Spectral properties of horocycle flows for compact surfaces of constant negative curvature

Transcription

1 Proyecciones Journal of Mathematics Vol. 36, N o 1, pp , March 217. Universidad Católica del Norte Antofagasta - Chile Spectral properties of horocycle flows for compact surfaces of constant negative curvature Rafael Tiedra de Aldecoa P. Universidad Católica de Chile, Chile Received : September 216. Accepted : October 216 Abstract We consider flows, called W u flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of W u flows and we show that W u flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than C 2. This generalises recent results on time changes of horocycle flows. 21 Mathematics Subject Classification : 37A25, 37A3, 37C1, 37C4, 37D2, 37D4, 58J51. Keywords : Horocycle flow, Anosov flow, strong mixing, continuous spectrum, commutator methods. Supported by the Chilean Fondecyt Grant and by the Iniciativa Cientifica Milenio ICM RC122 Mathematical Physics from the Chilean Ministry of conomy.

2 96 Rafael Tiedra de Aldecoa 1. Introduction Horocycle flows for compact surfaces of constant negative curvature, and their generalisations, are a classical object of study in dynamical systems. Since the papers of G. A. Hedlund [8, 9] and. Hopf [1] in the 193 s, many properties of these flows have been put into evidence (far too many to be listed here). As for spectral properties, in particular the structure of the essential spectrum, not that many results are available. In 1953, O. S. Parasyuk [19] has shown that the classical horocycle flows for compact surfaces of constant negative curvature have countable Lebesgue spectrum. In 1974, A. G. Kushnirenko [13] has shown that some small time changes of the classical horocycle flows for compact surfaces of constant negative curvature are mixing, and thus have purely continuous spectrum in the orthocomplement of the constant functions. In 1977, B. Marcus [15] has shown that a large class of minimal W u flows associated with codimension one Anosov flows (in particular, a large class of reparametrisations of the classical horocycle flows for compact surfaces of negative curvature) are mixing. Finally, more recently, G. Forni and C. Ulcigrai [6], and the author [23, 25] have shown that sufficiently smooth time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions (in [6] Lebesgue spectrum is also obtained, and in [23] surfaces of finite volume are also considered). The purpose of this paper is to extend these last results on the absolutely continuous spectrum of time changes of horocycle flows to a very general class of time changes, namely, time changes in a regularity class slightly less than C 2. Our approach is the following. We consider as B. Marcus a continuous minimal W u flow, that is, a continuous minimal flow whose orbits are the unstable manifolds of a C 1+ε codimension one Anosov flow on a compact connected Riemannian manifold (minimality is required for the W u flow to be uniquely ergodic). Under some regularity assumptions on the W u flow and the Anosov flow,wegiveashortproofofthe strong mixing property of the W u flow. Then, under some additional regularity assumption, we construct a self-adjoint operator, called conjugate operator, satisfying a suitable positive commutator estimate with the selfadjoint generator of the W u flow (a Mourre estimate). Finally, we deduce from this positive commutator estimate and commutator methods that the generator of the W u flow has purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain

3 Spectral properties of horocycle flows for compact surfaces of that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than C 2. Here is a description of the content of the paper. In Section 2, we recall some definitions and results on codimension one Anosov flows and minimal W u flows, we introduce our regularity assumptions (Assumptions 2.1 and 2.4), and we give a proof of the strong mixing property of the W u flow (Theorem 2.6). In Section 3, we recall the needed facts on commutators of operators and regularity classes associated with them. Then, under some additional regularity assumption (Assumption 3.4), we construct the conjugate operator (Proposition 3.5), we prove the positive commutator estimate (Proposition 3.7), and we show that the generator of the W u flow has purely absolutely continuous spectrum in the orthocomplement of the constant functions (Theorem 3.8). Finally, we present the application of this result to time changes of the classical horocycle flows for compact surfaces of constant negative curvature (Remark 3.9). Acknowledgements. The author thanks D. Krejčiřík for interesting discussions and for his warm hospitality at the Department of Theoretical Physics of the Nuclear Physics Institute in Řež in January 215. The author also thanks the referees for various useful remarks. 2. Strong mixing In this section, we recall some definitions and results on codimension one Anosov flows and minimal W u flows, and we present a short proof of the strong mixing property of a class of minimal W u flows. We follow fairly closely the presentation and notations of B. Marcus [15], but we refer to the review papers [2, 17, 2] for additional information. A C 1+ε Anosov flow on a compact connected Riemannian manifold M with distance d : M M [, ) isac 1+ε flow {f t } t R with ε>, without fixed points, satisfying the following property: through each point x M pass three submanifolds W u (x), W s (x), and Orb(x) whose tangent spaces u x, s x,and x (respectively) vary continuously with x M and satisfy T x M = u x s x x. The regularity assumption on {f t } t R means that the function R M 3 (t, x) 7 f t (x) M is of class C 1+ε. So, {f t } t R has a vector field X f

