 L, the unit tangent space to L x at x is not contained in M, i.e. T 1 x L ∩ M is a nonempty closed proper subset of T 1 x L. In any case, by identifying in the usual way T 1 x L with the set of points at infinity of the universal coverL x of L x , we may think of T 1 x L ∩ M as a subset K x of the circle at infinity ofL x . Notice that K x does not depend on x but only on the leaf L x , since M is B-invariant. Lemma 1. Let L be a compact, minimal, hyperbolic surface lamination endowed with a metric of constant curvature -1, and let M be a minimal set for the right action of the affine group on its unit tangent bundle T 1 L. If for every x ∈ L the set K x (defined as above) has at least two points, then M is the whole of T 1 L; namely, the B-action is minimal.
Proof of Lemma 1 (due toÉtienne Ghys): Let us start by considering the simpler case in which all leaves are hyperbolic planes.
Assume M is a nontrivial minimal set.
For every x ∈ L, letK x be the convex hull of K x in L x . (Namely, consider all geodesics in L x joining pairs of points in K x , and take the convex hull of their union. This set isK x .) It is possible to do this because K x has at least two points.
Let f : L → [0, +∞) be the function defined by where d is the hyperbolic distance on the leaf passing through y.

Remark 1. The function f is measurable.
Proof of the remark: Let E ≃ D × T be a compact foliated chart of L, where D is a closed disk in C and T is a topological space. In E, theK x form a semicontinuous family of compact sets parametrized by T , and the function f is the distance on each fiber D × {t} to the corresponding compact set. Therefore f is measurable. △ In [4], the authors prove that the axiom of choice is not needed to prove the existence of minimal sets. This implies that the function f can be defined without using the axiom of choice, and this is another reason why f must be measurable (see [25]).
Let µ be an ergodic harmonic measure on L. For every n ∈ N, we define The sequence {A n } is increasing and µ(∪ n A n ) = 1, therefore there exists an n ∈ N for which µ(A n ) > 0. The ergodic theorem tells us that for µ-almost every x ∈ L and almost every continuous path ω on L x that starts at x where χ An is the characteristic function of the set A n . In spite of this, for every x ∈ L, the set of continuous paths ω(t) on L x which start at x and which converge, when t → ∞, to a point outside K x has positive Wiener measure, and for any of these paths Since this contradicts (2), we have proved the lemma when all leaves of L are hyperbolic planes.
If there is a leaf L x of L which is not simply connected, we can still definê K x on its universal cover. For points on L x , we define f (y) as the distance from y to the projection ofK x , and use the same argument as before to complete the proof.

Remark 2.
Let h be the Poincaré disk model of the hyperbolic plane and x ∈ h be the origin. Consider the measure on the circle at infinity ∂h defined by saying that the probability of A is the probability that a Brownian path starting at x ends in A. In the preceding proof we have used the fact that this measure is the Lebesgue measure on ∂h = S 1 , and therefore it is positive in open subsets of ∂h.
If we consider the disk with a Riemannian metric of variable negative curvature, Brownian paths starting at any point x converge almost surely to the circle at infinity; see [23]. The measure ν x they induce on the circle at infinity is in a measure class which is independent of x, which follows from Harnack's inequality. Furthermore, when starting points x converge to a boundary point ξ, the measures ν x converge weakly to the Dirac measure at the point ξ. (See [1], [26] or [13].) This implies that the support of the measures ν x is the whole circle at infinity Remark 3. If the metric on L is of negative but not necessarily constant curvature, the proof of Lemma 1 still holds, since it depends on properties of the geodesic flow which are true on surfaces of negative curvature. Furthermore, we can consider a continuous right action of the affine group on T 1 L such that

by virtue of equation (1). Consequently, the proof of lemma 1 does not even need rephrasing.
We therefore have: Lemma 2. Let L be a surface lamination endowed with a Riemannian metric of negative curvature. If M ⊂ T 1 L is nonempty, closed and invariant under both the geodesic flow and the foliation by stable horocycles, and K x = T 1 x L ∩ M has at least two points for every x ∈ L, then M = T 1 L.
In fact, there is an apparently weaker condition that can be substituted for the condition that K x has at least two points for every x: We say that a subset U of L has total probability if it has full measure according to all harmonic measures on L. Observe that as long as the set does not have total probability, the proof of Lemma 1 is valid, and therefore the B-action is minimal.
The function d : x −→diameter(K x ) is upper semicontinuous, so the set U = d −1 (0) (which is leaf-saturated) must be either residual or empty.
An interesting consequence of the minimality of the B-action is the following:

