Time-Changes of Horocycle Flows

We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the spectrum of smooth time-changes and show that the spectrum is absolutely continuous with respect to the Lebesgue measure on the real line and that the maximal spectral type is equivalent to Lebesgue.


Introduction
The classical horocycle flow is a fundamental example of a unipotent, parabolic (non-hyperbolic) flow. Its dynamical properties have been studied in great detail. It is known that the flow is minimal [10], uniquely ergodic [6], has Lebesgue spectrum and is therefore strongly mixing [16], in fact mixing of all orders [14], and has zero entropy [9]. Its finer ergodic and rigidity properties, as well as the rate of mixing, were investigated by M. Ratner is a series of papers [17], [18], [19], [20] (for results on the rate of mixing of the geodesic as well as horocycle flows see also the paper by C. Moore [15]). In joint work with L. Flaminio [5], the first author has proved precise bounds on ergodic integrals of smooth functions. In the case of finite-volume, noncompact surfaces, the horocyle flow is not uniquely ergodic and the classification of invariant measures is due to Dani [4]. The asymptotic behaviour of averages along closed horocycles has been studied by D. Zagier [24], P. Sarnak [21], D. Hejal [11] and more recently in [5] and by A. Strömbergsson [22]. Horocycle flows on general geometrically finite surfaces have been studied by M. Burger [3].
Not much is known for general smooth parabolic flows, not even for smooth perturbations of classical horocycle flows in the compact case. In fact, even the dynamics of non-trivial smooth time-changes is poorly understood. Our paper addresses the latter question. By the classification of horocycle invariant distributions [5] and by the related results on asymptotic of ergodic averages for classical horocycle flows (see [5], [2]), it is known that smooth time-changes which are measurably trivial form a subspace of countable codimension, so that the generic smooth time-change is not even measurably conjugate to the horocycle flow. It is therefore interesting to know, perhaps as a step towards a better understading of parabolic dynamical systems, to what extent the dynamical properties of the horocycle flow persist after a smooth time-change. The most important result to date is the proof by B. Marcus more than thirty years ago that all time-changes satisfying a mild differentiability conditions are mixing [14]. Marcus results generalized earlier work by Kushnirenko who proved mixing for all time-changes with sufficiently small derivative in the geodesic direction [13].
A. Katok and J.-P. Thouvenot have conjectured that "Any flow obtained by a sufficiently smooth time change from a horocycle flow has countable Lebesgue 1 spectrum" (see [12], Conjecture 6.8). In fact, the question on the spectral type of smooth time changes of horocycle flows was already asked in Kushnirenko's paper [13]. There the author is able to prove the relative absolute continuity of the spectrum of (restricted) smooth perturbations of skew-shifts, but cannot extend his results to time-changes of horocyle flows. In our paper we prove sharp bounds on the rate of equidistribution and mixing of smooth time-changes of the classical horocycle flow on the unit tangent bundle of a compact hyperbolic surface (see Theorem 2 and Theorem 3 in Sections 3 and 4 respectively). We then derive results on the spectrum of smooth time-changes (in Section 6), most notably we prove that the spectrum is absolutely continuous with respect to the Lebesgue measure (Theorem 6, in Section 6.2). We finally prove that the maximal spectral type is indeed equivalent to Lebesgue (Theorem 7, in Section 6.3).
The guiding idea of our work is that Marcus' mixing mechanism can be made quantitative by the more recent quantitative results on the rate of equidistribution for horocycle flows (see [3], [5], [2]). In fact, Marcus argument is based on the equidistribution of long horocycle-like arcs, that is, arcs which are long in the horocycle direction and bounded in the complementary directions. Sharp results on the rate of equidistributions of horocycle-like arcs were recently obtained in [2] as a refinement of earlier results for horocycle arcs [3], [5]. Finally, our estimates on the rate of mixing for time-changes would be far from optimal, and definitely too weak to derive any significant spectral result, without a key bootstrap trick. Thanks to this bootstrap trick we can prove that decay of correlations of the horocycle flows are indeed stable under any smooth time-change.
Spectral results are derived from square-mean bounds on twisted ergodic integrals of smooth functions which are equivalent to bounds on the Fourier transform of the spectral measures. A well-known difficulty in this approach is that the decay of correlations of a general smooth function under the horocycle flow is not square-integrable, so that it would seem hopeless to prove absolute continuity of the spectrum in this way. However, our results on decay of correlations of timechanges are precise enough, thanks to the bootstrap trick, to give optimal, and hence square-integrable, decay of correlations for smooth coboundaries. Once it is established that all smooth coboundaries have absolutely continuous spectral measures, it follows (for instance by a density argument) that the spectrum is purely absolutely continuous. Our estimates on decay of correlations of coboundaries are also crucial in the proof that the maximal spectral type is Lebesgue.