4 98 Rafael Tiedra de Aldecoa which is Hölder continuous with exponent ε. The submanifolds W u (x), W s (x), and Orb(x) are called unstable manifolds, stable manifolds, and orbits (respectively), ½ and are characterised by ¾ W u (x) = y M lim t d f t (x),f t (y) =, ½ ¾ W s (x) = y M lim t + d f t (x),f t (y) =, n o Orb(x) = f t (x) t R. The following two facts are well-known [2]: (i) The families {W u (x)} x M and {W s (x)} x M each form a continuous foliation of M (that is, if y W u (x) thenw u (y) =W u (x), and W u (x) varies locally continuously with x M). (ii) f t W u (x) = W u f t (x) and f t W s (x) = W s f t (x) for all t R and x M. In this paper, we consider a codimension one Anosov flow {f t } t R such that {W u (x)} x M is a one-dimensional orientable continuous foliation of M which defines a continuous minimal flow {φ s } s R whose orbits are the unstable manifolds. Such a flow {φ s } s R is called a minimal W u flow or aminimalw u parametrisation, and it is uniquely ergodic with respect to a Borel probability mesure µ on M [5, 14]. However, the unique invariant measure µ is in general not absolutely continuous with respect to the volume element [15, Sec. 6] [16]. Since f t W u (x) = W u f t (x) for all t R and x M, there exists a function s : R R M R such that (2.1) f t φ s (x) = φ s (t,s,x) f t (x) for all s, t R and x M. This commutation relation, which describes how the Anosov flow {f t } t R expands W u orbits, is the starting point of our analysis. It generalises the well-known commutation relation [4, Rem. IV.1.2] between the geodesic flow and the classical horocycle flow on the unit tangent bundle of compact orientable surfaces of constant negative curvature. We recall three facts in relation with (2.1): (iii) The family {W u (x)} x M admits a uniformly expanding parametrisation, that is, a continuous parametrisation { e φ s } s R such that f t e φ s = eφ λ t s f t for some constant λ>1andalls, t R [14, Rem. 1.8 & Prop. 2.1] (the constant λ is equal to e h(f 1),withh(f 1 ) the topological entropy of f 1 ).

5 Spectral properties of horocycle flows for compact surfaces of (iv) Since { e φ s } s R is a continuous reparametrisation of {φ s } s R, and since {φ s } s R is uniquely ergodic with respect to the measure µ, the flow { e φ s } s R is uniquely ergodic with respect to a measure eµ given in terms of µ (however, the measure eµ is in general not absolutely continuous with respect to the measure µ [3, 3&4]). (v) The measure eµ is invariant under the Anosov flow {f t } t R [15, Rem. 6.4]. In order to be able to define a self-adjoint generator for the flow {φ s } s R and to have a simple relation between the measures µ and eµ, weassume the following regularity condition: Assumption 2.1. The flow {φ s } s R is of class C 1,and{ e φ s } s R is a C 1 reparametristation of {φ s } s R. Under Assumption 2.1, the flow {φ s } s R has a continuous vector field X φ, and there exists a function τ C 1 (M R; R) such that φ s (x) = eφ τ(x,s) (x), τ(x, ) =, τ(x, ) is strictly increasing and τ(x, s + t) = τ(x, s)+τ φ s (x),t for all s, t R and x M [3, 1] (the function τ(x, ) can be chosen strictly increasing for all x M because M is connected). These properties imply in particular that 2 τ (x, ) > for all x M. Therefore, the function 1 ρ : M R, x 7 ( 2 τ)(x, ), is well-defined and belongs to C M;(, ),theflow { φ e s } s R has continuous vector field ρx φ,andthemeasureeµ satisfies [11, Prop. 3]: Z (2.2) eµ = µ/eρ with eρ := ρ dµ ρ 1. Also, one verifies that the pullback operators in H := L 2 (M,µ) associated with the flow {φ s } s R, M U φ s ϕ := ϕ φ s, s R, ϕ H, define a strongly continuous 1-parameter group of unitary operators with Us φ C 1 (M) C 1 (M)foralls R. Thus, Nelson s criterion [21, Thm. VIII.1] implies that the generator of the group {Us φ } s R, H φ ϕ := s lim is 1 Us φ 1 ϕ, s

6 1 Rafael Tiedra de Aldecoa ½ ϕ D(H φ ):= ϕ H lim s 1 ¾ Us φ 1 ϕ <, s is essentially self-adjoint on C 1 (M) andgivenby H φ ϕ = ix φ ϕ, ϕ C 1 (M). On another hand, the pullback operators associated with the flow {f t } t R, U f t ϕ := ϕ f t, t R, ϕ H, are not unitary if ρ 6 1, but they define a strongly continuous 1-parameter group of bounded operators: Lemma 2.2. Suppose that Assumption 2.1 is satisfied. Then, Ut f B(H) for all t R, Us f Ut f = Us+t f for all s, t R, U f =1,andlim ε Ut+ε f ϕ =for all t R and ϕ H. U f t Proof. A direct calculation using (2.2) and the fact that eµ is invariant under {f t } t R implies for t R and ϕ H that Z Ut f ϕ 2 Z = deµ eρ ϕ f t 2 = deµ eρ f t ϕ 2 Thus, U f t (2.3) Z = B(H) with M M dµ ρ f t ρ M ϕ 2 max(ρ) min(ρ) kϕk2. s Ut f max(ρ) min(ρ). The group properties Us f Ut f = Us+t f for s, t R and U f =1areevident. Toshowthelastproperty,taket R and ϕ C(M). Then, the continuity of ϕ f t+ε ϕ f t and Lebesgue dominated convergence theorem imply that lim U f ε t+ε U f t ϕ 2 Z = dµ lim ϕ ft+ε ϕ f t 2 =. M ε Since C(M) isdenseinh and U f t+ε U f t B(H), this implies that lim ε Ut+ε f U t f ϕ =forallt R and ϕ H. 2 For the next lemma, we need the following result of B. Marcus:

