Lyapunov spectrum of a relativistic stochastic flow in the Poincaré group
aa r X i v : . [ m a t h . P R ] M a r Lyapunov spectrum of a relativistic stochastic flow in the Poincar´egroup.
Camille Tardif ∗ June 7, 2018
Abstract
We determine the Lyapunov spectrum and stable manifolds of some stochastic flows on thePoincar´e group associated to Dudley’s relativistic processes.
Key words:
Relativistic processes. L´evy processes in Lie groups. Poincar´e group. Lyapunovspectrum. Hyperbolic dynamics.
AMS Subject Classification:
Contents G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Asymptotic behavior of the G -component . . . . . . . . . . . . . . . . . . . . . . . 83.3 Asymptotic behavior of the R ,d -component . . . . . . . . . . . . . . . . . . . . . . 133.4 Geometric description of the convergence . . . . . . . . . . . . . . . . . . . . . . . 14 H d × R ,d of the stable manifolds . . . . . . . . . . . . . . . . . . . . 21 In 1966 Dudley [6] defined a class of relativistic processes with Lorentzian-covariant dynamics inthe framework of special relativity. Such a process ξ t with values in Minkowski space-time R ,d ,is differentiable and has velocity smaller than the speed of light. So it can be parametrized byits proper time, which amounts to impose to the velocity ˙ ξ t to be an element of the unit pseudosphere H d of R ,d . The restriction to the tangent space of H d of Minkowski ambient pseudo-metric turns H d into a Riemannian manifold of constant negative curvature. The invariance ofthe process ( ˙ ξ t , ξ t ) by the natural action of the set of Lorentz transforms on H d × R ,d imposesto the laws of ˙ ξ t to be invariant by the action of the isometries of H d . Among this class of ∗ [email protected] Universit´e du Luxembourg. elativistic processes, there is essentially only one which is continuous. It corresponds to the casewhere ˙ ξ t is a Riemannian Brownian motion in the hyperbolic space and in this case ( ˙ ξ t , ξ t ) iscalled Dudley diffusion . Forty years after this seminal work, Franchi and Le Jan [8] extendedDudley diffusion to the framework of any Lorentz manifold. They defined relativistic processeswith Lorentzian-covariant dynamics on generic Lorentzian manifolds by rolling without slipping aDudley diffusion on the unit tangent space. They studied the asymptotic behavior of such diffusionin the Schwarzschild space-time. Bailleul [3] succeeded to compute the Poisson boundary of Dudleydiffusion in Minkowski space-time and showed that it coincides with the causal boundary of R ,d .The asymptotic behavior of relativistic diffusions was investigated in other non flat Lorentzianmanifolds ([1], [7]) with the aim of describing how the asymptotic behavior of the diffusion reflectsthe asymptotic geometry of the manifold.In this work we ask a new question concerning these processes dealing with the asymptoticbehaviour of some stochastic flow associated to it. As Brownian motion on a Riemannian manifold, the relativistic diffusion [8] is obtained by projecting a diffusion process with values in theorthonormal frame bundle, solution of a stochastic differential equation. This SDE generates astochastic flow which, in our Lorentzian framework, consists in a stochastic perturbation of thegeodesic flow. Existence and computation, for example, of the Lyapunov spectrum and stablemanifolds of these flows may be investigated in the same way as it was done by Carverhill andElworthy [5] for the canonical stochastic flow in the Riemannian framework. The main difficultyto study the flow of relativistic processes comes from the fact that the orthonormal frame bundleof a Lorentz manifold is never compact. Nevertheless in this article we provide a study of theasymptotic dynamics of the stochastic flow generated by Dudley processes in the Minkowskispace-time (without restricting ourselves to the diffusion case). Precisely, in this framework,the orthonormal frame bundle is identified with the Poincar´e group e G := P SO (1 , d ) ⋉ R ,d anddenoting φ t the left invariant stochastic flow associated to one of Dudley’s processes in e G weobtain the description of the Lyapunov spectrum and the stable manifolds of ϕ t . Precisely weobtain the following two results. Theorem 1 (Lyapunov spectrum) . There exist a constant α > and two asymptotic random Liesub-algebras V −∞ ⊂ V ∞ of Lie( e G ) such that for some norm k · k on Lie( e G ) and e X ∈ Lie( e G ) wehave for almost every trajectory t log k dϕ t (Id)( e X ) k ϕ t (Id) −→ t → + ∞ α if e X ∈ Lie( e G ) \ V ∞ e X ∈ V ∞ \ V −∞ − α if e X ∈ V −∞ \ { } Theorem 2 (Stable manifolds) . Denote by V −∞ := exp( V −∞ ) and d the distance associated to aleft invariant and Ad( SO ( d )) -invariant Riemanian metric in e G . Then for any two distinct points ˜ g ′ and ˜ g in e G we have • If ˜ g ′ ∈ ˜ g V −∞ then t log d ( ϕ t (˜ g ) , ϕ t (˜ g ′ )) −→ t → + ∞ − α. • If ˜ g ′ / ∈ ˜ g V −∞ then lim inf t →∞ d ( ϕ t (˜ g ) , ϕ t (˜ g ′ )) > . We begin by constructing, in section 2, Dudley processes as projections of left L´evy processeson the Poincar´e group e G , identified with the orthonormal frame bundle of the Minkowski space-time. These L´evy processes are solutions of stochastic integral equations and induce a left invariantstochastic flow ϕ t in e G . In section 3 we find the asymptotic behavior of Dudley processes andexhibit the asymptotic random variables ( θ ∞ , λ ∞ ) ∈ S d − × R ∗ + . Finally in section 4 we proveTheorems 1 and 2 and explicit the projection of the stable manifold in H d × R ,d by showing thatit corresponds to a skew product of a horosphere by a line.Note that stochastic flows generated by L´evy processes on semi-simple Lie groups were inten-sively studied by Liao ([12], [11], [13]). But his results cannot be used directly here since ourL´evy processes lie in the Poincar´e group which is not semi-simple. Moreover in our work we uppose only that the Levy measure is integrable at infinity whereas Liao [13] request the entireintegrability of it.Our work is also strongly inspirited by the work of Bailleul and Raugi [4] where the authorsused Raugi’s methods [14] to find the Poisson boundary of Dudley diffusion. We present in this section the geometrical framework of special relativity and define a naturalclass of relativistic Markov processes with Lorentzian-covariant dynamics introduced by Dudleyin [6]. They are obtained by projecting left L´evy processes with values in the Poincar´e group andare described by two parameters: a diffusion coefficient σ ∈ R and a jump intensity L´evy measure ν on R ∗ + . The Minkowski space-time R ,d is R × R d endowed with the Lorentz quadratic form q defined by ∀ ξ = ( ξ , ξ , . . . , ξ d ) ∈ R × R d , q ( ξ ) = (cid:0) ξ (cid:1) − (cid:0) ξ (cid:1) − · · · − (cid:0) ξ d (cid:1) . We denote by ~ξ := ( ξ , . . . , ξ d ) t the space component of ξ .Set Q = Diag(1 , − , . . . , − q in the canonical basis ( e , e , . . . , e d ). Time orientation is given by the constantvector field e and some ξ ∈ R ,d is said to be future oriented when q ( ξ, e ) >
0. A path γ s in R ,d is said to be time-like when it is differentiable almost everywhere and q ( ˙ γ s ) > q ( ˙ γ s , e ) >
0. The Poincar´e group is the group of affine q -isometries which preserve orientationand time-orientation. It is the semi-direct product connected group e G := P SO (1 , d ) ⋉ R ,d where G := P SO (1 , d ) denotes the group of linear q -isometries which preserve orientation andtime-orientation. An element ˜ g = ( g, ξ ) ∈ e G is made up of its linear part g ∈ G and its translationpart ξ . We identify G with the sub-group of e G which fixes 0. By this way, we identify R ,d withthe homogeneous space e G/G . The identity element of e G and G is denoted by Id (thus for usId = (Id , g = ( g, ξ ) ∈ e G we associate the affine frame (( g ( e ) , g ( e ) , . . . , g ( e d )); ξ ) of R ,d and e G is identified with the bundle of q -orthonormal, oriented and time-oriented, frames over R ,d . We denote by ˜ π : e G −→ R ,d ˜ g = ( g, ξ ) ξ the projection associated to this trivial fibration and G = ˜ π − { } . The canonical basis being fixedwe identify G with the matrix group G = (cid:8) g ∈ SL( R d +1 ) , gQg t = Q, q ( g ( e ) , e ) > (cid:9) , and its Lie algebra isLie( G ) = (cid:8) X ∈ M d +1 ( R ) , XQ − QX t = 0 (cid:9) = (cid:26)(cid:18) b t b C (cid:19) , b ∈ R d , C ∈ M d ( R ) s.t C = − C t (cid:27) . We have Lie( e G ) = Lie( G ) × R ,d and for e X = ( X, x ) , e Y = ( Y, y ) ∈ Lie( e G )[ e X, e Y ] = ([ X, Y ] , Xy − Y x ) . e identify Lie( G ) with Ker( d Id ˜ π ) and its elements are vertical for the fibration ˜ π . We set V i := e e ti + e i e t i = 1 , . . . , dV ij := [ V i , V j ] = e i e tj − e j e ti j > i. Moreover we set H := (0 , e ) ∈ Lie( e G )which is horizontal for the fibration ˜ π . Notation
For e X ∈ Lie( e G ) we denote by e X l the left invariant vector field in e G associated.Denote by K the subgroup of G made of the rotations of R d . We have K := (cid:26)(cid:18) k (cid:19) , k ∈ SO ( d ) (cid:27) , and K is also the stabilizer of e under the action of G on R ,d . The homogeneous space G/K can be identified with the orbit of e under the action of G which is the unit pseudo sphere H d := { ξ ∈ R ,d , q ( ξ ) = 1 , ξ > } and is a Riemannian manifold of constant negative curvaturewhen its tangent space is endowed with the restriction of q on it.For r ∈ R + and θ ∈ S d − ⊂ R d , define S ( r, θ ) := exp r d X i =1 θ i V i ! = (cid:18) cosh( r ) sinh( r ) θ t sinh( r ) θ Id + (cosh( r ) − θθ t (cid:19) . Each g ∈ G can be decomposed in polar form g = S ( r ( g ) , θ ( g )) R where R ∈ K . In this paragraph we define the relativistic processes introduced by Dudley in [6]. These processesenjoy two natural properties: • they are e G -invariant i.e their dynamics are invariant by a change of q -orthonormal frame • their trajectories in R ,d are time-like: they are almost everywhere differentiable,and thetangents vectors are time-like and time oriented.First remark that no Markov processes with values in R ,d is ˜ G -invariant. Indeed, the law atsome time t > G -invariant probability measure in R ,d which is necessary trivial by the following lemma. Lemma 1.
