Sergi Burniol
email: sergi.burniolclotet@upmc.fr

Clotet Lpsm

Unique ergodicity of the horocyclic flow on nonpositively curved surfaces

On the unit tangent bundle of a nonflat compact nonpositively curved surface, we prove that there is a unique probability Borel measure invariant by the horocyclic flow which gives full measure to the set of rank 1 vectors recurrent by the geodesic flow. If we assume in addition that the surface has no flat strips, we show that the horocyclic flow is uniquely ergodic. These results are valid for any parametrization of the horocyclic flow.

Introduction

The horocyclic flow defined on a compact hyperbolic surface is a dynamical system of high interest, for its relation with the geodesic flow and because it exhibits properties such as minimality [START_REF] Hedlund | Fuchsian groups and transitive horocycles[END_REF] or unique ergodicity. In 1973, H. Furstenberg proved the unique ergodicity of the horocyclic flow on compact surfaces with constant negative curvature using techniques from harmonic analysis [START_REF] Furstenberg | The unique ergodicity of the horocycle flow[END_REF]. In 1975 it was generalized to compact surfaces with negative variable curvature by B. Marcus [START_REF] Marcus | Unique ergodicity of the horocycle flow: variable negative curvature case[END_REF]. We show in this article how to extend Marcus's result to the setting of nonpositve curvature. A horocyclic flow is, by definition, a continuous flow whose orbits are horocycles.

Theorem A. Let M be an oriented nonflat compact Riemannian surface with nonpositive curvature. Let h s be a horocyclic flow on the unitary tangent bundle T 1 M of M and let Σ 0 denote the set of horocycles having a rank 1 vector recurrent under the action of the geodesic flow. Then there is a unique Borel probability measure on T 1 M invariant by the flow h s giving full measure to Σ 0 .

As we will see, the presence of flat strips is probably the main obstruction that we can encounter. At the end of the article we study the case of nonpositively curved compact surfaces without flat strips, which is a class of manifolds that satisfies interesting properties [START_REF] Coudène | Generic measures for geodesic flows on nonpositively curved manifolds[END_REF]. On these surfaces, we prove the best result one could expect.

Theorem B. Let M be an orientable nonpositively curved compact surface without flat strips and let h s be any horocyclic flow on T 1 M . Then the flow h s is uniquely ergodic.

The parametrization of the horocycles by their arc-length is one of the most natural horocyclic flows to consider. We call this the Lebesgue parametrization of the horocyclic flow. Nevertheless, Marcus's method relies on the Margulis parametrization. This Margulis flow h M s is characterized by the commutative law

g t • h M s = h M se δt • g t ,
where g t is the geodesic flow on T 1 M and δ is its topological entropy. The Margulis flow is well defined on surfaces of strictly negative curvature. It can also be defined in the more general setting of Anosov flows with one dimensional stable leaves [START_REF] Marcus | Unique ergodicity of the horocycle flow: variable negative curvature case[END_REF]. However, there is no such parametrization on the whole unit tangent bundle of a general surface with nonpositive curvature, essentially due to the lack of hyperbolicity of the geodesic flow on some regions. But we can at least define a Margulis-like parametrization for the subset of horocycles which do not cross the non-hyperbolic regions and manage to deduce the results from this Margulis horocyclic flow. In Proposition 3.4, we give quite optimal conditions concerning the set on which the Margulis flow can be defined.

In our previous article [START_REF] Burniol | Equidistribution of horospheres in nonpositive curvature[END_REF], we already proved that the Margulis parametrization is well defined on the set Σ 0 of Theorem A and that there is a unique Borel probability measure on Σ 0 invariant by the Margulis flow. We first study how to translate this property from the Margulis parametrization to the Lebesgue parametrization. In fact, we prove that, given a certain subset where the Margulis parametrization is defined, there is a correspondence between the probability Borel measures invariant by both flows and supported on that subset. The proof relies on the fact that the two parametrizations are related by a continuous family of measures on the whole space, not only on the subset. Then we can generalize the unique ergodicity to any other horocyclic flow, as stated in Theorem A, thanks to the compactness of the space.

We want to clarify that there is no direct relation between Theorem A and B, since the set Σ 0 is strictly contained in T 1 M even if M is a nonpositively curved compact surface without flat strips. To prove Theorem B, we will observe that, when there are no flat strips, the Margulis flow is well defined everywhere. So the proof is reduced to working with this parametrization. We can then leverage arguments from [START_REF] Burniol | Equidistribution of horospheres in nonpositive curvature[END_REF] or [START_REF] Coudène | A short proof of the unique ergodicity of horocyclic flows[END_REF] concerning the equidistribution of horocycles under the action of the geodesic flow, to prove the unique ergodicity of the horocyclic flow.

The results of this article demonstrate the great resemblance between geodesic flows on negatively and nonpositively curved manifolds, despite the fact that the second ones are not uniformly hyperbolic systems. Our work relies on tools such as the boundary of the universal cover, the Busemann functions [START_REF] Ballmann | Lectures on spaces of nonpositive curvature[END_REF] and the Patterson-Sullivan theory [START_REF] Link | Ergodic geometry for non-elementary rank one manifolds[END_REF][START_REF] Link | Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry[END_REF]. These provide a good description of the Bowen-Margulis measure [START_REF] Knieper | The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds[END_REF] and are still available in nonpositive curvature. Other well known properties of the geodesic flow on negative curvature such as the uniqueness of the measure of maximal entropy and the asymptotics of the number of closed geodesics have been recently generalized in the absence of conjugate points [START_REF] Climenhaga | Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points[END_REF][START_REF] Climenhaga | Closed geodesics on surfaces without conjugate points[END_REF].

Let us finally give a sketch of the organization of the paper. In Section 2 we recall some basic facts about rank 1 manifolds with nonpositve curvature, we introduce the Bowen-Margulis measure and we show a continuity property of the Margulis measures on the horocycles for the case of surfaces. In Section 3, we study the relationship between invariant measures of two different parametrizations of the horocyclic flow on a subset of the unit tangent bundle and we deduce that unique ergodicity is preserved. In Section 4 we deal with the case of nonpositively curved surfaces without flat strips.

2 On the Bowen-Margulis measure and the measures on the horospheres First, we set up the notation and terminology throughout the paper, as we explain the structure of the manifolds that we work on, [START_REF] Ballmann | Lectures on spaces of nonpositive curvature[END_REF][START_REF] Ballmann | Manifolds of nonpositive curvature[END_REF] are good general references on these subjects. Then we introduce the Bowen-Margulis measure on the unit tangent bundle and measures on the horospheres that are related to this measure. Under the right hypothesis, we prove that in dimension 2 these measures depend continuously on the horocycle. We will need to apply this fact later, since the measures are involved in the definition of the Margulis parametrization.

Geometry of nonpositively curved manifolds

Let M be a C ∞ connected complete Riemannian manifold. The unitary tangent bundle T 1 M of M is the set of unit length tangent vectors on M . We denote by π : T 1 M → M the projection of tangent vectors to their base points. For a unitary tangent vector v ∈ T 1 M , write γ v : R → M for the unit speed geodesic generated by v, i.e. γ v (0) = v.