7 Spectral properties of horocycle flows for compact surfaces of Lemma 2.3 (Lemma 3.1 of [15]). Let {φ s } s R be a minimal W u flow on M. Then, lim s s 1 s (t, s, x) =λ t uniformly in x M and t in a given compact interval of R. We also need to introduce as B. Marcus in [15, Sec. 4] the following regularity condition on the function s : Assumption 2.4. The derivative u t,s (x) := 1 2 s (t, s, x) exists and is continuous in s, t R and x M. Lemma 2.5. Suppose that Assumption 2.4 is satisfied. Then, Z (2.4) lim s s 1 1 s (t, s, x) = dµ u t, =ln(λ)λ t uniformly in x M and t in a given compact interval of R. Proof. Let I R be a compact interval, and take r, s R, t I and x M. Using successively the relation 1 s (t,,x) =, Assumption 2.4 and the cocycle equation s (t, r + s, x) =s (t, r, x)+s (2.5) t, s, φ r (x), we obtain 1 s (t, s, x) = Z s dr d dr M 1 s (t, r, x) = = Z s Z s dr u t, φ r (x). dr d d s dt ds s= (t, r + s, x) Therefore, it follows by the unique ergodicity of {φ s } s R that (2.6) lim s s 1 1 s (t, s, x) = lim s s 1 Z s Z dr u t, φ r (x) = dµ u t, M uniformly in x M and t I (the uniformity in t follows from the continuity of u t, in t). To show the second equality in (2.4), we note from Lemma 2.3 that

8 12 Rafael Tiedra de Aldecoa lim s s 1 s (t, s, x) =λ t uniformly in x M and t I. We also note from (2.6) that s 1 1 s (t, s, x) converges to some function of t uniformly in x M and t I. So,itfollows by a uniform convergence argument that lim s s 1 1 s (t, s, x) = d dt lim s s 1 s (t, s, x) = d dt λt =ln(λ)λ t uniformly in x M and t I. 2 In the following theorem, we give a short proof, inspired by [24, Thm. 4.1], ofthestrongmixingpropertyoftheflow {φ s } s R. It can be viewed as a simplified version of the proof of B. Marcus [15, Sec. 4] in the case Assumption 2.1 is satisfied (the proof of B. Marcus works without Assumption 2.1). Theorem 2.6 (Strong mixing). Suppose that Assumptions 2.1 and 2.4 are satisfied. Then, lim s Dψ, Us φ ϕ =for all ψ H and ϕ ker(h φ ). In particular, the flow {φ s } s R is strongly mixing with respect to the measure µ. Proof. (2.1) that Take ϕ C 1 (M), s, t R and x M. Then, we know from ϕ f t φ s (x) = ϕ φ s (t,s,x) f t (x). and using the relation s t= (,s,x)=s, we Applying the derivative d dt obtain X f ϕ φ s (x) = dt d ϕ φ s t= (t,s,x) f t (x) =lim ε ε 1n o ϕ φ s (ε,s,x) f ε (x) ϕ φ s (,s,x) f ε (x) +lim ε ε 1n o ϕ φ s f ε (x) ϕ φ s f (x) = 1 s (,s,x) X φ ϕ φ s (x) + X f (ϕ φ s ) (x), which is equivalent to (2.7) 1 s (,s, )U φ s H φ ϕ = iu φ s X f ϕ ix f U φ s ϕ. So, for ψ C 1 (M), we infer from Lemma 2.5 that

9 Spectral properties of horocycle flows for compact surfaces of lim s Dρ 1 ψ, Us φ H φ ϕ =ln(λ) 1 lim s Dρ 1 ψ, ln(λ)us φ H φ ϕ D =ln(λ) 1 lim s ρ 1 ψ, s 1 1 s (,s, )Us φ H φ ϕ =ln(λ) 1 lim s s 1D ρ 1 ψ, ius φ X f ϕ ln(λ) 1 lim s s 1D ρ 1 ψ, ix f Us φ ϕ = ln(λ) 1 lim s s 1D ψ, iρ 1 X f Us φ ϕ. But, the operator iρ 1 X f is symmetric on C 1 (M). So, lim s s 1D ψ, iρ 1 X f Us φ ϕ = lim s 1D iρ 1 X f ψ, U φ s s ϕ =, and thus D (2.8) lim ρ 1 ψ, U φ s s H φ ϕ =. Moreover, the set ρ 1 C 1 (M) isdenseinh because C 1 (M) isdensein H and ρ 1 : H H is an homeomorphism, and the set H φ C 1 (M) is dense in ker(h φ ) because C 1 (M) isacoreforh φ and H φ D(H φ )isdense in ker(h φ ). Therefore, (2.8) implies that lim s Dψ, Us φ ϕ =forall ψ H and ϕ ker(h φ ). 2 Under Assumptions 2.1 and 2.4, the result of Theorem 2.6 applies in particular to the case of reparametrisations of classical horocycle flows on the unit tangent bundle of compact connected orientable surfaces of constant negative curvature (see [15, Cor. 4.2]). 3. Absolutely continuous spectrum We know from Theorem 2.6 that, under Assumptions 2.1 and 2.4, the flow {φ s } s R is strongly mixing with respect to the measure µ. Therefore, its generator H φ has purely continuous spectrum in R \{}. Our goal in this section is to show that the spectrum of H φ is even purely absolutely continuous in R \{} under some additional regularity assumption. For this, we first need to recall some results on commutator methods borrowed from [1, 22] (see also the original paper [18] of É. Mourre).