The only G -invariant probability measure in R ,d is the Dirac measure at .Proof. Let µ be a G -invariant probability measure in R ,d . First suppose that the support of µ is not contained in the q -orthogonal hyperplane of some light-like line { u ( e + ˆ θ ) , u ∈ R } (ˆ θ := P di =1 ˆ θ i e i ∈ S d − ). So there exist a compact set C in the complement of this hyperplanesuch that µ ( C ) >
0. For g = S ( r, ˆ θ ), r > ξ = ( ξ , ~ξ ) ∈ R ,d , denoting k · k the Euclideannorm in R ,d we have k g ( ξ ) k = 2 q ( g ( ξ ) , e ) − q ( g ( ξ )) = 2 q ( ξ, g − ( e )) − q ( ξ ) = 2 (cid:16) cosh( r ) ξ − sinh( r )ˆ θ · ~ξ (cid:17) − q ( ξ )= 2 (cid:16) cosh( r )( ξ − ˆ θ · ~ξ ) + e − r ˆ θ · ~ξ (cid:17) − q ( ξ )= 2 (cid:16) cosh( r ) q ( ξ, e + ˆ θ ) + e − r ˆ θ · ~ξ (cid:17) − q ( ξ ) . Since C is a compact set in the complement of the q -orthogonal hyperplan to { u ( e + ˆ θ ) , u ∈ R } then inf ξ ∈ C | q ( ξ, e + ˆ θ ) | > r can be chosen such that inf ξ ∈ C k g ( ξ ) k is arbitrary large.Thus g can be chosen such that C and g ( C ) are disjoint. Furthermore q ( g ( ξ ) , e + ˆ θ ) = q ( ξ, g − ( e +ˆ θ )) = e r q ( ξ, e + ˆ θ ) and thus g ( C ) belongs to the complement of the hyperplan q -orthogonal to e + ˆ θ . By iteration we can find a sequence g k = S ( r k , ˆ θ ), k ∈ N such that the compact sets g k ( C ) re pairwise disjoint. Thus we obtain a contradiction writing 1 ≥ µ ( ∪ k g k ( C )) = P k µ ( g k ( C )) = P k µ ( C ) = + ∞ .Now if the support of µ is contained in some hyperplan tangent to the light-cone and is notrestricted to 0, we can find a compact set C in this hyperplan with 0 / ∈ C and µ ( C ) > R ∈ K such that R ( C ) is not in the hyperplan. Thus µ ( C ) = µ ( R ( C )) = 0and it gives a contradiction. So we proved that µ is necessary the Dirac measure in { } .Thus R ,d = e G/G cannot be the space of states of some non-trivial e G -invariant Markov process.But e G -homogeneous spaces of the form e G/ e K where e K is a compact subgroup of e G have some e K -invariant probability measure and may be endowed with some e G -invariant Markov processes(see [13] or [9] ). The smallest spaces of states we can consider correspond to the maximal compactsub-group of e G . Thus it is natural to consider the space of states e G/K ≃ H d × R ,d . The group K is seen as the subgroup of e G which stabilize 0 and e under the action of e G on R ,,d .We denote by π : e G H d × R ,d ≃ e G/K the canonical projection ∀ ˜ g = ( g, ξ ) ∈ e G π (˜ g ) = ( g ( e ) , ξ ) . The following Proposition exhibit all the relativistic processes in H d × R ,d . It is essentiallyan application of a result of Liao ( Theorem 2.1 and 2.2 p 42 in [13] ). Proposition 1.
The Markov processes on H d × R ,d , starting at ( ζ , ξ ) , which are e G -invariantand whose trajectories are time-like are of the form ( ζ s , ξ s ) where ζ s is a G -invariant Markovprocess on H d and ξ t = ξ + a R t ζ s ds ; a being some positive constant. For such a process thereexist σ > and a measure ν on R + satisfying Z + ∞ min(1 , r ) ν ( dr ) < + ∞ , such that ( ζ t , ξ t ) = π (˜ g t ) in law where ˜ g t is a left Levy process on e G starting at ˜ g s.t ( ζ , ξ ) = π (˜ g ) of which generator ˜ L is defined by ∀ f ∈ C ( e G ) e L f (˜ g ) = aH l f (˜ g ) + σ d X i =1 ( V li ) f (˜ g )+ Z + ∞ Z S d f (˜ gS ( r, θ )) − f (˜ g ) − r r ∈ [0 , d X i =1 θ i V li f (˜ g ) ! ν ( dr ) dθ. Definition 1 (Dudley processes) . When a = 1 then ˙ ξ t = ζ t and ξ t is parametrized by its propertime , i.e q ( ˙ ξ t ) = 1 . When moreover σ or ν is non trival we call ( ˙ ξ t , ξ t ) a Dudley process and weconsider exclusively these processes in the sequel. When ν = 0 ( ˙ ξ t , ξ t ) is continuous and is called Dudley diffusion . Remark 1.
