On topological and measurable dynamics of unipotent frame flows for hyperbolic manifolds
OON TOPOLOGICAL AND MEASURABLE DYNAMICS OFUNIPOTENT FRAME FLOWS FOR HYPERBOLIC MANIFOLDS
FRANC¸ OIS MAUCOURANT, BARBARA SCHAPIRA
Abstract.
We study the dynamics of unipotent flows on frame bundles ofhyperbolic manifolds of infinite volume. We prove that they are topologi-cally transitive, and that the natural invariant measure, the so-called ”Burger-Roblin measure”, is ergodic, as soon as the geodesic flow admits a finite mea-sure of maximal entropy, and this entropy is strictly greater than the codi-mension of the unipotent flow inside the maximal unipotent flow. The latterresult generalises a Theorem of Mohammadi and Oh. Introduction
Problem and State of the art.
For d ≥
3, let Γ be a Zariski-dense, dis-crete subgroup of G = SO o ( d, N be a maximal unipotent subgroup of G (hence isomorphic to R d − ), and U ⊂ N a nontrivial connected subgroup (henceisomorphic to some R k in R d − ). The main topic of this paper is the study of theaction of U on the space Γ \ G . Geometrically, this is the space FM of orthonormalframes of the hyperbolic manifold M = Γ \ H d , and the N (and U )-action movesthe frame in a parallel way on the stable horosphere defined by the first vector ofthe frame. There are a few cases where such an action is well understood, fromboth topological and ergodic point of view.1.1.1. Lattices.
If Γ has finite covolume, then Ratner’s theory provides a completedescription of closures of U -orbits as well as ergodic U -invariant measures. If Γ hasinfinite covolume, while it no longer provide information about the topology of theorbits, it still classifies finite U -invariant measures. Unfortunately, the dynamicallyrelevant measures happen to be of infinite mass. In the rest of the paper, we willalways think of Γ as a subgroup having infinite covolume.1.1.2. Full horospherical group.
If one looks at the action of the whole horosphericalgroup U = N , a N -orbit projects on T M onto a leaf of the strong stable foliationfor the geodesic flow , a well-understood object, at least in the case of geometricallyfinite manifolds. In particular, the results of Dal’bo [5] imply that for a geometri-cally finite manifold, such a leaf is either closed, or dense in an appropriate subsetof T M .From the ergodic point of view, there is a natural good N -invariant measure,the so-called Burger-Roblin measure , unique with certain natural properties. Recallbriefly its construction. The measure of maximal entropy of the geodesic flowon T M , the Bowen-Margulis-Sullivan measure , when finite , induces a transverseinvariant measure to the strong stable foliation. This transverse measure is oftenseen as a measure on the space of horospheres, invariant under the action of Γ. a r X i v : . [ m a t h . D S ] M a y FRANC¸ OIS MAUCOURANT, BARBARA SCHAPIRA
Integrating the Lebesgue measure along these leaves leads to a measure on T M ,which lifts naturally to FM into a N -invariant measure, the Burger-Roblin measure .In [26], Roblin extended a classical result of Bowen-Marcus [4], and showedthat, up to scalar multiple, when the Bowen-Margulis-Sullivan measure is finite,it induces (up to scalar multiple) the unique invariant measure supported on thisspace of horospheres, supported in the set of horospheres based at conical (radial)limit points.In particular, if the manifold M is geometrically finite, this gives a completeclassification of Γ-invariant (Radon) measures on the space of horospheres, or equiv-alently of transverse invariant measures to the strong stable foliation. In general,Roblin’s result says that there is a unique (up to scaling) transverse invariant mea-sure of full support in the set of vectors whose geodesic orbit returns i.o. in acompact set.It is natural to try to ”lift” this classification along the principal bundle FM → T M , since the structure group is compact. This was done by Winter [32], whoproved that, up to scaling, the only N -invariant measure of full support in theset of frames whose A -orbit returns i.o. in a compact set is the Burger-Roblinmeasure, i.e. the natural M -invariant lift of the above measure. On geometricallyfinite manifolds, this statement is simpler: the Burger-Roblin is the unique (up toscaling) N -invariant ergodic measure of full support.1.1.3. A Theorem of Mohammadi and Oh.
However, if one considers only the actionof a proper subgroup U ⊂ N , the situation changes dramatically, and much less isknown, because ergodicity or conservativeness of a measure with respect to a groupdoes not imply in any way the same properties with respect to proper subgroups. Inthis direction, the first result is a Theorem of Mohammadi and Oh [23], which statesthat, in dimension d = 3 (in which case dim ( U ) = 1) and for convex-cocompactmanifolds, the Burger-Roblin measure is ergodic and conservative for the U -actionif and only if the critical exponent δ Γ of Γ satisfies δ Γ > Dufloux recurrence results.