10 14 Rafael Tiedra de Aldecoa 3.1. Commutators and regularity classes Let H be a Hilbert space with scalar product h, i antilinear in the first argument, denote by B(H) the set of bounded linear operators on H, and write k kboth for the norm on H and the norm on B(H). Let A be a self-adjoint operator in H with domain D(A), and take S B(H). For any k N, wesaythats belongs to C k (A), with notation S C k (A), if the map (3.1) R 3 t 7 e ita S e ita B(H) is strongly of class C k.inthecasek = 1, one has S C 1 (A) ifandonlyif the quadratic form D D D(A) 3 ϕ 7 ϕ, SAϕ Aϕ,Sϕ C is continuous for the topology induced by H on D(A). We denote by [S, A] the bounded operator associated with the continuous extension of this form, or equivalently i times the strong derivative of the function (3.1) at t =. If H is a self-adjoint operator in H with domain D(H) and spectrum σ(h), we say that H is of class C k (A) if(h z) 1 C k (A) forsome z C \ σ(h). So, H is of class C 1 (A) if and only if the quadratic form D(A) 3 ϕ 7 D ϕ, (H z) 1 Aϕ D Aϕ,(H z) 1 ϕ C extends continuously to a bounded form defined by the operator [(H z) 1,A] B(H). In such a case, the set D(H) D(A) isacoreforh and the quadratic form D D D(H) D(A) 3 ϕ 7 Hϕ,Aϕ Aϕ,Hϕ C is continuous in the topology of D(H) [1, Thm (b)]. This form then extends uniquely to a continuous quadratic form on D(H) whichcanbe identified with a continuous operator [H, A] fromd(h) to the adjoint space D(H). In addition, the following relation holds in B(H): h i (3.2) (H z) 1,A = (H z) 1 [H, A](H z) 1. Let H ( ) denote the spectral measure of the self-adjoint operator H, and assume that H is of class C 1 (A). If there exist a Borel set I R, a number a> and a compact operator K B(H) such that (3.3) H (I)[iH, A] H (I) a H (I)+K,

11 Spectral properties of horocycle flows for compact surfaces of then one says that H satisfies a Mourre estimate on I and that A is a conjugate operator for H on I. Also, one says that H satisfies a strict Mourre estimate on I if (3.3) holds with K =. One of the consequences of a Mourre estimate is to imply spectral results for H on I. We recall here these spectral results in the case where H is of class C 2 (A) (see[1, Sec ] and [22, Thm..1] for more details). Theorem 3.1. Let H and A be self-ajoint operators in a Hilbert space H, with H of class C 2 (A). Suppose there exist a bounded Borel set I R, a number a> and a compact operator K B(H) such that (3.4) H (I)[iH, A] H (I) a H (I)+K. Then, H has at most finitely many eigenvalues in I, each one of finite multiplicity, and H has no singular continuous spectrum in I. Furthermore, if (3.4) holds with K =, then H has only purely absolutely continuous spectrum in I (no singular spectrum) Absolutely continuous spectrum We show in this section that the spectrum of H φ is purely absolutely continuous in R \{} under some additional regularity assumption. We start with two technical lemmas. Lemma 3.2. Suppose that Assumptions 2.1 and 2.4 are satisfied. (a) If s R and ϕ C 1 (M), thenus φ X f U sϕ φ = X f ϕ + R s u dr, φ r X φ ϕ. (b) If z C \ R and ϕ C 1 (M), thenx f H φ z 1ϕ H. (c) If X f (ρ) C(M), thenu, =ln(λ)+ρ 1 X f (ρ). Proof. (a) Take s R and ϕ C 1 (M). Then, (2.7), the identity and Assumption 2.4 imply Us φ X f U sϕ φ = X f ϕ + 1 s (,s, )X φ ϕ = X f ϕ + (3.5) Z s 1 s (,, ) dr u,r X φ ϕ.

12 16 Rafael Tiedra de Aldecoa On another hand, the cocycle equation (2.5) implies for all r, s, t R that s (t, s, )=s t, s r, φ r ( ) s t, r, φ r ( ) = Ur φ s (t, s r, ) s t, r, ) U r. φ Deriving with respect to t and s, wethusobtain u t,s = U φ r u t,s r U φ r = u t,s r φ r. In particular, we have u,r = u, φ r, and the claim follows from (3.5). (b)wegivetheproofinthecaseim(z) > since the case Im(z) < is analogous. Take ϕ C 1 (M). Then, the formula H φ z 1ϕ = i R dr eirz Ur φ ϕ and point (a) imply X f H φ z 1ϕ R = i lim t dr eirz t 1 Ut f 1 U r φ ϕ (3.6) R = i lim t dr eirz t 1 R t ds U s f X f Ur φ ϕ R = i lim t dr eirz t 1 R t ds U s f Ur φ X f + R r dq u, φ q X φ ϕ. r Now, we know from (2.3) that Us f max(ρ) min(ρ) for all s R. Thus, we have e irz t 1 R t ds U s f Ur φ X f + R r dq u, φ q X φ ϕ r e rim(z) max(ρ) Xf min(ρ) ϕ + rku, k L (X,µ) X φ ϕ L 1 [, ),dr, and we can apply Lebesgue dominated convergence theorem to (3.6) to obtain X f H φ z 1ϕ = i R dr eirz lim t t 1 R t ds U s f Ur φ X f + R r dq = i R dr eirz Ur φ X f + R r dq u, φ q X φ ϕ H. u, φ q X φ ϕ (c) The proof is inspired by a result of L. W. Green in the case of the classical horocycle flows on the unit tangent bundle of compact connected