Let us remark that while in this paper we only deal with horocycle flows for compact hyperbolic surfaces, most of the methods and results can presumably be extended to the non-compact, finite volume case with appropriate modifications.
Structure of the paper. In Section 2 we introduce basic definitions, notation and properties of smooth time-canges of the classical horocycle flow. In Section 3 we recall the results on the invariant distributions for the classical horocycle flow from [5] and, from the results on the asymptotics of ergodic integrals in [5,2], we derive analogous results for smooth-time changes (Theorem 2). These quantitative equidistribution results are used in Section 4 together with a key bootstrap trick to make quantitative Marcus'mixing argument for smooth-time changes and derive the quantitative mixing result in Theorem 3. In Section 5 we prove mean-square bounds on twisted ergodic integrals of smooth functions. Finally, in Section 6 we prove our main spectral results. We first prove a local estimate (Theorem 5 in § 6.1), then absolute continuity if the spectrum (Theorem 6 in § 6.2) and finally that the maximal spectral type is Lebesgue (Theorem 7 in § 6.3).

Time-changes of horocycle flows
Let {U, V, X} the basis of the Lie algebra sl(2, R) of P SL(2, R) given by the generators U and V of the stable and unstable horocycle flows and by the generator X of the geodesic flow, respectively. The following commutation relations hold: Let {h U t } and {h V t } denote respectively the stable and unstable horocyle flows and let {φ X t } denote the geodesic flow on a compact homogeneous space M := Γ\P SL(2, R). They are defined respectively by the multiplicative action on the right of the 1-parameter subgroups of the group P SL(2, R) listed below: A smooth time-change of the (stable) horocycle flow is a flow {h α t } on M defined as follows. Let τ : M × R → R be a smooth cocycle over the flow {h U t }, that is, a function with the property that We denote by α : M → R + the infinitesimal generator of the cocycle τ : M ×R → R, that is, the function defined as follows: The time-change {h α t } is the flow on M generated by the smooth vector field (3) U α =: U/α .
One can check that {h α t } is given by the formula The flow {h α t } preserves the (smooth) volume form vol α := αvol; in fact L Uα vol α = ı Uα vol α = ı U vol = L U vol = 0 .
We will assume below that the function α : M → R is everywhere strictly positive and is normalized so that The time-change {h α t } is parabolic, in fact the infinitesimal divergence of trajectories is at most quadratic with respect to time (as is the case for the standard horocycle flow). The tangent flow {Dh α t } on T M is described as follows. Lemma 1. The tangent flow {Dh α t } on T M is given by the following formulas: Proof. For any vector field W on M , let us write It follows that the function (a t , b t , c t ) satisfies the following system of O.D.E.'s: the initial condition is (a 0 , b 0 , c 0 ) = (0, 1, 0), hence the unique solution of the Cauchy problem is given by the functions: If W = X, the initial condition is (a 0 , b 0 , c 0 ) = (0, 0, 1), hence the unique solution of the Cauchy problem is given by the functions: Formula (6) is therefore proved.

Cohomological equation and quantitative equidistribution
It is a general fact that all properties of the cohomological equation and of the asymptotics of ergodic integrals for all smooth time-changes of any smooth flow can be read from the corresponding properties of the flow itself.
Let L 2 (M ) := L 2 (M, vol) denote the Hilbert space of square-integrable function with respect to the standard volume form and , for any r ≥ 0 let W r (M ) ⊂ L 2 (M ) denote the standard Sobolev spaces on the compact manifold M and let W −r (M ) denote the dual spaces.
