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J. Math. Pures Appl. 80, 2 (2001) 153–175 ? 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0021-7824(00)01182-X/FLA CENTRAL LIMIT THEOREM FOR THE GEODESIC FLOW ASSOCIATED WITH A KLEINIAN GROUP, CASE ? > d/2 N. ENRIQUEZ a,1, J. FRANCHI b,2, Y. LE JAN c,3 a Laboratoire de Probabilités de Paris 6, 4 place Jussieu, tour 56, 3ème étage, 75252 Paris cedex 05, France b Université Louis Pasteur, I.R.M.A., 7 rue René Descartes, 67084 Strasbourg cedex, France c Université Paris Sud, Mathématiques, Bâtiment 425, 91405 Orsay, France Manuscript received 1 June 2000 ABSTRACT. – Let ? be a geometrically finite Kleinian group, relative to the hyperbolic space H=Hd+1, and let ? denote the Hausdorff dimension of its limit set, that we suppose here strictly larger than d/2. We prove a central limit theorem for the geodesic flow on the manifold M := ? \H, with respect to the Patterson–Sullivan measure. The argument uses the ground-state diffusion and its canonical lift to the frame bundle, for which the existence of a potential operator is proved. ? 2001 Éditions scientifiques et médicales Elsevier SAS Keywords: Geodesic flow, Hyperbolic manifold of infinite volume, Diffusion process, Stable foliation, Spectral gap, Patterson–Sullivan measure, Central limit theorem.
Voir
- geodesic flow
- patterson–sullivan measure
- group ?
- when ?
- see also
- euclidian rotations
- stable horocycle
- state diffusion
- smooth function
J. Math. Pures Appl. 80, 2 (2001) 153175 2001 Éditions scienti ques et médicales Elsevier SAS. All rights reserved S0021-7824(00)01182-X/FLA
CENTRAL LIMIT THEOREM FOR THE GEODESIC FLOW ASSOCIATED WITH A KLEINIAN GROUP, CASE δ>d/ 2
N. ENRIQUEZ a , 1 , J. FRANCHI b , 2 , Y. LE JAN c , 3 a Laboratoire de Probabilités de Paris 6, 4 place Jussieu, tour 56, 3ème étage, 75252 Paris cedex 05, France b Université Louis Pasteur, I.R.M.A., 7 rue René Descartes, 67084 Strasbourg cedex, France c Université Paris Sud, Mathématiques, Bâtiment 425, 91405 Orsay, France Manuscript received 1 June 2000
A BSTRACT . Let Γ be a geometrically nite Kleinian group, relative to the hyperbolic space H = H d + 1 , and let δ denote the Hausdorff dimension of its limit set, that we suppose here strictly larger than d/ 2. We prove a central limit theorem for the geodesic ow on the manifold M := Γ \ H , with respect to the PattersonSullivan measure. The argument uses the ground-state diffusion and its canonical lift to the frame bundle, for which the existence of a potential operator is proved. 2001 Éditions scienti ques et médicales Elsevier SAS Keywords: Geodesic ow, Hyperbolic manifold of in nite volume, Diffusion process, Stable foliation, Spectral gap, PattersonSullivan measure, Central limit theorem. AMS classi cation : 37A50, 37D40, 58J65, 60J60, 60F05
1. Introduction Consider the hyperbolic space H = H d + 1 , endowed with some geometrically nite Kleinian group Γ . The Hausdorff dimension δ ∈ [ 0 , d ] of its limit set (see [13,17] or [18]) plays a fundamental role. When δ is larger than d/ 2, δ(δ − d) is the highest eigenvalue of the Laplacian on a fundamental domain. The associated eigenstate Φ plays an important role in the study of the quotient M = Γ \ H and of its geodesic ow. The corresponding ground-state diffusion Z tΦ , which we call “ Φ -diffusion, is then also a natural object and tool in this framework: see [17, 24]. As in [3], where the problem of the asymptotic law of windings in cusps of hyperbolic surfaces