The process ξ t is differentiable and ˙ ξ t is c`adl`ag.Proof of Proposition 1. Let ( ζ t , ξ t ) a Markov (Feller) process on H d × R ,d starting at ( ζ , ξ ),which is e G -invariant and whose trajectories are time-like. By choosing ˜ g such that ( ζ , ξ ) = π (˜ g )and considering the Markov process ˜ g − ( ζ t , ξ t ) it remains to prove the proposition in the case where( ζ , ξ ) = ( e , H i := (0 , e i ) ∈ Lie( e G ) , i = 1 , . . . , d. The family { H , H i , V i , V ij } i
0. The push forward of e Π by π is supported on G/K ≃ H d and is K -invariant. Thus Π can be chosen of the form ∀ f ∈ C ( G ) Π f = Z + ∞ Z S d − f ( S ( r, θ )) ν ( dr ) dθ where ν is a Levy measure on R ∗ + ( i.e satisfying R min(1 , r ) ν ( dr ) < + ∞ ).Denote by g t the G -component of ˜ g t . Thus g t ( e ) = ˙ ξ t and ξ t = R t g s ( e ) ds . By definition g t is a G -valued left Levy process, K -right invariant, generated by L defined by ∀ f ∈ C ( G ) L f ( g ) = σ d X i =1 ( V li ) f ( g )+ Z + ∞ Z S d f (˜ gS ( r, θ )) − f (˜ g ) − r r ∈ [0 , d X i =1 θ i V li f (˜ g ) ! ν ( dr ) dθ. Denote by Π the Levy measure supported on G defined byΠ f = Z + ∞ Z S d − f ( S ( r, θ )) ν ( dr ) dθ. Define U := { g ∈ G, r ( g ) ≤ } which is a K invariant neighborhood of Id in G . For f ∈ C ( G )we have the following Itˆo formula (see [2]) for g t f ( g t ) = f (Id) + σ d X i =1 Z t V li f ( g s − ) dB is + σ Z t d X i =1 ( V li ) f ( g s − ) ds + Z t Z U ( f ( g s − h ) − f ( g s − )) ˜ N ( ds, dh )+ Z t Z U f ( g s − h ) − f ( g s − ) − r ( h ) d X i =1 θ i ( h ) V li f ( g s − ) ! ds Π( dh ) (1)+ Z t Z ( U ) c ( f ( g s − h ) − f ( g s − )) N ( ds, dh ) , where B t is a Brownian motion of R d , N is a Poisson random measure on R × G of intensity measure dt ⊗ Π and ˜ N ( ds, dh ) := N ( ds, dh ) − ds Π( dh ) is the compensated random measure associated. In this section we determine the asymptotic behavior of π (˜ g t ) = ( g t ( e ) , ξ t ) under an integrabilitycondition on the jump intensity measure ν (Ass.1). Writing g t = n t a t k t in some Iwasawa decom-position of G we first prove (Prop.2), applying the Itˆo formula (1) and the law of large number,that the abelian term a t = exp( α t V ) is positively contracting , α t t converges almost surely to a ositive constant α depending explicitly on σ and ν . Next we prove (Prop.4) that the nilpotentterm n t converges almost surely to an asymptotic random variable n ∞ and this convergence isexponentially fast with rate α . Then we investigate (Prop.5) the asymptotic behavior of ξ t in R ,d . Geometrically, seen in the projective space, the H d -valued process g t ( e ) converges to alimit angle θ ∞ ∈ ∂ H d ≃ S d − of which n ∞ is a stereographic projection. Moreover, the process ξ t is asymptotic to some affine hyperplane q -orthogonal to θ ∞ of which position is fixed by anotherasymptotic random variable λ ∞ ∈ R ∗ + . Figure 1 sum up the asymptotic results. H d Asymptotic angle Asymptotic affine hyperplane λ ∞ > θ ∞ ∈ S d − θ ∞ ∈ S d − Figure 1: Asymptotic behavior of a Dudley diffusion G Although a polar decomposition of G was used to introduce g t (defining the K invariant mea-sure Π), Iwasawa decomposition seems to be more adapted to describe its asymptotic dynamics.Introduce briefly this decomposition.The maximal abelian subalgebra contained in Vect { V , . . . , V d } , which is the orthogonal sub-space of Lie( G ) of K for the Killing form, is of dimension one. Let choose A := Vect { V } one ofthem. The linear endomorphism ad( V ) of Lie( G ) is diagonalisable with eigenvalues − , N = { X ∈ Lie( G ) , ad( V ) X = − X } N = { X ∈ Lie( G ) , ad( V ) X = X } the eigenspace corresponding respectively to the eigenvalue − N = Vect { V i − V i , i = 2 , . . . , d } and N = Vect { V i + V i , i = 2 , . . . , d } . The eigenspace corresponding to 0 is
A ⊕ M where M is the sub algebra of elements of K whichcommute with the elements of A . Explicitly M = Vect { V ij , i, j = 2 . . . d } = C , C ∈ so( d − . The subspace N is a nilpotent Lie algebra (even abelian since [ N , N ] = 0). The correspondingIwasawa decomposition of Lie( G ) is Lie( G ) = N ⊕ A ⊕ K . or X ∈ Lie( G ) we denote by { X } N (resp. { X } A and { X } K ) its projection in N (resp. A and K ) thus X = { X } N + { X } A + { X } K . Denoting by A := exp( A ), N := exp( N ) the subgroupcorresponding we obtain the corresponding Iwasawa decompositions of GG = N AK.
Moreover, the mapping from N × A × K to G which maps ( n, a, k ) to nak is an analytic diffeo-morphism. For g ∈ G we denote by g = ( g ) N ( g ) A ( g ) K its decomposition in Iwasawa coordinates.To simplify notations set n t := ( g t ) N , a t := ( g t ) A and k t = ( g t ) K , thus g t = n t a t k t .Note that we have other Iwasawa decompositions like Lie( G ) = N ⊕ A ⊕ K (with G = NAK ). Iwasawa and polar coordinates .For g ∈ G written in polar form g = exp (cid:16) r P di =1 θ i ( V i − V i ) (cid:17) R where R ∈ K , we have q ( e , g ( e )) = cosh( r ).Now denoting by b ∈ R d − and u ∈ R such that ( g ) N = exp (cid:16)P di =2 b i ( V i − V i ) (cid:17) and ( g ) A =exp ( uV ) we can compute explicitly q ( e , g ( e )) in terms of b and u and we obtaincosh( r ) = (cid:18) k b k (cid:19) cosh( u ) + k b k u ) . (2)Moreover u can be expressed in term of θ and r via e u = cosh( r ) + θ sinh( r ) (3) G -component The aim of this section is to show that n t converges almost surely to an asymptotic N -valued ran-dom variable n ∞ and that the convergence is exponentially fast. This result, stated in Proposition4, appears to be a consequence of the contracting property of a t . For this we need the followingintegrability condition on ν . The group G is semi simple and the tools used in this section arevery closed from those of Liao [13]. Nevertheless we present a self-contained proof in our specificframework and our results are established under a weaker assumption than the ones of [13] ( seeremark 3 ). Namely we suppose that the following integrability condition is satisfied. Assumption 1. Z + ∞ rν ( dr ) < + ∞ . The following proposition computes explicitly the linear drift of a t which appears to be positive.The proof is essentially a consequence of the law of large number. Proposition 2.
Let denote by α t the R -valued process such that a t = exp( α t V ) . Then thefollowing convergence holds almost surely α t t −→ t →∞ α > . The positive constant α is α := d − σ + Z + ∞ r cosh( r ) − sinh( r )sinh( r ) ν ( dr ) . Proof.