Let L 2 (M, vol α ) denote the Hilbert space of square-integrable function with respect to the {h α t }-invariant volume form and let Let D ′ (M ) be the space of distributions on M . For any distribution D ∈ D ′ (M ), let D α be the distribution defined as follows: The distribution D α is well-defined and belongs to the dual Sobolev space W −r (M ) whenever α ∈ W r (M ) and D ∈ W −r (M ) for any r > 3/2. In fact, the Sobolev space W r (M ), endowed with the standard structure of Hilbert space and with the standard product of functions, is a Banach algebra for r > dim(M )/2, which is here the case since M is a 3-dimensional manifold. The subspace I −r α (M ) ⊂ W −r (M ) of invariant distributions for the time-change {h α t } can be described in terms of the subspace I −r U (M ) ⊂ W −r (M ) of invariant distributions for the horocycle flow (described in [5]). Lemma 2. Let r > 3/2 and let α ∈ W r (M ). The following holds: Let us recall that we say that a function f on M is a coboundary for the flow {h α t } if there exists a function u on M , called the transfer function, such that U α f = u. The subspace of coboundaries for the time-changes of the horocycle flow is described by the following dictionary. Proof. It follows immediately from the definition of coboundary recalling that by definition U α = U/α.
By the above lemmas the theory of the cohomological equation for time-changes is reduced to that for the classical horocycle flow, developed in [5]. We state the main results below for the convenience of the reader.
By the theory of unitary representations for SL(2, R) (see [1], [7], [8]), the Sobolev spaces W r (M ) split as direct sums of irreducible sub-representations and each irreducible sub-representations characterized up to unitary equivalence by the spectral value µ ∈ σ( ) of the restriction of the Casimir operator (a normalized generator of the center of the enveloping algebra), that is, for all r ∈ R, the following splitting holds: Non-trivial irreducible unitary representations belong to three different series: the principal series (µ ≥ 1/4), the complementary series (0 < µ < 1/4) and discrete series (µ ≤ 0). We recall that the positive spectral values of the Casimir operator coincide with the eigenvalues of the Laplace-Beltrami operator on the (compact) hyperbolic surface, while the non-positive spectral values are given by the set of non-positive integers {−n 2 + n|n ∈ Z + }.
Let α ∈ W r (M ) for any r > 3/2 be any strictly positive function. For every Casimir parameter µ ∈ R, let By the above splitting (7), there is also a splitting The subspace I −r α (H µ ) has dimension 2 for all irreducible sub-representations of the principal and complementary series (µ > 0) and for irreducible sub-representations of the discrete series it has dimension 1 if r > 1+ In case a solution u ∈ H µ exists then u ∈ W s (H µ ) for all s < r − 1 and the following a priori bound holds: there exists a constant C s,r > 0, indipendent on µ ∈ σ( ), such that The quantitative equidistribution for time-changes, and in fact the complete asymptotics of ergodic averages, can also be derived from the corresponding results for the classical horocycle flow, derived in [5] and [2]. By change of variable there exists a function T : M × R → R such that, for any function f on M , By the above formula for the constant function f = 1, it follows that the function T : M × R → R is given by the following identity: for all (x, T ) ∈ M × R, The asymptotics of the function T : M × R → R for large T ∈ R, uniformly with respect to x ∈ M , can be derived by the quantitative equidistribution result of M. Burger [3] (see also [5]).
Let µ 0 > 0 be the smallest strictly positive eigenvalue of the Casimir operator (that, as we remarked, coincide with the smallest eigenvalue for the hyperbolic Laplacian on the compact surface Γ\{z ∈ C| Im(z) > 0}). Let ν 0 ∈ [0, 1) and ǫ 0 ∈ {0, 1} be the parameters defined as follows: Proof. Let us assume µ 0 = 1/4. The argument in the case µ 0 = 1/4 is similar. By the normalization condition (5), it follows from the identity (10) and from [5], Theorem 1.5, that there exists a constant C ′ r > 0 such that, for all (x, T ) ∈ M × R, . Thus by formula (12), for all x ∈ M and for all T ≥ T σ , The argument is thus completed.
By Lemma 4, the asymptotics of ergodic integrals for smooth time-changes of horocycle flows is entirely analogous and can be derived from the corresponding results for horocycle flows, proved in [5] (see also [2]).

Quantitative Mixing
In this section we show that the following result on quantitative mixing for smooth time-changes of horocycle flows can be derived by combining quantitative equidistribution results with B. Marcus' proof of mixing [14].
Let us denote · X the graph norm of the densely defined Lie derivative operator L X : L 2 (M ) → L 2 (M ), that is, for all functions g ∈ L 2 (M ) which belong to the maximal domain dom(L X ) ⊂ L 2 (M ) of L X , g X := ( g 2 0 + Xg 2 0 ) 1/2 . Theorem 3. For any r > 11/2 and for any α ∈ W r (M ), there exists a constant C ′′ r (α) > 0 such that the following holds. For any zero-average function f ∈ W r (M ) ∩ L 2 0 (M, vol α ), for any function g ∈ dom(L X ) and for any t ≥ 1, The rest of this section is devoted to the proof of Theorem 3. The key idea of Marcus' method is to consider the push-forward under the flow of a geodesic arc.