The geodesic flow g t :

T 1 M → T 1 M is defined by g t (v) = γ v (t), t ∈ R, v ∈ T 1 M .
We say that the rank of a geodesic γ of M is the dimension of the set of parallel Jacobi fields along γ. The rank of v ∈ T 1 M is the rank of the geodesic γ v . We will assume that the manifold M has at least one geodesic of rank 1, in other words, a geodesic whose unique parallel Jacobi field is the tangent field. Such a manifold is said to have rank 1. In most part of the article, we will need in fact to assume the stronger hypothesis that M has a closed geodesic of rank 1. We are interested in manifolds M that have nonpositive sectional curvature in every tangent plane.

The universal cover of M will be denoted by M and it is equipped with the pullback of the metric of M . The lift of a vector v ∈ T 1 M to the tangent space of the universal cover T 1 M is denoted by ṽ. The same notation can be used for the lift of objects from M to M . However, the geodesic flow of T 1 M is simply denoted by g t , since no confusion can arise. Let Γ denote the group of covering transformations of the projection from M to M . The set Γ is a subgroup of the group of isometries of M , which can also act on T 1 M . The distance associated to the Riemannian metric on M is denoted by d, and we use d 1 for the Sasaki distance on T 1 M .

The universal cover M of M is a nonpositively curved simply connected complete manifold. It is then diffeomorphic to R n , where n is the dimension of M . The classical construction of the boundary of M allows us to understand better the structure of M . Two geodesic rays σ 1 , σ 2 : [0, +∞) → M are asymptotic if there exists C > 0 such that for all t ≥ 0, we have d(σ 1 (t), σ 2 (t)) ≤ C. Since being asymptotic is an equivalence relation, we can consider the set ∂ M of equivalence classes of geodesic rays on M . Given a vector v ∈ T 1 M , there is a positive and a negative geodesic ray generated by v, we set the notation v + and v -, respectively, for the asymptotic classes of these rays in ∂ M . The union M ∪ ∂ M has a topology which extends that of M and makes the space compact and the projections from T 1 M to ∂ M , continuous.

Another geometric tool that is defined in nonpositive curvature are the Busemann functions. Given an asymptotic class ξ ∈ ∂ M and two points x, y ∈ M , we define

β ξ (x, y) = lim t→+∞ d(x, σ(t)) -d(y, σ(t))
where σ is a geodesic ray in the class ξ. For fixed ξ ∈ ∂ M and x ∈ M , the map β ξ (x, •) : M → R is of class C 2 and is called the Busemann function at ξ centered at x. We also know that β ξ (x, y) depends continuously on (ξ, x, y) ∈ ∂ M × M × M . A horosphere in M is a level set of a Busemann function β ξ (x, •). We will rather work with horospheres in T 1 M , which are sets of unit vectors normal to a horosphere of M . More precisely, we define the unstable horosphere of v ∈ T 1 M as the set

H(v) = {-grad y β v -(π(v), •) | y ∈ M , β v -(π(v), y) = 0}.
All the negative geodesic rays generated by the vectors in H(v) are in the same class v -, i.e. w -= v -for all w ∈ H(v).

It is worth mentioning that horospheres are embedded C 1 -submanifolds of T 1 M of dimension n -1. The set H of all unstable horospheres of T 1 M is a continuous foliation of T 1 M . More precisely, if we write B k for the unit ball of R k , for any vector v ∈ T 1 M , there is a homeomorphism

ϕ from B n-1 × B n to a neighborhood U of v such that, for every z ∈ B n , (i) ϕ(B n-1 × {z}) is the connected component of H(ϕ(0, z)) ∩ U containing ϕ(0, z), (ii) ϕ( • , z) is a C 1 diffeomorphism onto its image, (iii) and z → ϕ( • , z) is continuous in the C 1 -topology.
Horospheres on the quotient T 1 M are defined as the projection of the horospheres on T 1 M , and they also form a foliation

H of T 1 M .
In what follows, we denote by R1 the subset of T 1 M of rank 1 vectors. The set of geodesic endpoint pairs is

E( M ) := {(v -, v + ) ∈ ∂ M × ∂ M | v ∈ T 1 M }
and it has a subset of rank 1 geodesic endpoint pairs

E 1 ( M ) := {(v -, v + ) ∈ ∂ M × ∂ M | v ∈ R1 }.
Finally, we consider the map

P : T 1 M -→ ∂ M × ∂ M × R v -→ (v -, v + , β v -(0, π(v))),
where 0 ∈ M is a fixed point.

The boundary of M is useful because rank 1 geodesics are determined by their endpoints in ∂ M .

Lemma 2.1.

The map P | R1 : R1 → E 1 ( M ) × R is a homeomorphism.
Proof. It is injective because if two vectors are positively and negatively asymptotic then they bound a flat strip, in particular they cannot have rank 1. Clearly P is also surjective. The continuity of P and P -1 follows from Section II.2 and Lemma III.3.1 of [START_REF] Ballmann | Lectures on spaces of nonpositive curvature[END_REF].

Construction of the measure

In this section, the nonpositively curved Riemannian manifold M is assumed to have a closed rank 1 geodesic. It is also assumed that M is non-elementary. This means that the limit set Λ(Γ) ⊂ ∂ M of Γ is infinite. Recall that Λ(Γ) is the set of accumulation points in M ∪ ∂ M of an orbit Γx of a point x ∈ ∂ M . We refer the reader who is not familiar with the Patterson-Sullivan theory and the construction of the Bowen-Margulis measure to [START_REF] Patterson | The limit set of a Fuchsian group[END_REF][START_REF] Sullivan | The density at infinity of a discrete group of hyperbolic motions[END_REF][START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]. The main results have been generalized to rank 1 manifolds of nonpositive curvature [START_REF] Knieper | The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds[END_REF][START_REF] Knieper | Hyperbolic dynamics and Riemannian geometry[END_REF], especially we refer to the ergodic properties and the Hopf-Tsuji-Sullivan theorem [START_REF] Link | Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry[END_REF][START_REF] Link | Ergodic geometry for non-elementary rank one manifolds[END_REF].

We briefly recall some useful facts from this theory. The Poincaré series of Γ at s ∈ R is γ∈Γ e -sd(x,γ(y)) , where x and y are two points of M . It is natural to wonder whether this series converges or diverges, once noted that the result does not depend on the choice of x and y. We also define the critical exponent δ(Γ) of Γ by the formula

δ(Γ) = lim sup R→+∞ 1 R log #{γ ∈ Γ | d(x, γ(y)) ≤ R},
which is a strictly positive number. The reader may check that the Poincaré series converges for s > δ(Γ) and diverges for s < δ(Γ). The value of the series at s = δ(Γ) establishes two completely different behaviors of the action of the geodesic flow on M . We will work under the assumption that the Poincaré series diverges at s = δ(Γ), and we will simply say that Γ is divergent.

A δ-dimensional Γ-invariant conformal density is a family of measures {σ x} x∈ M on ∂ M with the following properties:

(i) ∀x ∈ M , supp σ x ⊂ Λ(Γ) (ii) ∀x, ỹ ∈ M , ∀ξ ∈ ∂ M , dσ ỹ dσ x (ξ) = exp (-δβ ξ (ỹ, x)) (iii) ∀x ∈ M , ∀γ ∈ Γ, γ * σ x = σ γ x
As it is shown in [START_REF] Link | Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry[END_REF], under the standing hypothesis there is only one of these objects with dimension δ = δ(Γ) (given by the Patterson construction). We fix a point 0 ∈ M and consider the measure σ 0 on ∂ M .