First define log : A → A ≃ R exp( uV ) uV and apply Itˆo formula (1) to the smoothmap f : g log( g ) A . Remark that for g, h ∈ G ( gh ) A = ( g ) A (( g ) K h ) A and f ( g s − h ) − f ( g s − ) = log ( k s − g ) A . oreover V li f ( g s − ) = ddt log( g s − e tV i ) A (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ddt log (cid:16) e tAd ( k s − ) V i (cid:17) A (cid:12)(cid:12)(cid:12)(cid:12) t =0 = { Ad ( k s − ) V i } A ( V li ) f ( g s − ) = ddt { Ad (cid:0) g s − e tV i (cid:1) K V i } A (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ddt { Ad (cid:16) e tAd ( k s − ) V i (cid:17) K Ad ( k s − ) V i } A (cid:12)(cid:12)(cid:12)(cid:12) t =0 = { [ { Ad ( k s − ) V i } K , Ad ( k s − ) V i ] } A . For k ∈ K we have Ad ( k ) V i = P dj =1 k ij V j where k = ( k ij ) ∈ K ≃ SO ( d ). Then we compute d X i =1 { [ { Ad ( k s − ) V i } K , Ad ( k s − ) V i ] } A = d X i =1 d X j =1 d X l =1 ( k s − ) ij ( k s − ) il { [ { V j } K , V l ] } A , since { V j } K = V j (and 0 for j = 1 ) and [ V j , V l ] = V δ jl we get d X i =1 { [ { Ad ( k s − ) V i } K , Ad ( k s − ) V i ] } A = d X j =2 d X i =1 (( k s − ) ij ) V = ( d − V . Thus Itˆo formula (1) can be writtenlog( a t ) = M t + d − σ t + Z t Z U c log( k s − h ) A N ( ds, dh )+ Z t Z U log( k s − h ) A − r ( h ) d X i =1 θ i ( h ) { Ad ( k s − ) V i } A ! ds Π( dh ) , where M t := σ P di =1 R t { Ad ( k s − ) V i } A dB is + R t R U ( k s − h ) A ˜ N ( ds, dh ) is a martingale. Its bracket is σ R t P di =1 k{ Ad ( k s − ) V i } A k ds + R t R U k log( k s − h ) A k ds Π( dh ) = t (cid:16) σ + R U k log( h ) A k Π( dh ) (cid:17) and thus we obtain that almost surely M t t −→ t → + ∞ . Moreover, making the change of variable h ′ = k s − hk − s − we obtain using the K -invariant byconjugation of Π Z t Z U log( k s − h ) A − r ( h ) d X i =1 θ i ( h ) { Ad ( k s − ) V i } A ! ds Π( dh )= Z t Z U log( h ′ ) A − r ( h ′ ) d X i =1 θ i ( k − s − h ′ ) { Ad ( k s − ) V i } A Π( dh ′ ) ds = Z t Z U log( h ′ ) A − r ( h ′ ) d X i =1 d X j =1 ( k s − ) ij θ j ( h ′ )( k s − ) i V Π( dh ′ ) ds = Z t Z U log( h ′ ) A − r ( h ′ ) θ ( h ′ ) V Π( dh ′ ) ds = t Z S d − Z log( S ( r, θ )) A − rθ V ν ( dr ) dθ By (3) we have log( S ( r, θ )) A = log(cosh( r ) + θ sinh( r )) V and the previous term equals t Z S d − Z log(cosh( r ) + θ sinh( r )) − rθ ν ( dr ) dθV = t Z (cid:18)Z − log (cosh( r ) + u sinh( r )) − ru du (cid:19) ν ( dr ) V = t Z r cosh( r ) − sinh( r )sinh( r ) ν ( dr ) t remains to consider the asymptotic behavior of the stochastic integral R t R U c log( k s − h ) A N ( ds, dh ).We know that there exist T i jump times of a Poisson process of intensity measure Π( U c ) and ran-dom variable h n i.i.d of law Π | U c / Π( U c ) and independent of ( T i ) i such that Z t Z U c log( k s − h ) A N ( ds, dh ) = X n,T n ≤ t log( k T − n h n ) A . Moreover, by invariance of Π under conjugation by K we check easily that the random variables h ′ n := Ad ( k T − n ) h n are i.i.d of common law Π | U c / Π( U c ) and we have Z t Z U c log( k s − h ) A N ( ds, dh ) = N t X n =1 log( h ′ n ) A , where N t is a Poisson process of intensity measure Π( U c ) and independent of ( h ′ n ) n . Moreoversince R + ∞ rν ( dr ) < + ∞ then log( h ′ n ) A is integrable and the law of large number ensures that1 t Z t Z U c log( k s − h ) A N ( ds, dh ) −→ t → + ∞ Π( U c ) E [log( h ′ n ) A ] . The proof is ended by checking that E [log( h ′ n ) A ] = 1Π( U c ) Z + ∞ r cosh( r ) − sinh( r )sinh( r ) ν ( dr ) V . Remark 2.
When R + ∞ rν ( dr ) = + ∞ we obtain E [ | log( h ′ n ) A | ] = + ∞ . Nevertheless E [ − min (log( h ′ n ) A , Z + ∞ Z − − min (cid:0) log(cosh( r + θ sinh( r )) , (cid:1) dθ ν ( dr )= Z + ∞ Z e − r − log( v ) dv r ) ν ( dr )= Z + ∞
12 sinh( r ) (1 − e − r ( r + 1)) ν ( dr ) < + ∞ . Now, applying a generalized law of large numbers we deduce that almost surely t Z t Z U c log( k s − h ) A N ( ds, dh ) −→ t → + ∞ + ∞ , and so α t t −→ t → + ∞ + ∞ . The following proposition establishes that g t is bounded in expectation on a finite time interval.This result is used to prove the convergence of n t in the next Proposition. Proposition 3.
Fix
T > . Then E " sup t ∈ [0 ,T ] r ( g t ) < + ∞ . Proof.
We cannot directly apply Itˆo formula to g r ( g ) since it is not regular at Id . But we canfind a smooth function ˜ r such that ˜ r ≥ r on U and ˜ r = r on U c . For such a function we have˜ r ( g t ) = ˜ r ( Id ) + ˆ m t + ˜ m t + I t + J t + Z t Z U c (˜ r ( g s − h ) − ˜ r ( g s − )) N ( ds, dh ) . (4) here ˆ m t := σ R t V li ˜ r ( g s − ) dB is and ˜ m t := R t R U (˜ r ( g s − h ) − ˜ r ( g s − )) ˜ N ( ds, dh ) are martingales, I t := σ R t P di =1 (cid:0) V li (cid:1) ˜ r ( g s − ) ds and J t := Z t Z r ∈ [0 , Z θ ∈ S d − ˜ r ( g s − S ( r, θ )) − ˜ r ( g s − ) − r d X i =1 θ i V li ˜ r ( g s − ) ! dθν ( dr ) ds are processes with finite variation.To prove the proposition we will bound the supremum on [0 , T ] of each of these five termsby means of k X l ˜ r k ∞ and k ( X l ) ˜ r k ∞ for some X ∈ Vect { V i , i = 1 , . . . , d } . Thus we need thefollowing lemma. Lemma 2.
For X ∈ P = Vect { V i , i = 1 , . . . , d } we have k X l ˜ r k ∞ < + ∞ and k ( X l ) ˜ r k ∞ < + ∞ .Proof of the lemma. U being a compact set it suffices to show that the supremum is finite on U c .Since ˜ r = r on U c it remains to prove that sup g ∈ U c | X l r ( g ) | < + ∞ and sup g ∈ U c | ( X l ) r ( g ) | < + ∞ .The polar decomposition of g can be written g = ˜ k exp( r ( g ) V ) k for some ˜ k, k ∈ K . Setting x ∈ R d such that X = P i x i V i we have r ( g exp( sX )) = r exp( r ( g ) V ) exp s d X i =1 ( kx ) i V i !! . Let θ ∈ S d − be such that ( kx ) i = k x k θ i . Then we compute explicitly, for g ∈ U c : r exp( r ( g ) V ) exp s d X i =1 ( kx ) i V i !! = (cosh) − (cid:0) cosh( r ( g )) cosh( s k x k ) + θ sinh( r ( g )) sinh( s k x k ) (cid:1) = r ( g ) + s k x k θ + s k x k cosh( r ( g ))sinh( r ( g )) (1 − ( θ ) ) + O ( s ) . Then it comes that X l r ( g ) = dds r ( g exp( sX )) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = k x k θ , and thus sup g ∈ U c (cid:12)(cid:12) X l r ( g ) (cid:12)(cid:12) ≤ k x k .Moreover ( X l ) r ( g ) = d ds r ( g exp( sX )) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = k x k cosh( r ( g ))sinh( r ( g )) (1 − ( θ ) )and so sup g ∈ U c (cid:12)(cid:12) ( X l ) r ( g ) (cid:12)(cid:12) ≤ k x k . Return to the proof of Proposition 3. By Doob’s norm inequalities (see [10]) we obtain E " sup t ∈ [0 ,T ] | ˆ m t | ≤ E (cid:2) ( ˆ m T ) (cid:3) ≤ σ E "Z T d X i =1 (cid:12)(cid:12) V li ˜ r ( g s − ) (cid:12)(cid:12) ds ≤ σ T d max i k V li ˜ r k ∞ < + ∞ . and E " sup t ∈ [0 ,T ] | ˜ m t | ≤ E (cid:2) ( ˜ m T ) (cid:3) ≤ E "Z T Z r ∈ [0 , Z θ ∈ S d − | ˜ r ( g s − S ( r, θ )) − ˜ r ( g s − ) | ν ( dr ) dθds ≤ dT Z r ν ( dr ) max i k V li ˜ r k ∞ < + ∞ . We have also E " sup t ∈ [0 ,T ] | I t | ≤ σ T d max i k ( V li ) ˜ r k ∞ < + ∞ . oreover applying a Taylor inequality to u ∈ [0 , ˜ r ( g s − S ( ur, θ )) it comes (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ r ( g s − S ( r, θ )) − ˜ r ( g s − ) − r d X i =1 θ i V li ˜ r ( g s − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ r sup θ ∈ S d − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i θ i V li ! ˜ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < + ∞ and thus E " sup t ∈ [0 ,T ] | J t | ≤ T (cid:18)Z r ν ( dr ) (cid:19) sup θ ∈ S d − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i θ i V li ! ˜ r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ < + ∞ . Finally, to bound the supremum of the last term of (4) we need the assumption 1. Indeed, since | ˜ r ( g s − S ( r, θ )) − ˜ r ( g s − ) | ≤ r sup θ ∈ S d − (cid:13)(cid:13)(cid:0)P i θ i V li (cid:1) ˜ r (cid:13)(cid:13) ∞ we obtain E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t Z U c ˜ r ( g s − h ) − ˜ r ( g s − ) ds Π( dh ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ T d (cid:18)Z + ∞ rν ( dr ) (cid:19) max i (cid:13)(cid:13) V li ˜ r (cid:13)(cid:13) ∞ < + ∞ . We can now state the main result of this section, the convergence of n t to some asymptoticrandom variable n ∞ and the speed of convergence. Proposition 4.