Remark 4. Let L be a compact surface lamination endowed with a Riemannian metric of negative curvature, which is minimal under the joint action of the geodesic and the stable horocycle flow. Then the laminated geodesic flow is topologically transitive in
The proof of this remark is standard. We will nevertheless write it down for the sake of completeness.
Proof of the remark: Let U and V be open subsets of T 1 L, and let x belong to U . A small segment of the stable horocycle passing through x is contained in U . That is, there is an injective curve κ : (−ε, ε) → T 1 L such that κ((−ε, ε)) ⊂ U , κ(0) = x and κ(t) belongs to the stable horocycle passing through x, for all t.
Let y be a point in the α-limit set of x (with respect to the geodesic flow), and let κ ′ be the stable horocycle passing through y. The minimality of the affine group action implies the fact that ∪ t∈R g t (κ ′ ) is dense in T 1 L, and therefore there exist a point z ∈ κ ′ and a time t ∈ R such that g t (z) ∈ V .
When suitably pushed backwards with the geodesic flow, κ approximates κ ′ . This means that there exist a point w ∈ κ((−ε, ε)) and a time s < 0 such that g s (w) is sufficiently close to z for g t (g s (w)) = g t+s (w) to belong to V . Therefore, for any two open sets U and V the orbit under the geodesic flow of U intersects V . Since the topology on T 1 L has a countable basis {U n }, this implies the existence of a dense residual set R = ∩ n∈N (∪ t∈R g t (U n )) of points having dense orbit.
In the proof of Theorem 3, it will be important for us to view T 1 L as a twodimensional lamination whose leaves are the orbits of the affine group B. The following theorem, due to Samuel Petite (see [19,Proposition 3.2] or the proof of [22,Lemma 3.1]) says that such an object does not admit holonomy-invariant measures. 1 Theorem 4. (Petite) A compact surface lamination whose leaves are the orbits of a continuous action of the affine group has no holonomy-invariant measures.

Proof of Theorem 3:
Let K be a minimal set for the horocycle flow h on T 1 L, and let us assume that K is not the whole of T 1 L. Let g t be the laminated geodesic flow. Consider the additive subgroup of R defined as The last equality holds because K is minimal for h. Since K = T 1 L, the group G must be discrete, and therefore cyclic or trivial.