We now detail the definition of the Bowen-Margulis measure. For every two points η, ξ ∈ ∂ M , let us consider the subset π(P -1 ({(ξ, η)} × R)) of M . It is either empty, a single geodesic or a flat totally geodesic submanifold of dimension at least 2. We denote by Vol ξ,η the induced volume measure on the submanifold π(P -1 ({(ξ, η)} × R)).

We define an auxiliary measure μ on E( M ) by its density dμ(ξ, η) = e δ(β ξ (0,p ξ,η )+βη(0,p ξ,η )) dσ 0 (ξ) dσ 0 (η), where (ξ, η) ∈ E( M ) and p ξ,η is any point in π(P -1 ({(ξ, η)} × R)), which is non-empty. The Bowen-Margulis measure is defined on the Borel σ-algebra of T 1 M by the formula

µ BM (A) = E( M ) Vol ξ,η (π(P -1 ({(ξ, η)} × R) ∩ A)) dμ(ξ, η) (1)
for all Borel subsets A ⊂ T 1 M . The measure µ BM is Γ and g t -invariant, so it passes to a g t -invariant measure µ BM on the quotient T 1 M . This definition was made by Knieper in [START_REF] Knieper | The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds[END_REF] and generalizes the classical Bowen-Margulis measure for negatively curved manifolds.

Still under the assumption of divergence of Γ, it is known that the measure-preserving dynamical system (T 1 M, µ BM , g t ) is ergodic and conservative. Also, the action of Γ on ∂ M with respect to the measure class of σ 0 is ergodic. This is part of the Hopf-Tsuji-Sullivan dichotomy for rank 1 nonpositively curved manifolds [START_REF] Link | Ergodic geometry for non-elementary rank one manifolds[END_REF].

Proposition 2.2. Let M be a connected complete non-elementary manifold with nonpositive curvature. We assume that M contains a rank 1 closed geodesic and that the fundamental group Γ is divergent. Then the rank 1 set has full Bowen-Margulis measure.

Proof. Since Γ is divergent, the Bowen-Margulis measure µ BM is ergodic [START_REF] Link | Ergodic geometry for non-elementary rank one manifolds[END_REF]. The rank 1 set is g t -invariant, so it has either full or zero measure. The proof reduces to see that the rank 1 set has positive measure.

Let v ∈ T 1 M be a rank 1 g t -periodic vector and ṽ ∈ T 1 M one lift. Since ṽ has rank 1, the set P -1 ({(ṽ -, ṽ+ )} × R) is the geodesic of ṽ and Vol ṽ-,ṽ

+ (π(P -1 ({(ṽ -, ṽ+ )} × R) ∩ A)) = Leb({t ∈ R | g t (ṽ) ∈ A}). Hence, we have µ BM ( R1 ) = E 1 ( M ) ∞ dμ(ξ, η).
We observe that the support of σ 0 is the limit set Λ(Γ), which is a consequence of the minimality of the action of Γ on Λ(Γ) [START_REF] Ballmann | Axial isometries of manifolds of nonpositive curvature[END_REF]Proposition 2.8]. We also know that E 1 ( M ) is open [START_REF] Ballmann | Lectures on spaces of nonpositive curvature[END_REF]: there exist disjoint open neighborhoods U -and U + of ṽ-and ṽ+ in ∂ M , respectively, such that U -× U + ⊂ E 1 ( M ). These neighborhoods have positive σ 0 -measure because ṽ-and ṽ+ are in Λ(Γ) = supp σ 0 because v is periodic. This implies that the μ-measure of E 1 ( M ) is positive, so µ BM ( R1 ) = ∞. When R1 is projected to the quotient T 1 M , it still has positive measure.

To our knowledge, the hypothesis of the closed rank 1 geodesic is necessary even for the Patterson construction to work. The integral of f with respect to µ BM has a simpler expression now. If f :

T 1 M → R is a function, for (ξ, η) ∈ E 1 ( M ) and t ∈ R we denote f (ξ, η, t) := f (P -1 (ξ, η, t)).
The next result says that the volume factors of the Bowen-Margulis measure that appear in [START_REF] Ballmann | Axial isometries of manifolds of nonpositive curvature[END_REF], and which are used in the cited works, can be forgotten.

Corollary 2.3. Let f : T 1 M → R be a Borel function. We have

T 1 M f dµ BM = E 1 ( M ) R f (ξ, η, t) dt dμ(ξ, η).
Let H be a horosphere in T 1 M . It can be described as

H = {w ∈ T 1 M | w -= v -, β v -(π(v), π(w)) = 0}
, where v is any vector in H. We consider the map P H : H → ∂ M , v → v + , which is the restriction of P to H. For every ξ ∈ ∂ M , let us consider the subset P -1 H (ξ) of H, which can be either empty, a single point or a totally geodesic submanifold of H. We denote by Vol ξ,H the induced volume measure of the submanifold π(P -1 H (ξ)).

If η ∈ P H (H), we choose w ∈ P -1 H (η) and write φ H (η) = e δβη(0,π(w)) , which in fact only depends on η, but not on w. We set φ

H (η) = 0 if η ∈ ∂ M \ P H (H). We define a measure µ H on H such that µ H (A) = ∂ M Vol η,H (π(P -1 H (η) ∩ A))φ H (η) dσ 0 (η),
where A is a Borel subset of H. This is how we defined a family of measure {µ H } H∈ H in [START_REF] Burniol | Equidistribution of horospheres in nonpositive curvature[END_REF], which are exponentially expanded by the geodesic flow, i.e. they satisfy

µ gtH = e δt g t * µ H .
The reader can also check that they are Γ-invariant:

γ * µ H = µ γH , ∀γ ∈ Γ.
As in the Bowen-Margulis measure, the volume factor can be removed if the integration is restricted to an appropriate subset. This simplification is a consequence of the next result. We define the singular subset of T 1 M as the set S = T 1 M \ R1 , which is invariant by the geodesic flow and by Γ. We write

S+ = {v + ∈ ∂ M | v ∈ S}.
Proposition 2.4. Let M be a connected complete non-elementary manifold with nonpositive curvature. We assume that M contains a rank 1 closed geodesic and that the fundamental group Γ is divergent. Then the set S+ is σ 0 -negligible.

Proof. In [START_REF] Link | Ergodic geometry for non-elementary rank one manifolds[END_REF]Proposition 4] it is shown that Γ acts ergodically on ∂ M when Γ is divergent. Since S, so S+ , is Γ-invariant, then σ 0 ( S+ ) = 0 or σ 0 ( S+ ) = σ 0 (∂ M ).

The system (T 1 M, µ BM , g t ) is conservative by [START_REF] Link | Ergodic geometry for non-elementary rank one manifolds[END_REF]Proposition 4]. Then the conclusion of the Poincaré recurrence theorem holds and, since T 1 M is a second-countable metric space, µ BM -almost every vector of T 1 M is g t -recurrent. The lift Rec of the set of recurrent points to T 1 M is a subset of full µ BM -measure.

We have seen in Proposition 2.2 that R1 is also a subset of full measure. Then the intersection R1 ∩ Rec has also full measure. We denote by E ⊂ ∂ M the set of endpoints of rank 1 vectors in T 1 M such that their projections to T 1 M are both positively and negatively recurrent under the geodesic flow, i.e.

E = {v + | v ∈ R1 ∩ Rec}. The σ 0 -measure of E has to be positive because P ( R1 ∩ Rec) ⊂ E × E × R.
Finally, we observe that E and S+ are disjoint: if v is a rank 1 recurrent vector, all the vectors on its stable horosphere have rank 1 [5, Lemma 2.3], so all the vectors pointing to v + have rank 1 also, which means that v + / ∈ S+ . We conclude that the complement of S+ has positive measure, so S+ is negligible.