Denote by b t = ( b it ) i =2 ,...,d the R d − -valued process such that n t = exp d X i =2 b i − t ( V i − V i ) ! . Then b t converges almost surely to b ∞ exponentially fast with rate α , i.e lim sup t → + ∞ t log k b t − b ∞ k ≤ − α. (5) As a consequence n t converges almost-surely to n ∞ := exp (cid:16)P di =2 b i − ∞ ( V i − V i ) (cid:17) and defining h t := e − tαV n − ∞ g t we obtain lim t → + ∞ t r ( h t ) = 0 a . s . (6) Proof.
Denoting by [ t ] the integer part of t we decompose b t = P [ t ] j =1 ( b j − b j − ) + b t − b [ t ] . Toprove that b t converge and (5) holds it is sufficient to verify thatlim sup j →∞ j log sup s ∈ [0 , k b j − b j + s k ≤ − α a . s . (7)For j ∈ N and s ∈ [0 ,
1] we have g − j g j + s = k − j a − j n − j n j + s a j + s k j + s and thus n − j n j + s = a j (cid:0) g − j g j + s k j (cid:1) N a − j . (8)Denote by ˜ b j,s = (˜ b ij,s ) i =2 ,...,d ∈ R d − such that (cid:0) g − j g j + s k j (cid:1) N = exp (cid:16)P di =2 ˜ b ij,s ( V i − V i ) (cid:17) . Then,(8) implies that b j + s − b j = e − α j ˜ b j,s . (9)Since g t is a Levy process, (cid:16) sup s ∈ [0 , r ( g − j g j + s ) (cid:17) j ∈ N are i.i.d random variables. Moreover, byProposition 3 their common expectation E [sup s ∈ [0 , r ( g s )] is finite. Thus, applying the law oflarge number it comes 1 j sup s ∈ [0 , r ( g − j g j + s ) −→ j →∞ . s . (10) ince r (( g − j g j + s ) N ) ≤ r ( g − j g j + s ) and, by (2), k ˜ b j,s k = 2 (cid:16) cosh (cid:16) r (cid:0) g − j g j + s k j (cid:1) N (cid:17) − (cid:17) wededuce from (10) that for ε > j such that for j > j k ˜ b j,s k ≤ e εj . (11)Since, by Proposition 2, α j = jα + o ( j ) thus (7) follows from (9) and (11). To finish the proof ofthe proposition we need to check (6). We have r ( h t ) = r ( e − αtV n − ∞ n t a t ) ≤ r ( e − αtV a t ) + r ( a − t n − ∞ n t a t ) , and r ( e − αtV a t ) = | α t − αt | = o ( t ) and we obtain from (2) ( since a − t n − ∞ n t a t ∈ N ), r ( a − t n − ∞ n t a t ) = cosh − (cid:18) e α t k b t − b ∞ k (cid:19) . So by (5) we have also r ( a − t n − t n ∞ a t ) = o ( t ) and (6) holds. Remark 3. • Without integrability condition, we can nevertheless show that g t satisfy the irreducibility and contraction conditions of [13] and deduce that α t converges almost surelyto + ∞ and n t converges to n ∞ . Nevertheless, by remark 2, we obtain in the case where R + ∞ rν ( dr ) = + ∞ , that almost surely α t t converges to + ∞ . • In [13] the author uses a stronger hypothesis to prove the rate of convergence of a L´evy processin a semi-simple group. It corresponds in our case to assume that R + ∞ rν ( dr ) < + ∞ . R ,d -component The linear endomorphism ξ exp( V ) ξ is diagonalisable with eigenvalues − , , +1. Denote by U − , U and U + the respective eigenspaces. Explicitly U − = Vect { e − e } , U = Vect { e , . . . , e d } and U + = Vect { e + e } . For ξ ∈ R ,d we denote by ( ξ ) − , ( ξ ) and ( ξ ) + its projection on eacheigenspace. Explicitly we obtain( ξ ) − = − q ( ξ, e + e )( e − e ) , ( ξ ) + = 12 q ( ξ, e − e )( e + e )( ξ ) = d X i =2 q ( ξ, e i ) e i . Recall that by definition, ξ t = R t g s ( e ) ds . The following proposition gives the asymptoticbehavior of ξ t . Proposition 5.
There exists an asymptotic random variable λ ∞ > such that ( n − ∞ ξ t ) − −→ t → + ∞ λ ∞ ( e − e ) , and moreover lim sup t → + ∞ t log k ( n − ∞ ξ t ) − − λ ∞ ( e − e ) k ≤ − α. (12) We also have lim sup t → + ∞ t log k ( n − ∞ ξ t ) k ≤ , and , lim sup t → + ∞ t log k ( n − ∞ ξ t ) + k ≤ α. (13) Proof.
We have − q ( n − ∞ ξ t , e + e ) = Z t − q ( n − ∞ n s a s ( e ) , e + e ) ds (14) nd the integrand can be written − q ( n − ∞ n s a s ( e ) , e + e ) = − q ( e − αsV n − ∞ n s a s ( e ) , e − αsV ( e + e ))= − e − αs q ( h s ( e ) , ( e + e )) , where h s = e − αs n − ∞ g s as defined in Proposition 4.Denote by (˜ r s , ˜ θ s ) ∈ R + × S d − the polar decomposition of h s ( e ) ∈ H d . So ˜ r = r ( h s ) and − e − αs q ( h s ( e ) , ( e + e )) = 12 e − αs (cid:16) cosh(˜ r s ) − ˜ θ s sinh(˜ r s ) (cid:17) ∈
12 [ e − ( αs +˜ r s ) , e − ( αs − ˜ r s ) ] . Proposition 4 ensures that ˜ r s = o ( s ) a.s, so fixing ε > s > s > s the integrand of (14) is positive and bounded by e − ( α − ε ) s . This ensures theconvergence of ( n − ∞ ξ t ) − to λ ∞ ( e − e ) with λ ∞ >
0. Moreover for t > s (cid:12)(cid:12)(cid:12)(cid:12) − q ( n − ∞ ξ t , e + e ) − λ ∞ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z + ∞ t e − ( α − ε ) s ds = 1 α − ε e − ( α − ε ) t , which prove (12).Now ( n − ∞ ξ t ) = d X i =2 Z t q ( n − ∞ n s a s ( e ) , e i ) dse i , and for i = 2 , . . . , d we have q ( n − ∞ n s a s ( e ) , e i ) = q ( e − αsV n − ∞ n s a s ( e ) , e i ) = ˜ θ is sinh(˜ r s ) so | q ( n − ∞ n s a s ( e ) , e i ) | ≤ e ˜ r s and this ensures that lim sup t → + ∞ t log k ( n − ∞ ξ t )0 k ≤ n − ∞ ξ t )+ = (cid:18)Z t q (cid:0) n − ∞ n s a s ( e ) , e − e (cid:1) ds (cid:19) ( e − e )and 12 q ( n − ∞ n s a s ( e ) , e − e ) = 12 e αs q ( h t ( e ) , e − e )= 12 e αs (cid:16) cosh(˜ r s ) − ˜ θ s sinh(˜ r s ) (cid:17) ∈