Let σ ∈ R + and (x, t) ∈ M × R. Let γ σ x,t : [0, σ] → M be the parametrized path defined as follows: We begin by computing the velocity of the path γ σ x,t and its length. Let Proof. The velocity of the geodesic path {φ X s (x)|s ∈ [0, σ]} is given at all points by the geodesic vector field X on M , hence dγ σ . The formula for the velocity of the path γ σ x,t then follows from Lemma 1. From the quantitative equidistribution result for the flow {h α t } we derive the following estimate on the velocity function.
Proof. The function v t (x, s)/t is given by an ergodic average along the trajectories of the time-change {h α t } (evaluated at the point φ X s (x) ∈ M ) of the function Xα/α−1 which has average equal to −1 with respect to the {h α t }-invariant volume vol α . In fact, the latter is given by the formula vol α = αvol, hence by the normalization condition (5) and the invariance of the volume vol under the geodesic flow, The result then follows from the quantitative equidistribution theorem stated above (see Theorem 2).
We then estimate the asymptotics (as t → +∞) of the integral LetÛ α be the 1-form on M uniquely defined by the conditions ı UαÛα = 1 and ı XÛα = ı VÛα = 0 .
Proof. By Lemma 5 and by the above definitions, The formula follows immediately.
The above formula can be refined by integration by parts: Lemma 9. For all s > 0 and all (x, t) ∈ M × R + , we have: Proof. By Lemma 1 and by formula (16) it follows that

By integration by parts we also have
as claimed in the statement of the lemma.
By the Sobolev embedding theorem, we have the following estimate: Lemma 10. For any r > 7/2, there exists a constant C r (α) > 0 such that the following holds. Let α ∈ W r (M ). For all (x, s) ∈ M × [0, σ] and for all t > 0, For any r > 7/2, let C r (α) > 0 be the constant of Lemma 10 and let By a bootstrap argument we derive the following bound.
Lemma 11. For any r > 7/2 and for any σ ∈ (0, σ r (α)), there exist a time t r,σ (α) > 0 and a constant C r,σ (α) > 0 such that, for any continuous function f on M , for any x ∈ M and for all t > t r,σ (α), Proof. By Lemmas 8 and 10, it follows that By definition C r (α)σ < 1 for any σ ∈ (0, σ r (α)), hence by Lemma 6 there exists t r,σ (α) > 0 such that, for all t > t r,σ (α), Thus we conclude that the statement holds if we set We recall that the path γ σ x,t is contained in a single leaf of the weak-stable foliation of the geodesic flow. By Lemma 5, its length is estimated below.
Lemma 12. For all r > 5/2 and for all α ∈ W r (M ), there exists a constant C ′ r (α) > 0 such that, for all σ > 0 and for all ( Proof. By Lemma 5 we have hence the first estimate in the statement is immediate, while the second estimate follows from the uniform linear bound on |v t (x, s)| established in Lemma 6.
From the results of [2] on the asymptotics of ergodic averages for the horocycle flow (see in particular [2], Theorem 1.3) we then derive the following bound: Lemma 13. Let r > 11/2. For any α ∈ W r (M ) and for any σ > 0, there exists a constant C r,σ (α) > 0 such that the following holds. For any zero-average function f ∈ W r (M ) ∩ L 2 0 (M, vol α ), for all x ∈ M , for all S ∈ (0, σ], and for all t > 0, Proof. Precise bounds on the integrals can be derived from [2], Theorem 1.3, if the function αf ∈ W r (M ) is supported on irreducible unitary components of the the principal and complementary series. By Lemma 12, it follows from [2], Theorem 1.3, that there exists a constant C r,σ (α) > 0 such that, for all S ∈ (0, σ] and for all t > 0, In case αf ∈ W r (M ) is supported on irreducible unitary components of the discrete series, since the path γ σ x,t is contained in a leaf of the weak stable foliation of the geodesic flow (tangent to the integrable distribution {X, U }) it follows from the methods of [2] that By the above formula and by Lemma 12 the argument is completed. Lemma 14. Let r > 11/2. For any α ∈ W r (M ) and σ ∈ (0, σ r (α)), there exists a constant C r,σ (α) > 0 such that the following holds. For any zero-average function f ∈ W r (M ) ∩ L 2 0 (M, vol α ), for all x ∈ M , for all S ∈ (0, σ], and for all t > t r,σ (α), Proof. By the bounds proved in Lemma 6 and Lemma 11, the statement follows immediately from the above Lemma 13 .