13 Spectral properties of horocycle flows for compact surfaces of orientable surfaces of negative curvature (see [7, q. (3.3) & Lemma 3.3]). Take ψ, ϕ C 1 (M). Then, the facts that { φ e s } s R has vector field ρx φ, that { φ e s } s R and {f t } t R preserve the measure eµ = µ/eρ and that f t φ e s = eφ λ t s f t imply D X φ ψ, X f ϕ = dt d D d ρ 1 ψ e φ s,ϕ f t t= ds s= = d D d dt ds ρ t= s= 1 ψ, ϕ f t φ e s = d D d dt ds ρ t= s= 1 ψ, ϕ φ e λ t s f t = dt d Dρ t= 1 ψ, λ t ρx φ ϕ f t = dt d Dψ f t,λ t= t X φ ϕ D D = X f ψ, X φ ϕ ψ, ln(λ)x φ ϕ. On another hand, point (a) and the symmetricity of iρ 1 X f on C 1 (M) imply D ψ, u,x φ ϕ = d ds D DρU sψ, φ ρ s= 1 X f U sϕ φ D = X φ ψ, X f ϕ + ρ 1 X f ρψ,x φ ϕ D D D = X φ ψ, X f ϕ + X f ψ, X φ ϕ + ψ, ρ 1 X f (ρ)x φ ϕ. Combining the two relations, we thus obtain D ψ, u, ln(λ) ρ 1 X f (ρ) X φ ϕ =, and we infer from the density of C 1 (M) inh and the density of X φ C 1 (M) in ker(h φ ) that (3.7) D ψ, u, ln(λ) ρ 1 X f (ρ) ϕ = for all ψ H and ϕ ker(h φ ). Now, Lemma 2.5, the fact that {f t } t R preserves eµ = µ/eρ, andtheergodicityof{φ s } s R imply that u, ln(λ) ρ 1 X f (ρ) ½ Z ¾ ϕ H dµ ϕ = =ker(h φ ). M So, we can set ψ =1andϕ = u, ln(λ) ρ 1 X f (ρ) in (3.7) to get Z M dµ u, ln(λ) ρ 1 X f (ρ) 2 =,

14 18 Rafael Tiedra de Aldecoa andtheninferthatu, ln(λ) ρ 1 X f (ρ) by the continuity of u, and ρ 1 X f (ρ). 2 Lemma 3.3. Let X be a C 1 vector field on M and g C 1 (M; R). Assume that the operator Aϕ := i(x + g)ϕ, ϕ C 1 (M), is symmetric in H. Then, A is essentially self-adjoint in H. Proof. Since X is of class C 1, X admits a C 1 flow {ζ s } s R [12, Thm. 3.43]. Thus, the operators R s V s ϕ := e dr (g ζr) ϕ ζ s, s R, ϕ C 1 (M), are well-defined operators in H. Simple calculations show that V s V t ϕ = V s+t ϕ and V ϕ = ϕ for s, t R and ϕ C 1 (M), that lim ε k(v s+ε V s )ϕk =fors R and ϕ C 1 (M), that V s C 1 (M) C 1 (M) fors R, and that d ds V sϕ = iav s ϕ for s R and ϕ C 1 (M). Furthermore, we have for s R and ψ, ϕ C 1 (M) the equalities D V s ψ, V s ϕ hψ, ϕi = = i Z s Z s dr d dr D V r ψ, V r ϕ D D dr AV r ψ, V r ϕ V r ψ, AV r ϕ =, due to the symmetricity of A. Therefore, the family {V s } s R satisfies on C 1 (M) the properties of a strongly continuous 1-parameter group of isometric operators in H with V s C 1 (M) C 1 (M) for all s R, andwith generator equal to A on C 1 (M). Since C 1 (M) isdenseinh, it follows that {V s } s R extends to a strongly continuous 1-parameter group of isometric (and thus unitary) operators in H with V s C 1 (M) C 1 (M) for all s R, and with generator equal to A on C 1 (M). Thus, Nelson s criterion [21, Thm. VIII.1] implies that A is essentially self-adjoint in H. 2 In the rest of the paper, in addition to Assumptions 2.1 and 2.4, we assume the following: Assumption 3.4. The vector fields X f and X φ are of class C 1, X f (ρ) C(M) and ρ 1 X f (ρ) C 1 (M).