12 [ e αs − ˜ r s , e αs +˜ r s ] . So k ( n − ∞ ξ t ) + k ≤ k ( n − ∞ ξ s ) + k + Z ts e ( α + ε ) s ds, and thus lim sup t → + ∞ t log k ( n − ∞ ξ t ) + k ≤ α. Denote by p : R ,d \ { } → P d R the projection onto the projective space of dimension d . Thehyperbolo¨ıd H d is mapped onto the interior of a projective ball and its boundary ( ∂ H d ≃ S d − )is the image of the q -isotropy cone ∂ H d := p ( { ξ, q ( ξ ) = 0 } \ { } ) . From the relation q ( ˙ ξ t , n t ( e + e )) = q ( g t ( e ) , n t ( e + e )) = q ( e , a − t ( e + e )) = e − α t −→ t →∞ , we deduce that all limit points of p ( ˙ ξ t ) are q -orthogonal to θ ∞ := p ( n ∞ ( e + e )). Since the onlypoint of p ( H ) which is q -orthogonal to θ ∞ is θ ∞ itself it comes that p ( ˙ ξ t ) converges to θ ∞ in P d R .Now, identifying P d R with its affine chart { ξ, ξ = 1 } we can consider that θ ∞ ∈ S d . From (2) e deduce that r t → + ∞ and since p ( ˙ ξ t ) = p ( e + θ t sinh( r t )cosh( r t ) ) it comes that θ t converges to θ ∞ in S d . The two asymptotic random variables θ ∞ and n ∞ are linked by p ( e + θ ∞ ) = p ( n ∞ ( e + e ))or more explicitly, b ∞ ∈ R d − (defined by n ∞ = exp (cid:16)P di =2 b i − ∞ ( V − V i ) (cid:17) ) is the stereographicprojection of θ ∞ θ ∞ = 11 + k b ∞ k (cid:18) − k b ∞ k b ∞ (cid:19) . Concerning the asymptotic behavior of ξ t , Proposition 5 ensures that q ( ξ t , n ∞ ( e + e )) con-verges to λ ∞ . Thus geometrically ξ t is asymptotic to an affine hyperplan which is q - orthogonalto n ∞ ( e + e ) (or e + θ ∞ ) and passing by λ ∞ ( e − e ) . The Levy process ˜ g t , with values in e G and starting at some ˜ g , can be obtained by solving thefollowing left invariant stochastic integro-differential equation in e G ∀ f ∈ C ( ˜ G ) , f (˜ g t ) = f (˜ g ) + σ d X i =1 Z t V li f (˜ g s − ) ◦ dB is + Z t H l (˜ g s − ) ds + Z t Z U ( f (˜ g s − h ) − f (˜ g s − )) ˜ N ( ds, dh )+ Z t Z U f (˜ g s − h ) − f (˜ g s − ) − r ( h ) d X i =1 θ i ( h ) V li f (˜ g s − ) ! ds Π( dh ) (15)+ Z t Z ( U ) c ( f (˜ g s − h ) − f (˜ g s − )) N ( ds, dh ) . This stochastic differential equation induces a stochastic flow ϕ t in e G which maps ˜ g on thesolution at time t and starting at ˜ g of (15). By left invariance ϕ t is also defined by ϕ t : e G −→ e G ˜ g ˜ g ˜ g t , where ˜ g t is starting at Id.Denote by k · k any norm on Lie( e G ) and by k · k ˜ g the left invariant (Finsler) metric associatedin e G on T ˜ g e G . For v ∈ T ˜ g e G we aim to investigate the asymptotic exponential rate of growth ordecay of k dϕ t (˜ g )( v ) k ϕ t (˜ g ) . Denote by L ˜ g the left translation by ˜ g in e G . By left invariance ofthe flow, k dϕ t (˜ g )( v ) k ϕ t (˜ g ) = k dϕ t (Id)( e X ) k ϕ t (Id) where e X := ( dL ˜ g ) − ( v ) ∈ T Id e G = Lie( e G ). For˜ g = ( g, ξ ) ∈ e G and e X, e Y ∈ Lie( e G ) it comesAd(˜ g )( e X ) = (Ad( g )( X ) , gx − Ad( g )( X ) ξ ) (16)ad( e Y )( e X ) = (ad( Y )( X ) , Y x − Xy ) . (17)The endomorphism e X ad( V , e X ) is diagonalisable on Lie( e G ). Its eigenvalues are − , , e U − , e U and e U + the eigenspaces associated.We can check that e X ∈ e U + ⇐⇒ X ∈ N and x ∈ U + e X ∈ e U ⇐⇒ X ∈ A ⊕ M and x ∈ U e X ∈ e U − ⇐⇒ X ∈ N and x ∈ U − . et ˜ g ∞ := ( n ∞ , λ ∞ ( e − e )) ∈ e G and V −∞ := Ad(˜ g ∞ ) (cid:16) e U + (cid:17) , V ∞ := Ad(˜ g ∞ ) (cid:16) e U + e U + (cid:17) .We denote by ˜ h t := (cid:0) e − tαV , (cid:1) ˜ g − ∞ ˜ g t , where we recall that ˜ g t := ϕ t (Id) is starting at Id. Theorem 1.
Let e X ∈ Lie( e G ) . For almost every trajectory t log k dϕ t (Id)( e X ) k ϕ t (Id) −→ t → + ∞ α if e X ∈ Lie( e G ) \ V ∞ e X ∈ V ∞ \ V −∞ − α if e X ∈ V −∞ \ { } Proof.
By left invariance of k · k , k dϕ t (Id)( e X ) k ϕ t (Id) = k Ad( ˜ g t − )( e X ) k . We set, for ˜ g ∈ e G k Ad(˜ g ) k := sup e X =0 k Ad(˜ g )( e X ) kk e X k . Let e X ∈ Lie( e G ). Writting ˜ g − t = (cid:16) ˜ h t (cid:17) − (cid:0) e − tαV , (cid:1) ˜ g − ∞ we deduce that k Ad (cid:0)(cid:0) e − tαV , (cid:1) ˜ g − ∞ (cid:1) ( e X ) kk Ad(˜ h t ) k ≤ k Ad(˜ g − t )( e X ) k≤ k Ad(˜ h t ) − kk Ad (cid:0)(cid:0) e − tαV , (cid:1) ˜ g − ∞ (cid:1) ( e X ) k . (18)Suppose for the moment that lim sup t → + ∞ t log (cid:13)(cid:13)(cid:13) Ad(˜ h t ) − (cid:13)(cid:13)(cid:13) ≤ t → + ∞ t log k Ad(˜ h t ) k ≤ . (20)Then we deduce from (18) that t log k Ad(˜ g − t )( e X ) k and t log k Ad (cid:0)(cid:0) e − tαV , (cid:1) ˜ g − ∞ (cid:1) ( e X ) k have thesame limit when t goes to ∞ . The linear isomorphism Ad (cid:0) e − tαV , (cid:1) is diagonalisable with eigen-values e − αt , 1 and e αt associated respectively to the eigenspaces e U + , e U and e U − . DecomposingAd(˜ g ∞ ) − ( e X ) in the direct sum e U − ⊕ e U ⊕ e U + and using a Euclidean norm k · k on Lie( e G ) forwhich this decomposition is orthogonal, we deduce easily the theorem (note that the convergenceis independant of the chosen norm).Thus it remains to prove (19) and (20). We have˜ h t = (cid:0) e − tαV , (cid:1) ˜ g − ∞ ˜ g t = (cid:0) e − tαV , (cid:1) (cid:0) n − ∞ , − λ ∞ ( e − e ) (cid:1) ( n t a t k t , ξ t )= (cid:0) h t , e − tαV (cid:0) n − ∞ ξ t (cid:1) − λ ∞ e tα ( e − e ) (cid:1) = (cid:0) Id , e − tαV (cid:0) n − ∞ ξ t (cid:1) − λ ∞ e tα ( e − e ) (cid:1) ( h t , ε >
0. By Proposition 4 we can find t > ∀ t > t r ( h t ) ≤ εt and by Proposition5 we have k e − tαV (cid:0) n − ∞ ξ t (cid:1) − λ ∞ e tα ( e − e ) k ≤ e αt k ( n − ∞ ξ t ) − − λ ∞ ( e − e ) k + k ( n − ∞ ξ t ) k + e − αt k ( n − ∞ ξ t ) + k≤ e εt Now using the following Lemma 3 we deduce easily (19) and (20).
Lemma 3.