Corollary 2.5. Let H be a horosphere in T 1 M and A a Borel subset of H. Then

µ H (A) = P H (H)\ S+ 1 A (P -1 H (η))φ H (η) dσ 0 (η) = P H (H∩ R1 ) 1 A (P -1 H (η))φ H (η) dσ 0 (η),
Proof. On the one hand, φ H vanishes outside P H (H) and S+ has 0 measure, so we have

∂ M Vol η,H (π(P -1 H (η)∩A))φ H (η) dσ 0 (η) = P H (H)\ S+ Vol η,H (π(P -1 H (η)∩A))φ H (η) dσ 0 (η).
On the other hand, for η ∈ P H (H ∩ R1 ), the set P -1 H (η) consists of a single point, so the volume factor is just the indicator function of A.

Since P H (H) \ S+ ⊂ P H (H ∩ R1 ), we obtain µ H (A) = P H (H)\ S+ 1 A (P -1 H (η))φ H (η) dσ 0 (η).
The second equality is true because P H (H) \ S+ has full measure in P H (H ∩ R1 ).

An immediate consequence is that H ∩ R1 is of full µ H -measure in H. Finally, we recall some known conditions for a rank 1 manifold to have a closed rank 1 geodesic. Proposition 2.6. [START_REF] Ballmann | Lectures on spaces of nonpositive curvature[END_REF][START_REF] Ballmann | Manifolds of nonpositive curvature[END_REF] Let M be a connected complete non-elementary rank 1 Riemannian manifold of nonpositve curvature. If we assume one of the following:

• the limit set Λ(Γ) of Γ is equal to the whole boundary ∂ M ,

• M has finite Riemannian volume; then M has a closed rank 1 geodesic.

Continuity of the measures on the horocycles

Let M be a connected complete non-elementary Riemannian surface of nonpositve curvature with a closed rank 1 geodesic. We assume additionally that the group Γ is divergent. The next result expresses the continuity of the measure µ H with respect to the horocycle H, and is proved from the simplified expressions of the measures.

A horocycle H in T 1 M is diffeomorphic to R, so it makes sense to speak of the open interval (v, w) of H between the vectors v and w of H. In the same way we can define the closed interval [v, w], the half-open intervals [v, w), (v, w].

Proposition 2.7. The map

{(v, w) ∈ T 1 M × T 1 M | w ∈ H(v)} -→ R (v, w) -→ µ H(v) ((v, w))
is continuous.

Proof. The measure of the interval (v, w) is

µ H(v) ((v, w)) = P H(v) ((v,w))\ S+ 1 (v,w) (P -1 H(v) (η))φ H(v) (η) dσ 0 (η).
The set

P H(v) ((v, w)) is an interval of ∂ M that satisfies (v + , w + ) ⊂ P H(v) ((v, w)) ⊂ [v + , w + ].
Since σ 0 has no point masses [17, Proposition 5], we can write

µ H(v) ((v, w)) = (v + ,w + )\ S+ φ H(v) dσ 0 .
Let f : T 1 M → R be the real continuous function given by f (u) = exp(δβ u + (0, π(u))). Because of the definition of φ H(v) and the fact that (v

+ , w + ) \ S+ ⊂ P H(v) (H(v) ∩ R1 ), we have ∀η ∈ (v + , w + ) \ S+ , φ H(v) (η) = e δβη(0,π(P -1 H(v) (η))) = f (P -1 H(v) (η)).
Let v ∈ T 1 M and w ∈ H(v). We want to show the continuity of the map in the statement at (v, w). Let us explain step by step the needed estimates and at the end we will apply them. We fix ε > 0.

Boundedness of the integrated function. Let K be a compact neighborhood of [v, w] in T 1 M . By the continuity of horocycles there are open neighborhoods

V 1 ⊂ K and W 1 ⊂ K of v and w in T 1 M such that if v ∈ V 1 and w ∈ W 1 ∩ H(v ) the interval (v , w ) is contained in K. The function f is bounded on K by a constant C > 0.
For all v ∈ V 1 , for all w ∈ W 1 ∩ H(v ), we have the inclusion

P -1 H(v ) ((v + , w + ) \ S+ ) ⊂ (v , w ) ⊂ K,
which says that, for all η ∈ (v + , w + ) \ S+ , the quantity f (P -1 H(v ) (η)) is bounded by C. Approximation of intervals. Since σ 0 has no point masses and is outer regular, there are open intervals A and B around v + and w + in ∂ M of arbitrarily small measure. We choose A and B such that σ 0 (A) < ε/16C and σ 0 (B) < ε/16C. By the continuity of the projection T 1 M → ∂ M to the endpoint, there are neighborhoods V 2 and W 2 of v and w such that V 2+ ⊂ A and W 2+ ⊂ B. For every v ∈ V 2 and w ∈ W 2 , we have 1). Continuity of the integrated function uniform with respect to η. First, we apply the inner regularity of σ 0 : there exists a compact subset

(v + , w + ) (v + , w + ) ⊂ [v + , v + ] ∪ [w + , w + ] ⊂ A ∪ B,
((v + , w + ) (v + , w + )) ≤ σ 0 (A) + σ 0 (B) < ε 8C (Figure
F ⊂ (v + , w + ) \ S+ such that σ 0 (((v + , w + ) \ S+ ) \ F ) < ε/8C. If v ∈ T 1 M and η ∈ P H(v ) (H(v ) ∩ R1 ) then (v -, η) ∈ E 1 ( M ). Since F ⊂ (v + , w + ) \ S+ ⊂ P H(v) (H(v) ∩ R1 ), one has that {v -} × F is a subset of E 1 ( M ). The subset E 1 ( M ) is open in ∂ M × ∂ M and {v -} × F is compact, so there exists a neighborhood A of v -in ∂ M such that A × F ⊂ E 1 ( M ). Let U be a neighborhood of v such that U + is contained in A. Now there is a well defined map U × F -→ T 1 M (v , η) -→ P -1 H(v ) (η) = P -1 (v -, η, β v -(0, π(v ))),
which is continuous, because P -1 is continuous on E 1 ( M ) × R and the Busemann function depends continuously on its three variables. Composing by f we obtain a continuous map from U × F to R. Since F is compact, this map is continuous at v uniformly with respect the second variable: there exists a neighborhood U 0 ⊂ U of v such that, for all v ∈ U 0 , for all η ∈ F ,

|f (P -1 H(v ) (η)) -f (P -1 H(v) (η))| < ε 2σ 0 ((v + , w + ))
.

Conclusion.

We choose a subset F of (v + , w + ) \ S+ as explained above. Thanks to the integral expressions of the measures we can write, for v ∈ T 1 M and w ∈ H(v ),

|µ H(v) ((v, w)) -µ H(v ) ((v , w ))| ≤ F ∩(v + ,w + ) |f (P -1 H(v) (η)) -f (P -1 H(v ) (η))| dσ 0 (η) ( * ) + (((v + ,w + )\F )∪((v + ,w + ) (v + ,w + )))\ S+ (|f (P -1 H(v) (η))| + |f (P -1 H(v ) (η))|) dσ 0 (η). ( * * )
We can now see that both terms are small if (v , w ) is close to (v, w).