15 Spectral properties of horocycle flows for compact surfaces of The conditions X f (ρ) C(M) andρ 1 X f (ρ) C 1 (M) areequivalent to the condition X f ln(ρ) C 1 (M). So, if one prefers, one can replace the two conditions X f (ρ) C(M) andρ 1 X f (ρ) C 1 (M) in Assumption 3.4 by the single condition X f ln(ρ) C 1 (M). In the next proposition we define and prove the self-adjointness of the conjugate operator. Intuitively, the conjugate operator is constructed as follows. First, we take the sum of the vector field 2iX f and its divergence iρ 1 X f (ρ) to get a symmetric operator on C 1 (M). Then,wetakethe Birkhoff average of the resulting operator along the flow {φ s } s R to take into account the unique ergodicity of {φ s } s R. Proposition 3.5 (Conjugate operator). Suppose that Assumptions 2.1, 2.4, and 3.4 are satisfied. Then, the operator A t ϕ := 1 t Z t ds U φ s 2iX f + iρ 1 X f (ρ) U sϕ, φ t >, ϕ C 1 (M), is essentially self-adjoint in H (and its closure is denoted by the same symbol). Proof. Since ρ 1 X f (ρ) =eρ 1 X f (eρ), the operator 2iX f + iρ 1 X f (ρ) is symmetric on C 1 (M). Therefore, the operator A t is also symmetric on C 1 (M) because U φ s C 1 (M) C 1 (M) for all s R. To show that A t is essentially self-adjoint on C 1 (M), we take ψ, ϕ C 1 (M). Then, Lemma 3.2(a) implies that D ψ, A t ϕ D = ψ, 1 R t t 2iX ds f +2i R s dr u, φ r X φ + i ρ 1 X f (ρ) φ s ϕ D = ψ, i 2X f + a t X φ + b t ϕ (3.8) with a t := 2 t Z t ds Z s dr u, φ r and b t := 1 t Z t ds ρ 1 X f (ρ) φ s. Furthermore, Assumption 3.4, Lemma 3.2(c) and Leibniz integral rule imply that X f, X φ, a t and b t are of class C 1. Therefore, we can apply Lemma 3.3 with X := 2X f + a t X φ and g := b t to conclude that A t is essentially self-adjoint in H. 2

16 11 Rafael Tiedra de Aldecoa Lemma 3.6. Suppose that Assumptions 2.1, 2.4, and 3.4 are satisfied. Then,wehavefort> (a) H φ i 1 C 1 (A t ) with h 1,At i 1ct 1 h 1,ct i i H φ i =2 H φ i H φ H φ i H φ i and c t := 1 Z t ds u, φ s, t (b) H φ i 1 C 2 (A t ). Proof. (a) Set A := 2iX f + iρ 1 X f (ρ) on C 1 (M) andtakeϕ C 1 (M). Then, we know from Lemma 3.2(b)-(c) that D 1ϕ, D 1ϕ H φ + i Aϕ Aϕ, H φ i D 1ϕ, = 2iX f + iρ 1 1 X f (ρ) H φ + i H φ i H φ i ϕ D 1 1ϕ H φ + i H φ + i ϕ, 2iX f + iρ 1 X f (ρ) H φ i = i ds d 1ϕ, nd2ix f + iρ s= 1 1 X f (ρ) H φ + i H φ i Us φ s ϕ D 1 H φ + i U s φ s ϕ, 2iX f + iρ 1 X f (ρ) H φ i = ds d 1ϕ, ndh φ + i 2X f + u, H φ i 1U φ s ϕ s= D 1ϕ, 1ϕ o H φ + i U φ s 2X f + u, H φ i = 2 ϕ, d 1 1U ds H φ i nx f H φ i φ s Us s= φ 1 o À X f H φ i ϕ ϕ, ds d 1 1U H φ i nu, H φ i φ s Us s= φ 1 o À u, H φ i ϕ. R For the first term, the equation H φ i 1 = i dr e r Ur φ (valid in the strong sense) and Lemma 3.2(a) imply 1 1U d ds H φ i nx f H φ i φ s Us s= φ 1 o X f H φ i ϕ = i d R ds s= dr e r Us φ 1 H φ i U sx φ f Us φ X f Ur φ ϕ = i ds d R s= dr e r Us φ 1 R H φ i s dt u, φ t Ur φ X φ ϕ = i R dr e r H φ i 1u, Ur φ X φ ϕ 1u, = i H φ i H φ H φ i 1ϕ. 1ϕ o

17 Spectral properties of horocycle flows for compact surfaces of For the second term, a direct calculation gives d 1 1U φ H φ i nu, H φ i ds s Us s= φ u, H φ i = h 1,u, i H φ i ϕ. So, putting together the last equations, we get D 1ϕ, D 1ϕ H φ + i Aϕ Aϕ, H φ i D n 1u, 1 h = ϕ, 2i H φ i H φ H φ i + i H φ i 1 o ϕ 1,u, io ϕ. Therefore, for each t>weobtain D D 1ϕ H φ + i 1ϕ, At ϕ A t ϕ, H φ i = 1 R nd t t ds φ H φ + i 1U s ϕ, AU sϕ φ D AU sϕ, φ o φ H φ i 1U s ϕ = 1 R D t t ds U sϕ, φ n 1u, 1 2i H φ i H φ H φ i h 1,u, io + H φ i U sϕ φ D n 1ct 1 h 1,ct io = ϕ, 2i H φ i H φ H φ i + i H φ i ϕ with c t = 1 R t t ds (u, φ s ), and the claim follows by the density of C 1 (M) in D(A t ). (b) We know from point (a) that H φ i 1 C 1 (A t )with h 1,At i i H φ i 1ct 1 h 1,ct i =2 H φ i H φ H φ i H φ i 1ct 1 h 1,ct i =2 H φ i 1ct +2i H φ i H φ i H φ i. So, it is sufficient to show that c t C 1 (A t ). For this, we note that c t C 1 (M) due to the assumption ρ 1 X f (ρ) C 1 (M), Lemma 3.2(c) and Leibniz D integral rule. D Then, we use (3.8) to get for ϕ C 1 (M) c t ϕ, A t ϕ A t ϕ, c t ϕ D = Dϕ, ic t 2X f + a t X φ + b t ϕ ϕ, i 2X f + a t X φ + b t c t ϕ D = ϕ, i 2X f (c t )+a t X φ (c t ) ϕ with 2X f (c t )+a t X φ (c t ) a bounded multiplication operator, and we note that this implies the claim because C 1 (M) isdenseind(a t ). 2