There exist positive constants α, β, γ such that for g ∈ G and ξ ∈ R ,d k Ad( g, k ≤ αe r ( g ) k Ad(Id , ξ ) k ≤ β k ξ k + γ. roof. All norms are equivalent and it suffices to check the inequalities for some particuliar norms.Let choose the following SO ( d )-invariant euclidean norm on Lie( e G ) k ( X, x ) k := p Tr( X t X ) + x t x. We obtain easily k Ad( g, k = e r ( g ) . Taking now k ( X, x ) k := p Tr( X t X ) + √ x t x we get k Ad(Id , ξ )( X, x ) k = p Tr( X t X ) + k x − Xξ k ≤ k ( X, x ) k + k Xξ k≤ k ( X, x ) k + p Tr( X t X ) max i | ξ i | ≤ k ( X, x ) k (cid:16) i | ξ i | (cid:17) . Thus k Ad(Id , ξ ) k ≤ α k ξ k for a constant α > ξ . First, remark that V −∞ and V ∞ are Lie sub-algebras of Lie( e G ). Denote by V −∞ := exp( V −∞ ) , and V ∞ := exp( V ∞ )the closed subgroup of e G associated.Fix now a euclidean norm k · k on Lie( e G ) which is Ad( K )-invariant. Such a norm is of theform (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:18) b t b C (cid:19) , x (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) := p κ b t b + β Tr ( C t C ) + γ ~x t ~x + δ ( x ) , for some positive constants κ , β , γ and δ . We denote by d the distance in e G associated to theleft invariant Riemanian metric induced by k · k . To simplify notations, we denote by d ( g, h ) thedistance between ( g,
0) and ( h,
0) for g, h ∈ G .The following result shows that the stable manifold associated to ϕ t is ϕ V −∞ . Theorem 2.
Let ˜ g and ˜ g ′ two distinct points in e G . • If ˜ g ′ ∈ ˜ g V −∞ then t log d ( ϕ t (˜ g ) , ϕ t (˜ g ′ )) −→ t → + ∞ − α. • If ˜ g ′ / ∈ ˜ g V −∞ then lim inf t →∞ d ( ϕ t (˜ g ) , ϕ t (˜ g ′ )) > . The properties of d we need in the proof of Theorem 2 are sum up in the following proposition Proposition 6. i) Left invariance ∀ ˜ g, ˜ h ∈ e G, d (˜ g, ˜ g ˜ h ) = d (Id , ˜ h ) . Thus d (Id , ˜ g − ) = d (Id , ˜ g ) and triangularity inequality writes: ∀ ˜ g, ˜ h ∈ e G, d (Id , ˜ g ˜ h ) ≤ d (Id , ˜ g ) + d (Id , ˜ h ) ii) K-right invariance ∀ ˜ g ∈ e G and k ∈ K, d (( k, , ˜ g ( k, d (Id , ˜ g ) iii) For e X ∈ Lie( e G ) d (cid:16) Id , exp( e X ) (cid:17) ≤ k e X k . v) There exists a neighborood O of in Lie( e G ) and a constant C > such that ∀ e X ∈ O , C k e X k ≤ d (cid:16) Id , exp( e X ) (cid:17) v) ∀ ( g, ξ ) ∈ e G d (Id , g ) ≤ d (Id , ( g, ξ )) vi) For g = S ( r, θ ) R and g ′ = S ( r ′ , θ ′ ) R ′ we have κ p κ + 2 β d ( S ( r, θ ) , S ( r ′ , θ ′ )) ≤ d ( g, g ′ ) vii) For all r ≥ and θ ∈ S d − d (Id , S ( r, θ )) = κr Proof of Proposition 6. i ) and ii ).The left and K -right invariance follows from the definition ofthe metric as being a left invariant Riemannian metric on e G defined from an Ad( K )-invariantinner product on Lie( e G ). Inequality iii ) is obtained remarking that the length of the path t ∈ [0 , exp (cid:16) t e X (cid:17) is equal to k e X k . iv ). Denote by d exp : Lie( e G ) → e G be the exponential map at Id induced by the metric k · k in e G : for ˜ X ∈ Lie( e G ), d exp( ˜ X ) = γ ˜ X (1) where t ∈ [0 , γ ˜ X ( t ) is the geodesic starting from Id inthe direction ˜ X . The differential at 0 of d exp is known to be identity and there exists a sufficientsmall neighborhood O ′ of 0 ∈ Lie( e G ) such that: ∀ ˜ X ∈ O ′ , k ˜ X k = d (cid:16) I, d exp( ˜ X ) (cid:17) . ( ∗ )Furthermore, the map d exp − ◦ exp can be defined in a neighborhood of 0 and its differential at 0is the identity: d exp − ◦ exp( ˜ X ) = ˜ X + o ( k ˜ X k ). So we can find O neighborhood of 0 and C > X ∈ O , C k ˜ X k ≤ k d exp − ◦ exp( ˜ X ) k ≤ C k ˜ X k . Taking O small enough so that d exp − ◦ exp( O ) ⊂ O ′ , we can apply ( ∗ ) to d exp − ◦ exp( ˜ X ), thus yielding k d exp − ◦ exp( ˜ X ) k = d (cid:16) I, exp( ˜ X ) (cid:17) for every ˜ X ∈ O . v ). Each path s ∈ [0 , ( g s , ξ s ) joining Id to ( g, ξ ) is of length R k ( g − s ˙ g s , g − s ξ s ) k ds whichis greater than R k ( g − s ˙ g s , k ds corresponding to the path s ∈ [0 , ( g s ,
0) joining Id to ( g, vi ). Consider a path s ∈ [0 , S ( r s , θ s ) R s joining g to g ′ . We compute, using dot notationfor dds R − s S ( r s , θ s ) − dds ( S ( r s , θ s ) R s ) = r s θ ts + sinh( r s ) ˙ θ st ˙ r s θ s + sinh( r s ) ˙ θ s (cosh( r s ) − (cid:16) ˙ θ s θ ts − θ s ˙ θ ts (cid:17) + R − s ˙ R s ! . Its length l := R k R − t S ( r t , θ t ) − ddt S ( r t , θ t ) R t k dt is larger than Z κ k ˙ r s θ s + sinh( r s ) ˙ θ s k ds = Z κ q ( ˙ r s ) + sinh( r s ) k ˙ θ s k ds. Moreover, the path s S ( r s , θ s ) which join S ( r, θ ) to S ( r ′ , θ ′ ) is of length Z r κ (cid:16) ( ˙ r s ) + sinh( r s ) k ˙ θ s k (cid:17) + β (cid:16) r ) − k ˙ θ s k (cid:17) ds (21) ≤ p κ + 2 β κ Z κ q ( ˙ r s ) + sinh( r s ) k ˙ θ s k ds. (22)Thus d ( S ( r, θ ) , S ( r ′ , θ ′ )) ≤ p κ + 2 β κ l and taking the infimum over all the path joining g to g ′ we obtain vi ). roof of Theorem 2. Let e Y ∈ Lie( e G ) \ { } be such that exp( e Y ) = ˜ g − ˜ g ′ . Then d ( ϕ t (˜ g ) , ϕ t (˜ g ′ )) = d (cid:16) Id , ˜ g − t exp (cid:16) e Y (cid:17) ˜ g t (cid:17) = d (cid:16) Id , exp (cid:16) Ad(˜ g − t )( e Y ) (cid:17)(cid:17) . By Theorem 1, if e Y ∈ Ad( g ∞ )( e U + ) (i.e. if g ′ ∈ g V −∞ ) then k Ad( g − t )( e Y ) k converge to 0 exponen-tially fast with rate α and so for large t it evolves in O . Thus, using iii ) and iv ) of Proposition 6we obtain the first point of the theorem as a direct consequence of Theorem 1.Set e X := Ad(˜ g − ∞ )( e Y ), thus e Y = Ad(˜ g ∞ ) (cid:18)(cid:16) e X (cid:17) + + (cid:16) e X (cid:17) + (cid:16) e X (cid:17) − (cid:19) , and write (cid:16) e X (cid:17) + = ( X + , x + ) , where X + ∈ N and x + ∈ U + (cid:16) e X (cid:17) = ( X , x ) , where X ∈ A ⊕ M and x ∈ U (cid:16) e X (cid:17) − = ( X − , x − ) , where X − ∈ N and x − ∈ U − . Now suppose that ˜ g ′ / ∈ ˜ g V −∞ which is equivalent to (cid:16) e X (cid:17) = 0 or (cid:16) e X (cid:17) − = 0.Suppose first that (cid:16) e X (cid:17) − = 0. Thus e Y ∈ Lie( e G ) \ V ∞ and by Theorem 1 k Ad(˜ g − t ) e Y k convergesto + ∞ exponentially fast. Now suppose by contradiction that lim inf t → + ∞ d (cid:16) Id , exp (cid:16) Ad(˜ g − t ) e Y (cid:17)(cid:17) = 0.Then we can find a s t such that d (cid:16) Id , exp (cid:16) Ad(˜ g − s t )( e Y ) (cid:17)(cid:17) converges to 0 and for large t Ad(˜ g − s t )( e Y ) lies in O . The inequality iv ) of Proposition 6 give us the contradiction and wehave proved the second point of the Theorem if (cid:16) e X (cid:17) − = 0.So we can suppose (cid:16) e X (cid:17) − = 0 and (cid:16) e X (cid:17) = 0. First case: X = . By v ) of Proposition 6, d (Id , g − t exp( Y ) g t ) ≤ d (cid:16) Id , ˜ g − t exp (cid:16) e Y (cid:17) ˜ g t (cid:17) and it remains to prove that lim inf t →∞ d (Id , g − t exp( Y ) g t ) is positive. But Y = Ad( n ∞ )( X ) and X = X + + X ∈ N ⊕ A ⊕ M \ N . Consider an Iwasawa decomposition of exp( X ) in G exp( X ) = ¯ nam, n ∈ N , a ∈ A, and m ∈ M, and am = Id. Since ∀ g, h d (Id , gh ) ≤ d (Id , g ) + d (Id , h ) we get d (Id , g − t exp( Y ) g t ) = d (Id , h − t e − tαV ¯ name tαV h − t ) ≥ d (Id , h − t e − tαV ame tαV h t ) − d (Id , h − t e − tαV ¯ ne tαV h t ) . Writting ¯ n = exp( Z ) , Z ∈ N and d (Id , h − t e − tαV ¯ ne tαV h t ) is dominated by e − tα + r ( h t ) k Z k (byLemma 3 and iii ) of Proposition 6) and converges exponentially fast to zero (recall that by Propo-sition 4 r ( h t ) = o ( t ) a.s. ). Thus it remains to prove that lim inf d (Id , h − t e − tαV ame tαV h t ) > Lemma 4.