By the invariance of the the standard volume under the geodesic flow and by integration by parts we derive the following formula.
It is immediate to derive from Lemma 14 and from Lemma 15 estimates on the decay of correlations for sufficiently smooth functions in Theorem 3.

Proof of Theorem 3. By Lemma 14 and Lemma 15, there exists a constant
Finally, by taking into account that, for all f , g ∈ W r (M ) and all t ≥ 0, αg > , the theorem follows from the estimate in formula (18). In fact, by the Sobolev embedding theorem, for all g ∈ dom(L X ), αg X ≤ 2 α r g X .

Twisted ergodic integrals: square-mean estimates
We prove below L 2 bounds on twisted ergodic integrals of smooth functions, that is, on integrals of the form for any function w ∈ L ∞ (R, C) and for any sufficiently smooth function f on M . In the next sections we will derive from these bounds estimates on spectral measures of smooth time-changes of the horocycle flow. The relevant twisted integrals are those with twist function equal to an exponential function which are related to the Fourier transforms of spectral measures.
We remark that the question on optimal uniform bounds for twisted ergodic integrals (with an exponential twist) is open even for the classical horocycle flow. Non-optimal, polynomial bounds can be derived from estimates on the rate of equidistribution and on the rate of mixing by an argument due to A. Venkatesh [23]. Such uniform bounds are closely related to estimates on the rate of equidistribution of time-T maps of horocycle flows.
For our main results on spectral measures of time-changes, uniform bounds on twisted ergodic integrals are not needed. In fact, the key step is to prove squaremean estimates of the type below.
Lemma 16. Let r > 11/2 and let α ∈ W r (M ). There exists σ r (α) > 0 such that for all σ ∈ (0, σ r (α)) the following holds. There exists a constant C r,σ (α) > 0 such that for any bounded weight function w ∈ L ∞ (R + , C), for any continuos function f ∈ dom(X) ⊂ L 2 (M, vol α ) and for all T > 0, Proof. By the invariance of the volume form under the reparametrized horocycle flow and by change of variables we have that For every fixed t ∈ R, let w t ∈ C 0 (R, C) be the bounded weight function defined as By the formula of Lemma 15 we have that The statement of the lemma then follows from Lemma 11 and from the estimate Theorem 4. Let r > 11/2. For any α ∈ W r (M ), there exists a constant C r (α) > 0 such that the following holds. For any bounded weight function w ∈ L ∞ (R + , C), for any zero-average function f ∈ W r (M ) ∩ L 2 0 (M, vol α ) and for all T > 0, Proof. By the equidistribution estimates proved in Lemma 13, for any σ > 0 there exists a constant C r,σ (α) > 0 such that for all f ∈ W r (M ) ∩ L 2 0 (M, vol α ) and for all x ∈ M and all τ > 0, The statement of the theorem then follows from Lemma 16 by integration.
We remark that more refined estimates can be proved for functions supported on finite codimensional subspaces orthogonal to irreducible components of the complementary series and for coboundaries. From the estimates for coboundaries, which will be fully carried out below, we will deduce that all smooth time-changes of the horocycle flow have absolutely continuous spectrum.

Spectral theory
In this final section we state and prove spectral results for smooth time changes of horocycle flows. In Section 6.1 we first show a local estimate on spectral measures of smooth functions (see Theorem 5). Exploiting the L 2 -bounds established in the previous Section 5, in Section 6.2 we prove the absolute continuity of the spectrum (with countable multiplicity) for all smooth time-changes of the classical horocycle flow (Theorem 6). Finally in Section 6.3 we show that the maximal spectral type is always equivalent to Lebesgue (Theorem 7). 6.1. Local estimates. Let µ f denote the spectral measures of a function f ∈ L 2 (M, vol α ). We recall that µ f is a complex measure on the real line. The main result derived in this section is the following.
Theorem 5. Let r > 11/2 and let α ∈ W r (M ). There exists a constant C r (α) > 0 such that, for any function u ∈ W r+1 (M ) and for any ξ ∈ R \ {0}, δ| log δ| ξ 2 , for all δ ∈ (0, |ξ|/2) , hence the measure µ u has local dimension 1 at all points ξ ∈ R \ {0}, that is, The above result implies that the local dimension of spectral measures of smooth functions is everywhere equal to 1, but it is off by a logarithmic term from the sharpest possible bound, which would imply that spectral measures of sufficiently smooth functions are absolutely continuous with bounded densities. In the following section ( §6.2) we nevertheless show how one can derive absolutely continuity from the mean-square bounds for ergodic integrals of coboundaries.