We set

V = U 0 ∩ V 1 ∩ V 2 and W = W 1 ∩ W 2 . Let v ∈ V and w ∈ W ∩ H(v )
, so that we can apply all the bounds found above. In the first integral ( * ), the integrand is bounded by ε 2σ 0 ((v + ,w + )) and the integrating set has measure at most σ 0 ((v + , w + )) (we suppose σ 0 ((v + , w + )) > 0 because if σ 0 ((v + , w + )) = 0 the integral is trivially 0). In ( * * ), the integrand is bounded by 2C and the measure of the integrating set is at most σ 0 ((v + , w + ) \ F ) + σ 0 ((v + , w + ) (v + , w + )) < ε/4C. So the result is less than ε.

Reparametrization of the horocyclic flow

Next we address the ergodic properties of the horocyclic flow. When one wants to study these properties, it often arises the question of the parametrization of the horocycles: while it is more natural to work with the Riemannian length parametrization, it may be required to use a parametrization more adapted to the dynamics of the flow. In [START_REF] Burniol | Equidistribution of horospheres in nonpositive curvature[END_REF] we proved a result of unique ergodicity of the horocyclic flow using the Margulis parametrization. In this section we will study how invariant measure are transformed by a change of parametrization. This will eventually allow us to conclude that the unique ergodicity also holds for the arc-length parametrization. We start by a general result of reparametrization of measures.

A general reparametrization result

Let X be a topological space. A flow on X is a map

X × R -→ X (x, t) -→ f t (x).
satisfying for all x ∈ X and t 1 , t 2 ∈ R, f 0 (x) = x and f t 1 +t 2 (x) = f t 1 (f t 2 (x)). By abuse of notation, we denote the flow by f t . We say that the flow f t is continuous if the map (x, t) → f t (x) is continuous. The orbit of x ∈ X is the set {f t (x)} t∈R . The point x is fixed if its orbit consists of a single point.

Let f t be a continuous flow on X. A Borel measure µ on X is said to be invariant by f t if for every Borel subset A of X we have, for all t ∈ R, µ(f t (A)) = µ(A). Let Mes(f t ) denote the set of locally finite Borel measures on X invariant by f t . Let Prob(f t ) be the subset of Mes(f t ) consisting of the probability measures. The flow f t is uniquely ergodic if the set Prob(f t ) consists of a single measure. The next result is due to M. Beboutoff and W. Stepanoff in 1940.

Theorem 3.1. [START_REF] Beboutoff | Sur la mesure invariante dans les systèmes dynamiques qui ne diffèrent que par le temps[END_REF] Let X be a separable metric space. Let f t be a continuous flow on X without fixed points. Let g t be another continuous flow on X with the same orbits than f t , i.e. for all x ∈ X, we have {g t (x)} t∈R = {f t (x)} t∈R . Then there is a bijective correspondence Φ : Mes(f t ) → Mes(g t ).

The correspondence Φ will be described in the next section in the setting of horocyclic flows. If X is a compact space, since every locally finite measure is in fact finite, the map Φ given in the theorem is an isomorphism between the spaces of finite Borel invariant measures of f t and g t . This induces a bijection between Prob(f t ) and Prob(g t ). In particular, f t is uniquely ergodic if and only if so is g t .

However, in a general separable metric space, we think that there is no reason why the map of the theorem would send finite measures to finite measures, and infinite measures to infinite measures.

Reparametrization of the horocyclic flow on a compact surface

In this section, we apply the ideas of Beboutoff and Stepanoff [START_REF] Beboutoff | Sur la mesure invariante dans les systèmes dynamiques qui ne diffèrent que par le temps[END_REF] to establish what happens to a measure invariant by a horocyclic flow under a change of parametrization. We specify the bijection Φ for a horocyclic flow on a compact surface.

In what follows, let M denote an oriented compact connected rank 1 surface with nonpositive curvature. Such a manifold M is complete, has a rank 1 closed geodesic, and the covering transformations group Γ is non-elementary and divergent [START_REF] Knieper | The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds[END_REF]Theorem 4.3]. Moreover, the Bowen-Margulis measure we defined in Section 2.2 is finite.

By horocyclic flow we mean a continuous flow h s on T 1 M whose orbits are the unstable horocycles, i.e ∀v ∈ T 1 M, {h s (v)} s∈R = H(v). The Lebesgue horocyclic flow h L s is given by the arc length of the horocycles: for any v ∈ T 1 M and s ∈ R, the vector h L s (v) is the vector of H(v) that we get by traveling a distance |s| on H(v) from v in the positive or negative direction accordingly to the orientation depending on the sign of s. We will say that the horocyclic flow h L s has the Lebesgue parametrization. A horocyclic flow h s on T 1 M is lifted to a horocyclic flow hs on the unitary tangent bundle T 1 M of the universal cover satisfying hs (γv) = γ hs (v) for every s ∈ R, v ∈ T 1 M , γ ∈ Γ. Conversely, a horocyclic flow hs on T 1 M with the property hs (γv) = γ hs (v) passes to the quotient T 1 M giving a horocyclic flow h s . Moreover, h s -invariant measures on T 1 M are in correspondence with hs -invariant measures on T 1 M which in addition are Γ-invariant.

The weak unstable manifold W wu (v) of a vector v ∈ T 1 M is the union of all horocycles along the geodesic generated by v,

W wu (v) = ∪ t∈R H(g t v) = {w ∈ T 1 M | w -= v -},
and has dimension 2. In the universal cover, the weak stable manifold W ws (v) = -W wu (-v) of a vector v ∈ T 1 M is a section in the sense of [START_REF] Beboutoff | Sur la mesure invariante dans les systèmes dynamiques qui ne diffèrent que par le temps[END_REF] in many cases, as explained in the next lemma.

Lemma 3.2. Let v ∈ T 1 M . Assume that the weak stable manifold W ws (v) of v has no rank 2 vectors. Then for every w ∈ T 1 M \ W wu (-v) = {u ∈ T 1 M | u -= v + }, there exists a unique time s ∈ R such that hs (w) ∈ W ws (v).
Proof. It is known that a nonflat compact surface M with nonpostive curvature satisfies the visibility axiom, which means that any two distinct points of the boundary ∂ M can be joined by a geodesic on M [START_REF] Eberlein | Surfaces of nonpositive curvature[END_REF]Proposition 2.5]. If v and w are as in the statement, the points v + and w -∈ ∂ M are distinct, so there is at least a geodesic between them. Therefore it exists a vector u in the unstable horocycle H(w) of w pointing to v + . This vector can be written as u = hs (w) for some s ∈ R and is in W ws (v).

To see the uniqueness, let us suppose that for some different reals s and s , hs (w) and hs (w) are in the weak stable manifold W ws (v). Then the vectors hs (w) and hs (w) are asymptotic both for positive and negative time, so the corresponding geodesics bound a flat strip. This would imply that these vectors have rank 2, which contradicts the hypothesis that W ws (v) has no such vectors.

The condition that W ws (v) has no rank 2 vectors is equivalent to the fact that v + is not in the set S+ of endpoints of vectors of rank 2. We know that S+ has zero σ 0measure, so its complement must be dense in ∂ M . This ensures that there are enough sections for the horocyclic flow.