Let a ∈ A and m ∈ M s.t am = Id . Then ∃ C > , ∀ g ∈ G, d (Id , g − amg ) > C .Proof of lemma 4. Consider the polar decomposition g = S ( r, θ ) R . Suppose first that a = Id and = Id. Then we get d (Id , g − mg ) = d (Id , S ( r, − θ ) mS ( r, θ )) = d ( S ( r, θ ) , mS ( r, θ )) = d ( S ( r, θ ) , S ( r, mθ ) m ) ≥ κ p κ + 2 β d ( S ( r, θ ) , S ( r, mθ )) by vi ) of Proposition 6 ≥ κ p κ + 2 β (cid:0) d ( S ( r, θ ) , S ( r, θ ) m − ) − d ( S ( r, θ ) m − , S ( r, mθ )) (cid:1) = κ p κ + 2 β (cid:0) d (Id , m − ) − d ( S ( r, θ ) , mS ( r, θ )) (cid:1) = κ p κ + 2 β (cid:0) d (Id , m ) − d (Id , g − mg ) (cid:1) Thus d (Id , g − mg ) ≥ κκ + √ κ +2 β d (Id , m ) > a = Id. Let u = 0 such that a = exp( uV ), then an explicit computation gives:cosh r ( g − amg ) = cosh( u ) (cid:0) cosh( r ) − (( mθ ) ) sinh( r ) (cid:1) − sinh( r ) d X i =2 θ i ( mθ ) i (23)= cosh( u ) + cosh( u )(1 − (( mθ ) ) ) − d X i =2 θ i ( mθ ) i ! sinh( r ) (24) ≥ cosh( u ) + (1 − θ t ( mθ )) sinh( r ) we used ( mθ ) = θ (25) ≥ cosh( u ) . (26)Then by vi ) and vii ) of Proposition 6 it comes d (Id , g − amg ) ≥ κ p κ + 2 β κu > . Return to the proof of Theorem 2.
Second case: X = = . So e X = ( X + , x + + x ) and explicitelyexp( e X ) = (exp( X + ) , x + x + + X + x , ξ )(exp( X + ) , , where we have set ξ := x + x + + X + x .Thus ˜ g − t exp( e Y )˜ g t = ˜ h − t ( e − tαV ,
0) exp( e X )( e tαV , h t = (Id , h − t e − tαV ξ )(exp(Ad( h − t e − tαV ) X ) , , and d (Id , ˜ g − t exp( e Y )˜ g t ) ≥ d (cid:0) Id , (Id , h − t e − tαV ξ ) (cid:1) − d (Id , (exp(Ad( h − t e − tαV ) X ) , . As done previously in the first case, d (Id , (exp(Ad( h − t e − tαV ) X ) , t →∞ d (cid:0) Id , (Id , h − t e − tαV ξ ) (cid:1) > . (27)Suppose by contradiction that we can find s t such that d (cid:0) Id , (cid:0) Id , h − s t e − s t αV ( ξ ) (cid:1)(cid:1) converges to0. By iv ) of Proposition 6 for large td (cid:0) Id , (Id , h − s t e − s t αV ( ξ )) (cid:1) ≥ k h − s t e − s t αV ( ξ ) k ince X + x ∈ U + we obtain directly that q ( ξ ) = q ( x ) which is negative since x is supposedto be non zero. But k h − s t e − s t αV ( ξ ) k = γ d X i =1 q (cid:0) h − s t e − s t αV ( ξ ) , e i (cid:1) ! + δ q (cid:0) h − s t e − s t αV ( ξ ) , e (cid:1) ≥ min( γ, δ ) d X i =0 q (cid:0) h − s t e − s t αV ( ξ ) , e i (cid:1) ! = min( γ, δ ) (cid:16) q ( h − s t e − s t αV ( ξ ) , e ) − q (cid:0) h − s t e − s t αV ( ξ ) (cid:1) (cid:17) ≥ − min( γ, δ ) q ( x ) > . H d × R ,d of the stable manifolds We explicit here the projection of V −∞ on H d × R ,d . Recall that by definition an element of V −∞ is of the form ˜ g ∞ exp( X, x )˜ g − ∞ where ( X, x ) ∈ N × U + . We deduce, since in this caseexp( X, x ) = (exp( X ) , x ), that an element of π ( V −∞ ) is of the form (cid:0) n ∞ exp( X ) n − ∞ ( e ) , un ∞ ( e + e ) + λ ∞ (cid:0) Id − n ∞ exp( X ) n − ∞ (cid:1) ( e − e ) (cid:1) , (28)where X lies in N and u ∈ R .Since exp( X )( e + e ) = e + e for X ∈ N we obtain q ( n ∞ exp( X ) n − ∞ ( e ) , n ∞ ( e + e )) = q ( n − ∞ e , e + e ) = q ( e , n ∞ ( e + e ))and thus when X describes N then n ∞ exp( X ) n − ∞ ( e ) draws the intersection between H d andthe affine hyperplan passing by e and q -orthogonal to n ∞ ( e + e ). This submanifold of R ,d isa parabolo¨ıd of codimension 2 and is mapped by p (the projection onto the projective space) ona sphere tangent at ∂ H d in θ ∞ and passing by p ( e ). It is called the horosphere tangent at θ ∞ and passing by e and is denoted by H ∞ .Moreover, since q ( n ∞ exp( X ) n − ∞ ( e − e ) , n ∞ ( e + e )) = q ( e − e , e + e ) = 0 , we get that when X describes N then n ∞ exp( X ) n − ∞ ( e − e ) describes the intersection betweenthe light cone { ξ, q ( ξ ) = 0 } and the hyperplan passing by e − e and q -orthogonal to n ∞ ( e + e ). Thus, when X describes N then (cid:0) Id − n ∞ exp( X ) n − ∞ (cid:1) ( e − e ) draws a parabolo¨ıd P ∞ in the hyperplan q -orthogonal to n ∞ ( e + e ). For each ˙ ξ in the horosphere H ∞ correspondsa unique X ˙ ξ ∈ N such that ˙ ξ = n ∞ exp( X ˙ ξ ) n − ∞ ( e ) and the one-to-one function ψ : ˙ ξ (cid:16) Id − n ∞ exp( X ˙ ξ ) n − ∞ (cid:17) ( e − e ) maps H ∞ on P ∞ .Then by (28), we obtain the following one-to-one map H ∞ × h n ∞ ( e + e ) i −→ π ( V −∞ )( ˙ ξ, ξ ) ( ˙ ξ, ξ + λ ∞ ψ ( ˙ ξ ))and π ( V −∞ ) is a skew product of the line h n ∞ ( e + e ) i with the horosphere H ∞ . b θ ∞ p ( e ) p ( H ∞ ) 0 h n ∞ ( e + e ) i P ∞ e Figure 2: π ( V −∞ ) is a skew-product of a horosphere with a line References [1] J. Angst.
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