18 112 Rafael Tiedra de Aldecoa In the next proposition, we show that H φ satisfies a strict Mourre estimate on compact subsets of (, ) and(, ). Weusetheoperator ln(λ)a t as conjugate operator on (, ) andtheoperatorln(λ)a t as conjugate operator on (, ). Proposition 3.7 (Strict Mourre estimate). Suppose that Assumptions 2.1, 2.4, and 3.4 are satisfied. (a) If I (, ) is a compact set with I σ(h φ ) 6=, then there exist t> and a> such that i H φ (I) hih φ, ln(λ)a t H φ (I) a H φ (I). (b) If J (, ) is a compact set with J σ(h φ ) 6=, thenthereexist t> and a> such that i H φ (J) hih φ, ln(λ)a t H φ (J) a H φ (J). Before the proof, we recall that the flow {φ s } s R is ergodic. Therefore, the spectrum of the operator H φ in R \{} is not empty, and thus there exist compact sets I (, ) such that I σ(h φ ) 6= and/or compacts sets J (, ) such that J σ(h φ ) 6=. Proof. (a) Let t>. Then, we know from Lemma 3.6(a) that C 1 ln(λ)a t with H φ i 1 h 1, i 1ct 1 i H φ i ln(λ)at = 2ln(λ) H φ i H φ H φ i h 1,ct i +ln(λ) H φ i. This, together with (3.2), implies i that H φ(i) hih φ, ln(λ)a t H φ(i) h 1, i = H φ i H φ(i) i H φ i ln(λ)at H φ i H φ(i) h =2ln(λ) H φ(i)c t H φ H φ(i) ln(λ) H φ i H 1,ct i φ(i) H φ i H φ i H φ(i) =2 ln(λ) 2Hφ H φ(i)+2ln(λ) H φ(j) c t ln(λ) H φ H φ(i) h i ln(λ) H φ i H φ(i) H φ i 1,ct ln(λ) H φ i H φ(i) a I H φ(i)+2ln(λ) H φ(i) c t ln(λ) H φ H φ(i) h i ln(λ) H φ i H φ(i) H φ i 1,ct ln(λ) H φ i H φ(i)

19 Spectral properties of horocycle flows for compact surfaces of with a I := 2 ln(λ) 2 inf(i) >. Furthermore, since {φs } s R is uniquely ergodic, we obtain from Lemma 2.5 that Z lim c t ln(λ) = dµ u, ln(λ) = t M uniformly on M. Therefore,ift> is large enough, there exists a (,a I ) such that i H φ (I) hih φ, ln(λ)a t H φ (I) a H φ (I). (b) The proof is similar to that of point (a). 2 The next theorem is the main result of the paper. Theorem 3.8 (Absolutely continuous spectrum). Suppose that Assumptions 2.1, 2.4, and 3.4 are satisfied. Then, H φ has purely absolutely continuous spectrum, except at, where it has a simple eigenvalue with eigenspace C 1. Proof. We know from Lemma 3.6(b) that H φ is of class C 2 ln(λ)a t for all t>. Moreover, we know from Proposition 3.7(a) that for each compact set I (, ) withi σ(h φ ) 6= there exist t>anda> such that i H φ (I) hih φ, ln(λ)a t H φ (I) a H φ (I). Therefore, it follows from Theorem 3.1 that H φ has purely absolutely continuous spectrum in (, ). Since the same holds for (, ), H φ has purely absolutely continuous spectrum, except at, where it has a simple eigenvalue with eigenspace C 1 due to the ergodicity of the flow {φ s } s R. 2 We conclude with some remarks on Theorem 3.8. Remark 3.9. (a) Under Assumptions 2.1, 2.4, and 3.4, the result of Theorem 3.8 applies, as in the case of Theorem 2.6, to reparametrisations of classical horocycle flows on the unit tangent bundle of compact connected orientable surfaces of constant negative curvature. Our regularity assumptions on the function ρ are X f (ρ) C(M) and ρ 1 X f (ρ) C 1 (M). Therefore, Theorem 3.8 extends the results of [6, 23, 25] on the absolutely continuous spectrum of time changes of the classical horocycle flows on the unit tangent bundle of compact orientable surfaces of constant negative curvature since the function corresponding to ρ in