has been treated, we approximate the PattersonSullivan measure m by the images under the geodesic ow θ t of a quasi-invariant measure ν , which is stationary for the lift of the ground-state diffusion Z tΦ to T 1 M . An important step, which was not needed in [3], but which extends the proof given in [9] for the nite volume case, is the existence of a potential operator V for the lift of the Φ -diffusion. Denote by L 0 the Lie derivative along the geodesic ow, and by L 1 , . . . , L d the Lie derivatives along the stable horocycle ows (which are no longer de ned on the tangent 1 E-mail: enriquez@ccr.jussieu.fr 2 E-mail: franchi@math.u-strasbg.fr 3 E-mail: yves.lejan@math.u-psud.fr
154 N. ENRIQUEZ ET AL. / J. Math. Pures Appl. 80 (2001) 153175 bundle T 1 M , but only on the frame bundle O M ). Let f 1 , . . . , f d denote the conjugate functions of any given function f having bounded derivatives on O M ; they are de ned by: f j := − 0 ∞ e − s L j f ( · θ s ) d s . Set f 0 := f for convenience. Let us also introduce the canonical projection π 2 from O M onto M , and the divergence operator K : d Kf := 12 d L j f j + ( L j log Φ ◦ π 2 )f j − d 2 f. j = 0 j = 0 Our main result, assuming that δ > d/ 2, is the following central limit theorem for the geodesic ow on T 1 M , relating to the PattersonSullivan measure m : T HEOREM . Let us x a real function f on T 1 M , such that f d m = 0 , and of class C 2 with bounded and Hölderian derivatives. Then for all a ∈ R we have : t l → im ∞ T 1 M exp at − 1 0 t f (ξ θ s ) d s d m(ξ ) = m(T 1 M ) × exp − a 2 2 V (f ) , where V (f ) := dj = 0 (f j + L j V Kf ) 2 d ν vanishes if and only if f is a L 0 -derivative. “ F Hölderian on O M precisely means: there exists some r > 0 such that dist (ξ, ξ ) − r | F (ξ ) − F (ξ ) | is bounded on { (ξ, ξ ) ∈ O M 2 | 0 < dist (ξ, ξ ) < 1 } . Equivalently, our result reads (with c(δ) given in Corollary 1): for all a ∈ R we have : f ) = ) − 1 × ( 2 π ) − 1 / 2 exp − s 2 / 2 d s. t li → m ∞ m ξ ∈ T 1 M 0 t f (ξ θ s ) d s a t V ( Φ 22 c(δ a −∞ The particular case of M being convex-cocompact, that is to say without cusp, or associated with a group Γ without parabolic element, can be handled by the coding method of [15]. See also [1,8] and [19]. The particular case of M having nite volume (corresponding to δ = d ) was handled in [9], the PattersonSullivan measure being in this case just the Liouville measure. Our result concerns the more general case d/ 2 < δ d , in which M may have both in nite volume and cusps. For dealing with this new case, we use here globally the same strategy as in [9], which consists roughly in comparing the geodesics with the paths of a diffusion on T 1 M , for which the existence of potentials has to be exhibited. But the in nite volume case is much more involved, since m is not invariant under the horocycle ows and is distinct of ν , which is only quasi-invariant under the geodesic and stable horocycle ows. Finally note that the remaining case δ d/ 2 cannot be handled in the same way, since in that case there does not exist any fundamental diffusion Z tΦ associated with δ on the base manifold M .
2. Notations and basic data Let H denote the hyperbolic space H d + 1 , with boundary ∂ H , unitary tangent bundle T 1 H , and orthonormal frame bundle O H . Let us identify H with its Poincaré half-space model R d × R ∗+ , and the current point z ∈ H with its canonical coordinates (x, y) = (x 1 , . . . , x d , y) ∈ R d × R ∗+ . Set e 0 := ( 0 , 1 ) . Recall that