We recall how an invariant measure is locally disintegrated. Let µ ∈ Mes( hs ) be an invariant measure. Given a Borel subset A of a section W ws (v), we consider the function

φ A : [0, 1] → R + ∪ {+∞} defined by φ A (s) = µ( h[0,s] (A)).
Let us assume that φ A (1) is finite. Then we have, for any integer number n ≥ 1,

φ A (0) = µ(A) ≤ µ( h[0,1/n) (A)) = 1 n n-1 k=0 µ( hk/n ( h[0,1/n) )) = 1 n µ( h[0,1) (A)) ≤ φ A (1) n ,
hence φ A (0) = 0. Moreover, for every two nonnegative numbers s, t such that s + t ≤ 1, we have φ A (s + t) = φ A (s) + φ A (t), thanks to the invariance of µ. Since φ A is monotonic, we deduce that it is linear, so there is a constant l A ≥ 0 such that φ A (t) = l A t for all t ∈ [0, 1]. We now define a measure µ W ws (v) on W ws (v) which associates the value l A = φ A (1) to the set A if φ A (1) is finite, and the value ∞ otherwise. It is not difficult to check that µ W ws (v) is a Borel locally finite measure and it is the same for two vectors on the same weak stable leaf. Furthermore, the measure µ is the product of the Lebesgue measure on each horocycle by the measure µ W ws (v) : for every Borel subset

E ⊆ T 1 M \ W wu (-v) we have µ(E) = W ws (v) R 1 E ( hs (u)) ds dµ W ws (v) (u).
Let h s be another horocyclic flow of T 1 M . The two horocyclic flows hs and h s on T 1 M are related by a change of time, given v ∈ T 1 M and s ∈ R, there exists a unique s = s(s , v) ∈ R such that hs (v) = h s (v). In fact, s as a function R×T 1 M → R satisfies:

(i) For all v ∈ T 1 M and s ∈ R, we have h s (v) = hs(s ,v) (v). (ii) s is continuous. (iii) For all v ∈ T 1 M , s( • , v) : R → R is strictly monotonic. (iv) For all v ∈ T 1 M , s(0, v) = 0. (v) For all v ∈ T 1 M and s 1 , s 2 ∈ R, s(s 1 + s 2 , v) = s(s 1 , v) + s(s 2 , hs(s 1 ,v) (v)). (vi) For all v ∈ T 1 M , s ∈ R and γ ∈ Γ, s(s , γv) = s(s , v)
The converse is also true: given a function s with the properties (ii)-(vi) we can define a new horocyclic flow by h s (v) := hs(s ,v) (v) which passes to the quotient T 1 M . As stated in Theorem 3.1, an invariant measure µ ∈ Mes( hs ) has a corresponding h s -invariant measure µ = Φ(µ) ∈ Mes( h s ), which is defined in [START_REF] Beboutoff | Sur la mesure invariante dans les systèmes dynamiques qui ne diffèrent que par le temps[END_REF] by the formula

µ (E) = W ws (v) R 1 E ( h s (u)) ds dµ W ws (v) (u), (2)
where v is any vector in T 1 M such that W ws (v) has no rank 2 vectors and E is any Borel subset of T 1 M \ W wu (-v). If the measure µ is in addition Γ-invariant, then so is the measure µ . We obtain, after normalization of measures, a correspondence between the sets Prob(h s ) and Prob(h s ), because T 1 M is compact. We also remark that the measures µ W ws (v) are independent of the parametrization of the horocyclic flow.

Horocyclic flows on a subset of T 1 M

Let Σ be Borel subset of T 1 M which is a union of horocycles and let Σ be its lift to T 1 M . We want to study the horocyclic flows on Σ, that is to say, the continuous flows defined on Σ whose orbits are horocycles. It is clear that a horocyclic flow h s on T 1 M is restricted to a horocyclic flow h s | Σ on Σ, and that a measure µ ∈ Mes(h s ) can be restricted to a measure µ| Σ ∈ Mes(h s | Σ ). In many situations, a certain parametrization of the horocyclic flow is just defined on a subset Σ and we would like to deduce ergodic properties for the flow on the whole space from this specific parametrization. The set Σ does not have to be compact, or even locally compact. Then, if we have two horocyclic flows h s and h s on Σ, we still have a bijection between Mes(h s ) and Mes(h s ) by Theorem 3.1, but we have no information about the subsets Prob(h s ) ⊂ Mes(h s ) and Prob(h s ) ⊂ Mes(h s ) or the relation between them.

Everything that has been said in the previous section regarding the decomposition of the measures is essentially true. If the set W ws (v) ∩ Σ is non-empty and W ws (v) has no rank 2 vectors, then W ws (v) ∩ Σ is a section for the flow hs and we can define a measure µ W ws (v)∩ Σ on this set.

Another horocyclic flow h

s on Σ is related to hs by a change of time s = s(s , v) as before. The h s -invariant measure µ = Φ(µ) associated to µ ∈ Mes( hs ) has a local expression like (2) as a product of the Lebesgue measures on horocycles by µ W ws (v)∩ Σ.

The Margulis parametrization

Some of the most relevant properties of the horocyclic flow are deduced thanks to the Margulis parametrization, which allows to apply the usual techniques of ergodic theory. As we mentioned, our result [START_REF] Burniol | Equidistribution of horospheres in nonpositive curvature[END_REF] establishes the unique ergodicity of this horocyclic flow for a certain class of nonpositively curved surfaces. The situation is different from negatively curved manifolds, because the Margulis parametrization can only be defined on a certain subset Σ of T 1 M . Here we will study the relation between the Margulis parametrization h M s on Σ and the Lebesgue parametrization h L s on T 1 M . We will see that the map Φ induces a bijection between Prob(h L s | Σ ) and Prob(h M s ), which is not trivial at all because no assumptions of compactness are made on Σ.

As we have seen in Section 2.2, the set of horocycles H of T 1 M admits a family of measures {µ H } H∈ H, which is exponentially expanded by the geodesic flow, µ gtH = e δt g t * µ H . To define the Margulis parametrization we parametrize each horocycle H by the measure µ H . Let us explore some further properties of these measures. Proof. By definition, µ H is the pullback of the measure σ 0 on ∂ M by some factor. Since σ 0 is finite and this factor is bounded on bounded sets, µ H is locally finite. We also know that σ 0 has no point masses, so neither does µ H .

The orientation of M induces an orientation on each horocycle H, so there are well defined positive and negative directions. A vector v ∈ H divides the horocycle into two infinite intervals, we write H R (v) for the one in the positive direction and H L (v) for the other.

Proposition 3.4. Let Σ be a Borel subset of T 1 M which is a union of horocycles. We assume that for every horocycle H ⊂ Σ we have:

(i) The measure µ H is of full support in H.

(ii) For one (hence for all) vector v ∈ H, the half horocycles H R (v) and H L (v) have infinite measure.

Then there exists a horocyclic flow hM s on Σ such that for all v ∈ Σ and s ∈ R, we have

µ H(v) ((v, hM s (v))) = |s|.
Moreover, the flow hM s satisfies, for every Borel subset A of the horocycle

H(v), µ H(v) (A) = Leb({s ∈ R | hM s (v) ∈ A}).
Proof. Let H be a horocycle in Σ and v ∈ H. We consider the function

m v : H → R defined by m v (w) = ±µ H(v) ((v, w)) if w ∈ H R/L (v) and m v (v) = 0.
(v))) = |m v ( hM s (v))| = |m v (m -1 v (s))| = |s|, (3)
from the definition of the flow. If we take the pullback m * v Leb of the Lebesgue measure by m v , we can see that m * v Leb coincides with µ H on the intervals (w 1 , w 2 ), so they are equal. In effect,

Leb(m v ((w 1 , w 2 ))) = Leb((s 1 , s 2 )) = |s 2 -s 1 | = µ H ((w 1 , hM s 2 -s 1 (w 1 ))) = µ H ((w 1 , w 2 ))
thanks to [START_REF] Ballmann | Manifolds of nonpositive curvature[END_REF], where w 1 = hM s 1 (v) and w 2 = hM s 2 (v). Then for every Borel subset

A of H, µ H (A) = Leb(m v (A)). But s ∈ m v (A) if and only if hM s (v) = m -1 v (s) ∈ A.
This proves the second property.