20 114 Rafael Tiedra de Aldecoa [6, 23, 25] is at least of class C 3 (in [23], the function ρ is of class C 2, but it satisfies another additional assumption). Also, the result of Theorem 3.8 answers the question on the regularity of the function ρ raised in [25, Rem. 3.4]. The result of [6] on Lebesgue spectrum and the result of [23] for surfaces of finite volume are of a different nature and are not covered by Theorem 3.8. (b) In the case of the classical horocycle flows on the unit tangent bundle of compact connected orientable surfaces of negative curvature, L. W. Green has shown in [7, Thm. B] that the curvature of the surface is necessarily constant if the flow {φ s } s R admits a C 2 uniformly expanding reparametrisation { φ e s } s R (and thus a function ρ of class C 1 ). Since it is possible that a similar argument also applies under our regularity assumptions (an anonymous referee signaled this to us), we prefer not to mention the case of compact surfaces of variable negative curvature. However, we hope that in some future the technics and the regularity assumptions presented in this manuscript could be improved in order to cover explicit examples of compact surfaces of variable negative curvature. (c) In Lemma 3.6, Proposition 3.7 and Theorem 3.8, we stated and proved the results using only the operator H φ under study, and not its square (H φ ) 2 as we did in [23, 25]. This allowed us to simplify the exposition in comparison to [23, 25]. References [1] W. O. Amrein, A. Boutet de Monvel, and V. Georgescu. C -groups, commutator methods and spectral theory of N-body Hamiltonians, volume 135 of Progress in Mathematics. Birkhäuser Verlag, Basel, (1996). [2] D. V. Anosov. Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, No. 9 (1967). Translated from the Russian by S. Feder. American Mathematical Society, Providence, R.I., [3] M. Beboutoff and W. Stepanoff. Sur la mesure invariante dans les systèmes dynamiques qui ne diffèrent que par le temps. Rec. Math. [Mat. Sbornik] N.S., 7 (49), pp , (194).

21 Spectral properties of horocycle flows for compact surfaces of [4] M.B.BekkaandM.Mayer.rgodic theory and topological dynamics of group actions on homogeneous spaces, volume 269 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, (2). [5] R. Bowen and B. Marcus. Unique ergodicity for horocycle foliations. Israel J. Math., 26(1), pp , (1977). [6] G. Forni and C. Ulcigrai. Time-changes of horocycle flows. J. Mod. Dyn., 6(2), pp , (212). [7] L. W. Green. Remarks on uniformly expanding horocycle parameterizations. J. Differential Geom., 13(2), pp , (1978). [8] G. A. Hedlund. Fuchsian groups and transitive horocycles. Duke Math. J., 2(3), pp , (1936). [9] G. A. Hedlund. Fuchsian groups and mixtures. Ann. of Math. (2), 4(2), pp , (1939). [1]. Hopf. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91, pp , (1939). [11] P. D. Humphries. Change of velocity in dynamical systems. J. London Math.Soc.(2), 7, pp , (1974). [12] M. C. Irwin. Smooth dynamical systems, volume17ofadvanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River dge, NJ, (21). Reprint of the 198 original, With a foreword by R. S. MacKay. [13] A. G. Kushnirenko. Spectral properties of certain dynamical systems with polynomial dispersal. Moscow Univ. Math. Bull., 29(1), pp , (1974). [14] B. Marcus. Unique ergodicity of the horocycle flow: variable negative curvature case. Israel J. Math., 21(2-3), pp , (1975). Conference on rgodic Theory and Topological Dynamics Kibbutz Lavi, (1974). [15] B. Marcus. rgodic properties of horocycle flows for surfaces of negative curvature. Ann. of Math. (2), 15 (1), pp , (1977).

22 116 Rafael Tiedra de Aldecoa [16] G. A. Margulis. Certain measures that are connected with U-flows on compact manifolds. Funkcional. Anal. i Priložen., 4 (1), pp , (197). [17] S. Matsumoto. Codimension one Anosov flows, volume27oflecture Notes Series. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, (1995). [18] É. Mourre. Absence of singular continuous spectrum for certain selfadjoint operators. Comm. Math. Phys., 78 (3), pp , 198/81. [19] O. S. Parasyuk. Flows of horocycles on surfaces of constant negative curvature. Uspehi Matem. Nauk (N.S.), 8 (3(55)): , [2] Y. B. Pesin. Geodesic flows with hyperbolic behavior of trajectories and objects connected with them. UspekhiMat.Nauk, 36 (4(22)), pp. 3 51, 247, (1981). [21] M. Reed and B. Simon. Methods of modern mathematical physics. I. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, second edition, (198). Functional analysis. [22] J. Sahbani. The conjugate operator method for locally regular Hamiltonians. J. Operator Theory, 38(2), pp , (1997). [23] R. Tiedra de Aldecoa. Spectral analysis of time changes of horocycle flows. J. Mod. Dyn., 6(2), pp , (212). [24] R. Tiedra de Aldecoa. The absolute continuous spectrum of skew products of compact lie groups. Israel J. Math., 28(1), pp , (215). [25] R. Tiedra de Aldecoa. Commutator methods for the spectral analysis of uniquely ergodic dynamical systems. rgodic Theory Dynam. Systems, 35(3), pp , (215). R. Tiedra de Aldecoa Facultad de Matemáticas Pontificia Universidad Católica de Chile Av. Vicuña Mackenna 486, Santiago Chile rtiedra@mat.puc.cl