We first estimate twisted ergodic integrals of coboundaries. Let us assume that f ∈ L 2 (M ) is a smooth coboundary for the time-change {h α t } on M , that is, there exists a function u ∈ W r (M ) such that  Proof. By the definition of the path γ σ x,t in formula (15) and by Lemma 5 we have (19) γ S x,t hence the statement of the lemma follows.
Corollary 1. Let r > 11/2 and let α ∈ W r (M ). There exists a constant C r (α) > 0 such that for any bounded weight function w ∈ L ∞ (R + , C), for all coboundaries f = U α u with a transfer function u ∈ W r+1 (M ) and for all T > 0, Proof. By Sobolev embedding theorem if u ∈ W r+1 (M ) then u, Xu ∈ L ∞ (M ) and the following estimate holds: there exists a constant C r > 0 such that The statement then follows by integration from Lemma 16 and Lemma 17.
Proof of Theorem 5. By the spectral theorem, for any function f ∈ L 2 (M, vol α ) and any ξ ∈ R , By a simple computation, there exists a constant C > 0 such that Let u ∈ dom(U α ) ⊂ L 2 (M, vol α ) and let f := U α u ∈ L 2 (M, vol α ). By the spectral theorem we have that hence there exists a constant C ′ > 0 such that, for all ξ ∈ R \ {0}, By formulas (20) and (21) and by Corollary 1, it follows that there exists a constant C r (α) > 0 such that, for all functions u ∈ W r+1 (M ) and for all T > 0, The statement of the theorem can readily be derived from formulas (23) and (24). 6.2. Absolute continuity. We show in this section that that the spectral measure µ f of any function f ∈ L 2 (M, vol α ) is absolutely continuous with respect to the Lebesgue measure on the real line and hence that any smooth time-change of the horocycle flow has absolutely continuous spectrum.
The Theorem is derived below from the following estimate on decay of correlations of coboundaries.
Lemma 18. Let r > 11/2 and let α ∈ W r (M ). There exist constants C r (α) > 0 and t r (α) > 0 such that the following holds. For any continuous coboundary f = U α u with a transfer function u ∈ L ∞ (M ) such that Xu ∈ L ∞ (M ) and for any g ∈ dom(X), for all t > t r (α), Proof. The statement follows readily from Lemma 11, Lemma 15 and Lemma 17. let µ f,g denote the joint spectral measure. By Theorem 6 the measure µ f,g is absolutely continuous with respect to Lebesgue. Whenever µ f,g has square-integrable density we have It follows from Lemma 18 that, whenever f = U α u is a coboundary with transfer function u ∈ C 1 (M ), then the Fourier transform of the spectral measure µ f,g , hence its density, is square-integrable, so that the identity in formula (26) holds. From formula (26) by translation under the geodesic flow and by integration we derive that, for any σ > 0 and for any function g ∈ W r (M ) ⊂ dom(X), By Lemma 18 the double integral in formula (27) is absolutely convergent, hence It follows from Lemma 19 that the function is bounded on M , hence it vanishes identically by formula (28) and by density of the subspace W r (M ) ⊂ L 2 (M ).
Let us prove the above commutation identity. For S = 0 it holds for all (r, s, t) ∈ R 3 . For all (r, s) ∈ R 2 let x r,s := (φ X s • h V r )(x). By Lemma 1, by differentiation of equation (34) with respect to S ∈ R, we find (X • φ X S • h α t )(x r,s ) = ∂T r,s ∂S (t, S) (U α • h α Tr,s(t,S) • φ X S )(x r,s ) + v Tr,s(t,S) (x r,s , S) (U α • h α Tr,s(t,S) • φ X S )(x r,s ) + (X • h α Tr,s(t,S) • φ X S )(x r,s ) . Since the above equation holds by the definition of the function T r,s in formula (32), the commutation relation (34) holds as well. We have thus proved the flow-box identities (33) from which the differentiation formulas (31) follow immediately.
By Lemma 20 and Lemma 21 we derive our conclusive spectral result.
Theorem 7. Let r > 11/2 and let α ∈ W r (M ). The maximal spectral type of the time-change {h α t } of the (stable) horocycle flow {h U t } with infinitesimal generator U α := U/α is equivalent to Lebesgue.