It remains to prove the continuity of the flow as a map from R × Σ → Σ. Fix a couple (s, v) ∈ R × Σ and consider a sequence ((s k , v k )) k of elements of R × Σ converging to (s, v). We need to show that hM s k (v k ) converges to hM s (v). We assume s ≥ 0, the other case being done analogously. We know that the horocycles H(w) depend continuously on w, so for each k there exists a vector w k ∈ H(v k ) such that the sequence {w k } k converges to hM s (v). By Lemma 2.7, we know that µ

H u (v k ) ((v k , w k )) converges to µ H u (v) ((v, hM s (v)))
= s when k tends to infinity. We deduce then that the measures of the intervals (w k , hM s k (v k )) go to 0. We claim that the distance between w k and hM s k (v k ) tends to 0. Assume, contrary to our claim, that, for some ε > 0, there is a subsequence k i such that the Riemannian distance d 1 (w k i , hM

s k i (v k i )) is greater than ε. Let us consider the points hL ε (w k i ), which are in the interval (w k i , hM s k i (v k i )). So the µ H(v k i ) -measure of (w k i , hL ε (w k i )
) also tends to 0. In the limit, we have

µ H(v) (( hM s (v), hL ε ( hM s (v)))) = 0
thanks again to the continuity of the measures. This is a contradiction because µ H(v) has full support in H(v) by hypothesis. Finally, since w k converges to hM s (v), the sequence hM s k (v k ) also converges to this point.

The point of this discussion is the following result that establishes a bijection between the finite invariant measures of the flows h L s | Σ and h M s .

Proposition 3.5. Let Σ be a subset of T 1 M satisfying the hypothesis of Proposition 3.4. Suppose that there are at least two distinct stable manifolds of vectors in Σ that do not have rank 2 vectors. Consider the horocyclic flow h M s on the set Σ ⊂ T 1 M there defined. The map Φ : Mes(h L s | Σ ) → Mes(h M s ) sends finite measures to finite measures, and infinite measures to infinite measures.

Proof. We first show that the image of a finite measure is finite. Let µ ∈ Mes( hL s | Σ) be a Γ-invariant measure whose projection to Σ ⊂ T 1 M is finite and let µ = Φ(µ) be its image in Mes( hM s ). Let D ⊂ T 1 M be a compact fundamental domain for the action of Γ. We want to show that µ (D ∩ Σ) is finite. Let v 1 , v 2 ∈ Σ two vectors such that W ws (v 1 ) and W ws (v 2 ) are distinct sections of the horocyclic flow (i.e. they do not contain any rank 2 vector). Take disjoint open neighborhoods A, B in ∂ M of the points v 1+ , v 2+ . By the continuity of the projection to the boundary we know that the sets

{w ∈ D | w -/ ∈ A}, {w ∈ D | w -/ ∈ B}
are closed in D, therefore compact. They form a cover of D. This reduces the proof to showing that µ (K ∩ Σ) is finite where K is a compact set such that v 0+ / ∈ K -, where v 0 is any vector of Σ whose stable manifold has no rank 2 vectors. Let us fix such a vector v 0 until the end of the proof.

By lemma 3.2, for every w ∈ K there exists a number s(w) such that hL s(w) (w) ∈ W ws (v 0 ) (Figure 2). We see s(w) as a continuous function from K to R. It is then bounded by some constant S > 0. The function w → hL s(w) (w) is also continuous, so the projection of K to W ws (v 0 ) is a compact set, say L. We have that The problem is reduced now to showing that sets of the form hL [-S,S] (L) ∩ Σ have finite µ -measure. Recall that our measure µ is the product of some measure µ W ws (v 0 )∩ Σ on W ws (v 0 ) ∩ Σ and the Lebesgue measures on the horocycles. Subsequently,

K ⊂ hL [-S,S] (L).
µ( hL [-S,S] (L) ∩ Σ) = 2S • µ W ws (v)∩ Σ(L ∩ Σ).
Since h[-S,S] (L) is covered by a finite number of images of D by elements of Γ, and D ∩ Σ has finite µ-measure by hypothesis, then the left hand side of the previous equation is finite, and so is µ W ws (v)∩ Σ(L ∩ Σ).

The measure µ of the set hL Proof. Every interval (v, w) on the horocycle H contains a rank 1 vector. Otherwise, all the vectors in [v, w] would be of rank 2, so the curvature would vanish everywhere on the geodesics they generate, and v and w would bound a flat strip. Since the rank 1 set is open, then (v, w) contains an interval of rank 1 vectors. Now we use the fact that µ H is a positive function times the projection of σ 0 on the rank 1 vectors (Corollary 2.5). Recall that σ 0 is supported on the limit set, which is the whole boundary ∂ M because M is compact, so the measure of (v, w) is strictly positive. This proves that µ H is fully supported.

To prove the infiniteness of the measures µ H on half-horocycles, we consider the map from T 1 M to R which sends v to the µ H(v) -measure of the horocyclic ball of center v and radius 1. This map is continuous, by Proposition 2.7, and T 1 M is compact, so it attains an absolute minimum α ∈ R. But the map is everywhere strictly positive, so the constant α is strictly positive too. Now, a half-horocycle H R/L in the tangent space of the universal cover M , contains infinitely many disjoint unstable balls of radius 1, each of them with measure at least α > 0. We conclude that the µ H -measures of the half-horocycles H R/L are infinite. This result ensures that the conditions of Proposition 3.4 are satisfied on the whole unitary tangent bundle. The Margulis parametrization can be defined everywhere for the class of nonpositively curved compact manifolds without flat strips. It passes to the quotient T 1 M as a horocyclic flow h M s . We deduce the main result of this section, namely the unique ergodicity of the horocyclic flow, thanks to the good properties of this parametrization. Theorem 4.2. Let M be an orientable nonpositively curved compact surface without flat strips. Then there is a unique Borel probability measure on T 1 M invariant by h M s . This measure is a constant multiple of the Bowen-Margulis measure.

We observe that the Bowen-Margulis measure µ BM is invariant by both the geodesic flow g t and the Margulis horocyclic flow h M s . The expanding property of the measures on the horocycles translates to the commuting property g t • h M s = h M se δt • g t between the flows. In this situation there is an argument of Coudène to show the unique ergodicity of h M s when µ BM is absolutely continuous with respect to the weak stable foliation [START_REF] Coudène | A short proof of the unique ergodicity of horocyclic flows[END_REF]. The geodesic flow of a nonpositively curved surfaces without flat strips does not meet this last requirement because there is not a local product structure on non-hyperbolic regions. Indeed, weak stable manifolds are tangent to unstable horocycles on rank 2 vectors. We thus need to adapt Coudène's argument.

Fortunately, the absolute continuity of µ BM with respect to the weak stable foliation only intervenes in the proof of the equicontinuity of averages along a horocycle pushed by the geodesic flow. The latter fact can be shown in our case using the disintegration of µ BM on the boundary ∂ M .