Proof of lemma 3: Suppose that
Consider the one-dimensional lamination in K whose leaves are the orbits of the horocycle flow. Bebutov's theorem guarantees that this flow admits local transversals so it is indeed a lamination; see [18, Theorem 2.14, pg.333]. It has a holonomy-invariant measure ν, since these measures are in one-to-one correspondence with measures invariant under the flow. (See [8], or simply apply Plante's theorem in [21].) The measure ν is defined on the union τ of some finite set of transversals τ 1 , . . . , τ m , which intersects every leaf (i.e. it intersects every horocycle orbit in K).
Let A = ∪ t∈R g t (K). It is a two-dimensional lamination whose leaves are orbits of the affine group B, and which is embedded in T 1 L. Notice that τ 1 , . . . , τ m are also transversals in A, and τ = τ 1 ∪ · · · ∪ τ m intersects every leaf of A.
Consider any path γ through a leaf of A which joins a pair of points in τ . Every leaf of A is simply connected, because h has no closed orbits and G is trivial. Therefore γ is homotopic to a segment of a horocycle orbit in K connecting the endpoints of γ, and this means that the holonomy transformation determined by γ preserves the measure ν (at least in a neighborhood in τ of γ(0), in principle). Since this holds for every path on a leaf of A joining points of τ , the measure ν is also a holonomy invariant measure for the lamination A. It can be extended by zero to a holonomy invariant measure of the lamination on the space T 1 L whose leaves are the orbits of the affine group B, which contradicts Petite's theorem (Theorem 4). This concludes the proof of the lemma. △ Therefore, G is cyclic. Let t 0 be its generator.
The minimality of the affine group action implies that and K is therefore a global transverse section of the geodesic flow g t , which every geodesic orbit intersects exactly at intervals of length t 0 . We call a closed set having this property a synchronized global transverse section. The function if g −t (x) ∈ K is well defined and it is a locally trivial fibration of T 1 L over S 1 . That is, the geodesic flow is a suspension. This was first noticed by Plante in [20]. The proof is completed by the following lemma: Proof of Lemma 4: Suppose K is a synchronized global transverse section for the geodesic flow on T 1 L.
Let T : T 1 L → T 1 L be the involution that leaves every unit tangent space T 1 x L, x ∈ L, invariant and takes a unit tangent vector v to −v. We can always assume that L is oriented; otherwise we take an orientable double covering. Under this assumption, T is homotopic to the identity Id in T 1 L. Let H u , u ∈ [0, 1] be an homotopy taking T = H 0 to Id = H 1 .
There exists an infinite cyclic covering ψ : R × K → T 1 L with the property that ψ(t+ s, x) = g t ψ(s, x) for all t, s. The flow f t in R× K defined by f t (s, x) = (t + s, x) is therefore the lifting of g t . Notice for all t ∈ R, and in particular the flows g t and g −t are topologically conjugated.
Let us compactify R × K by adding two points "to the left" and "to the right". Namely, the compactification is X = (R × K) ∪ {L, R}; a neighborhood of L is a set containing V a = {(t, x) : t < a}, for some a ∈ R and neighborhoods of R are defined analogously.
The flow f t can be continuously extended to a flowf t in X that has L and R as fixed points and that satisfies that for every x ∈ X\{L, R}. Likewise, the homotopy H can be lifted to a homotopyH : [0, 1] × X → X such thatH 1 =H(1, ·) is the identity in X. Then, each mapH u =H(u, ·) must satisfyH u (L) = L,H u (R) = R. Nevertheless,H 0 conjugates f t to f −t , which combined with equation (3) implies that H 0 (L) = R and H 0 (R) = L.
The fact that we have reached a contradiction proves the lemma and therefore Theorem 3. Corollary 1. Let L be a compact minimal surface lamination endowed with a Riemannian metric of negative curvature, on whose unit tangent bundle the affine group action is minimal. Then the laminated geodesic flow is topologically mixing in T 1 L.
Proof of the corollary: Let V be an open subset of T 1 L. For N ∈ N, define F N as the set of points in T 1 L whose orbit under the unstable horocycle flow h − does not intersect V between times −N and N . The F N form a decreasing family of closed, and therefore compact, subsets of T 1 L. Since their intersection is empty, one of them must be empty, which means that there exist a time N 0 such that for every This segment of unstable horocycle orbit is exponentially expanded by the geodesic flow g, and therefore there is a T > 0 for which {h − s (g t (x)) : −N 0 ≤ s ≤ N 0 } ⊂ g t (U ), for all t > T . By definition of N 0 , g t (U ) ∩ V = φ for every t > T , which proves the corollary.
We will finish by giving another proof of Lemma 4, that is based on the same idea but uses slightly different language.
Assume K is a synchronized global transverse section for the geodesic flow on T 1 L. Without loss of generality, we can say that t = 1 is the first return time of K to itself.
Let T be, as above, the involution in T 1 L that leaves each fiber T 1 x L invariant and sends a unit tangent vector v to −v.
Lemma 5. There exists s ∈ R such that g s (K) = T (K).
Proof of Lemma 5: Let x ∈ T 1 L be a point whose orbit Γ = {g t (x) : t ∈ R} is dense.
We know that g t • T = T • g −t , and this implies that g t (T (x)) ∈ T (K) if and only if t ∈ Z. Also notice that T (Γ) = {g −t (T (x)) : t ∈ R}, and therefore T (x) also has a dense orbit. Furthermore, since Γ ∩ K is dense in K, T (Γ) ∩ T (K) = {g t (T (x)) : t ∈ Z} is dense in T (K). These remarks together with the fact that T (K) is closed imply that all points in T (K) return to T (K) exactly at times t ∈ Z.
On the other hand, if s ∈ [0, 1] is such that T (x) ∈ g s (K), the orbit of T (x) intersects g s (K) exactly at times t ∈ Z. Moreover, g s (K) is the closure of {g t (T (x)) : t ∈ Z}, and therefore g s (K) = T (K).
Since L is oriented, there is a continuous action of S 1 on T 1 L whose orbits are the unit tangent spaces to the leaves of L. We will write it as T e 2πiθ : T 1 L → T 1 L, θ ∈ [0, 1].
Then T 1 = Id and T −1 = T .
Since π is differentiable in the sense of laminations, F is differentiable when restricted to each leaf. We can assume, without loss of generality, that 1 is a regular value of Ψ (otherwise we can use instead of K K ǫ , for ǫ small, and then use Sard's theorem). This means that the "band" F ([0, 1/2] × R) is transverse to K. Since x, g 1 (x), g −u (T (x)) and g 1−u (T (x)) belong to K, their image under F is 1. By transversality, F −1 ({1}) contains two arcs a and b, where a connects x with g −u (T (x)) and b connects g 1−u (T (x)) with g 1 (x). The arcs a and b together with the curves γ 0,x and γ 0,g−u(T (x)) determine an oriented rectangle Q that represents a 2-chain relative to the couple (T 1 L, K), whose boundary is represented by the cycles (mod K) γ 0,x and γ 0,g−u(T (x)) . Therefore [γ 0,x ] = −[γ 0,g−u(T (x)) ] and π * ([γ 0,g−u(T (x)) ]) = −1, which is what we had to prove. This second proof shows the cohomological nature of both proofs of our main theorem. If there existed a synchronized section, then the associated fibration π : T 1 L → S 1 would represent a nontrivial element α of H 1 (T 1 L, Z) (identifying, as usual, homotopy classes of maps of a space to the circle with the first cohomology group, with integer coefficients, of the space). The map T defined above is isotopic to the identity and conjugates the geodesic flow g t with g −t and this would imply that α = −α, i.e. α = 0 which would be a contradiction.