Figure 1 :

 1 Figure 1: Positions of the endpoints of the vectors v, w, v and w .

Lemma 3 . 3 .

 33 The measures µ H are locally finite and have no point masses.

Figure 2 :

 2 Figure 2: Definition of s(w).

  [-S,S] (L) ∩ Σ can also be decomposed asµ ( hL [-S,S] (L) ∩ Σ) = L∩ Σ s M (v,S) s M (v,-S) ds dµ W ws (v)∩ Σ(u),where s M is the change of time between the Lebesgue and the Margulis flows. If the quantities s M (v, -S) and s M (v, S) are bounded on the set L ∩ Σ, then the last integral is bounded and we obtain the desired result.

Proposition 4 . 1 .

 41 Let M be an oriented nonpositively curved compact surface without flat strips. Then for every horocycle H in T 1 M the measure µ H is of full support in H and the µ H -measure of each half-horocycle is infinite.

Lemma 4 . 3 .

 43 Let M be an orientable nonpositively curved compact surface without flat strips. Let f : T 1 M → R be a continuous function. Write M t (f )(v) :=

1 0

 1 f (g t (h M s (v)))ds for every v ∈ T 1 M and t ∈ R. The family of functions {M t (f )} t>0 is equicontinuous at every point v ∈ T 1 M .

  It is continuous because of Proposition 2.7, strictly increasing because µ H has full support in H and is surjective because both half-horocycles have infinite measure. Since H is homeomorphic to R, the map m v is a homeomorphism. We then define hM s (v) = m -1 v (s) for all s ∈ R. It is clear that the orbit of v is the horocycle H. Also, µ H has no point masses by Lemma 3.3, so we have m hM

	s (v) (w) = m v (w) -s. This leads to the additive property s+s . To see the equality of measures, we observe that s = hM s • hM hM
	µ H ((v, hM s

Acknowledgment

The author wishes to thank his Ph.D. advisor Yves Coudène for the many helpful suggestions.

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No. 754362.

It is not difficult to compute s M (v, ±S) = ±µ H(v) ((v, hL ±S (v))). Thanks to the continuity of the measure on the horocycles the functions v → µ H(v) ((v, hL ±S (v))) are continuous (and globally defined). They are then bounded on L because of the compactness. This completes the proof of the first implication.

Let µ ∈ Mes( hL s | Σ) any Γ-invariant measure and assume that its image µ = Φ(µ) in Mes( hM s ) induces a finite measure on the quotient Σ ⊂ T 1 M . We need to show that the µ-measure of D ∩ Σ is finite, where D is a compact fundamental domain. Similarly to the first situation we can reduce the problem to showing that the sets of the form hL [-S,S] (L) ∩ Σ, where S > 0 and L is a compact subset of W ws (v) and v ∈ Σ is a vector whose stable manifold has no rank 2 vectors, have finite µ-measure. Since µ( hL

We know that hL

[-S,S] (L) ∩ Σ has finite µ -measure, because it can be covered by finitely many images of D, and D ∩ Σ has finite measure. This measure is the integral of the function s M (v, S) -s M (v, -S) over L ∩ Σ with respect to µ W ws (v)∩ Σ. But the function s M (v, S) -s M (v, -S) is strictly positive because S > 0, and that implies that µ W ws (v)∩ Σ(L ∩ Σ) is finite, otherwise we would obtain an infinite integral. This proves that µ( hL [-S,S] (L)) is finite, as we wanted.

Corollary 3.6. There is a bijection between Prob(h L s | Σ ) and Prob(h M s ) given by the map µ → Φ(µ)/Φ(µ)(Σ).

We finally apply this result to the subset Σ 0 of T 1 M defined as the union of horocycles containing a rank 1 g t -recurrent vector,

This set satisfies the hypothesis of 3.4, so the Margulis parametrization of the horocyclic flow can be defined. In [START_REF] Burniol | Equidistribution of horospheres in nonpositive curvature[END_REF]Theorem 3.6], we proved that the Margulis flow on Σ 0 is uniquely ergodic. Thanks to the work in this paper, we now know that the same holds for the Lebesgue parametrization.

Theorem 3.7. Let M be an oriented rank 1 compact connected Riemannian surface with nonpositive curvature. The flow given by the parametrization by arc length of the horocycles restricted to Σ 0 is uniquely ergodic.

Unique ergodicity on compact surfaces without flat strips

We now look back to Proposition 3.4. There seem to be two difficulties in defining the Margulis parametrization on a certain horocycle. One is a rather technical difficulty, the fact that half-horocycles have infinite measure. We are not sure to what extent this could fail. Both in [START_REF] Burniol | Equidistribution of horospheres in nonpositive curvature[END_REF] and here, we have worked under hypothesis where it is true. In contrast, the fact that a horocycle H contains an interval of µ H -zero measure is a clear obstruction to the definition of the parametrization on H. This in fact can only happen if the interval consists of rank 2 vectors. This phenomenon is produced by flat strips.

A flat strip on the universal cover is a totally geodesic submanifold isometric to the space R × [0, r] for some r > 0. Given a horocycle H in T 1 M , an interval [v, w] ⊂ H consists only of rank 2 vectors if and only if the geodesics generated by the vectors in [v, w] form a flat strip. In this case, we say that H cuts a flat strip.

In the following we prove that, if M has no flat strips, the Margulis parametrization is defined on the whole unitary tangent bundle. With the help of this parametrization, we show that the horocyclic flow is uniquely ergodic using standard techniques.

Proof. We will rather work on the universal cover. Let f be a real continuous Γ-invariant function on T 1 M . Since Γ is cocompact, the absolute value of f is bounded by a real constant C > 0. From the definition of hM s , the average of f is

which is also written as

Given two vectors v, w ∈ T 1 M , we observe

The term ( * ) is independent of t and tends to 0 when w tends to v. This is because the function

. The function P -1 H(w) •P H(v) of the term ( * * ) is defined at least on the set P -1 H(v) (P H(w) (H(w))\ S+ ), which has full measure. The map (w, u) → P -1 H(w) • P H(v) (u) = P -1 (w -, u + , β w -(0, π(w))) is continuous on its domain. Given δ > 0, we can take w in a neighborhood of v such that the distance between u and P -1 H(w) • P H(v) (u) is less than δ for all u ∈ [v, hM 1 (v)] where the function is defined. Since u and P -1 H(w) • P H(v) (u) are on the same weak stable manifold, the distance between them is non-increasing when we apply g t . The distance between g t (u) and g t • P -1 H(w) • P H(v) (u) is therefore less than δ for all positive t. In this way, the term ( * * ) is bounded by the modulus of continuity ω f (δ) of f , which goes to 0 if δ → 0 because f is uniformly continuous. This proves the equicontinuity of the functions {M t (f )} t>0 at v.

It is straightforward to see that nonwandering rank 1 vectors are contained in the support of µ BM from the definition. But every vector is nonwandering and the rank 1 set is dense in T 1 M , because M is compact. So µ BM is fully supported. It is also known that the horocyclic foliation is transitive [START_REF] Eberlein | Geodesic flows on negatively curved manifolds[END_REF]Theorem 5.2]. These are the remaining ingredients of Coudène's theorem.

Finally, as a corollary, we can deduce the unique ergodicity of the horocyclic flow for other parametrizations, for example by the arc-length.