In [7, 8], Dufloux investigates the case of smallcritical exponent. Without any assumption on the manifold, when the Bowen-Margulis-Sullivan measure is finite (assumption satisfied in particular when Γ isconvex-cocompact, but not only, see [25]), he proves in [7] that the Bowen-Margulis-Sullivan is totally U -dissipative when δ Γ ≤ dim N − dim U , and totally recurrentwhen δ Γ > dim N − dim U . In [8], when the group Γ is convex-cocompact, he provesthat when δ Γ = dim N − dim U , the Burger-Roblin measure is U -recurrent.1.1.5. Rigid acylindrical 3-manifolds.
There is one last case where more is know onthe topological properties of the U -action, in fact in a very strong form. Assuming M is a rigid acylindrical 3-manifold, McMullen, Mohammadi and Oh recently man-aged in [22] to classify the U -orbit closures, which are very rigid. Their analysisrelies on their previous classification of SL(2 , R )-orbits [21].Unfortunately, their methods relies heavily on the particular shape of the limitset (the complement of a countable union of disks), and such a strong result iscertainly false for general convex-cocompact manifolds. NIPOTENT FRAME FLOWS 3
Results.
The results that we prove here divide in two distinct parts, a topo-logical one, and a ergodic one. Although they are independent, the strategy of theirproofs follow similar patterns, a fact we will try to emphasise.1.2.1.
Topological properties.
Let A ⊂ G be a Cartan Subgroup. Denote by Ω ⊂FM the non-wandering set for the geodesic flow (or equivalently, the A -action),and by E the non-wandering set for the N -action. For more precise definitions anddescription of these objects, see section 2.Using a Theorem of Guivarc’h and Raugi [13], we show: Theorem 1.1.
Assume that Γ is Zariski-dense. The action of A on Ω is topologi-cally mixing. This allows us to deduce:
Theorem 1.2.
Assume that Γ is Zariski-dense. The action of U on E is topologi-cally transitive. Both results are new. Note that, for example in the case of a general convex-cocompact manifold with low critical exponent, the existence of a non-divergent U -orbit is itself non-trivial, and was previously unknown.1.2.2. Ergodic properties.
We will assume that Γ is of divergent type, and denoteby µ the Bowen-Margulis-Sullivan measure - or more precisely, its natural lift to FM , normalised to be a probability. We are interested in the case where µ is afinite measure. Denote by ν the Patterson-Sullivan measure on the limit set, and λ the Burger-Roblin measure on FM . More detailed description of these objectsis given in section 4.The following is a strengthening of the Theorem of Mohammadi and Oh [23]. Theorem 1.3.
Assume that Γ is Zariski-dense. If µ is finite and δ Γ + dim( U ) >d − , then both measures µ and λ are U -ergodic. The hypothesis that µ is finite is satisfied for example when Γ is geometricallyfinite see Sullivan [29]. But there are many other examples, see [25], [2]. Note thatthe measure µ is not U -invariant, or even quasi-invariant; in this case, ergodicitysimply means that U -invariant sets have zero of full measure. Apart from the useof Marstrand’s projection Theorem, our proof differs significantly from the one of[23], and does not use compactness arguments, allowing us to go beyond the convex-cocompact case. It is also, in our opinion, simpler. Note that the work of Dufloux[7] uses the same assumptions as ours.For the opposite direction, we prove: Theorem 1.4.
Assume that Γ is Zariski-dense. If µ is finite with δ Γ + dim( U ) FRANC¸ OIS MAUCOURANT, BARBARA SCHAPIRA Topological transitivity. The proof of the topological transitivity can be sum-marised as follows. • The U -orbit of Ω is dense in E (Lemma 3.6). • The mixing of the A -action (Theorem 1.1) implies that there are couples( x , y ) ∈ Ω , generic in the sense that their orbit by the diagonal action of A by negative times on Ω is dense in Ω . • But one can ”align” such couples of frames so that x and y are in the same U -orbit, that is x U = y U (Lemma 3.4).These facts easily imply topological transitivity of U on E (see section 3.7).1.3.2. Ergodicity of µ and λ . In the convex-cocompact case, the Patterson-Sullivan ν is Ahlfors-regular of dimension δ Γ . To go beyond that case, we will need toconsider the lower dimension of the Patterson-Sullivan measure:dim ν = infess lim inf r → log ν ( B ( ξ, r ))log r , which satisfies the following important property. Proposition 1.5 (Ledrappier [16]) . If µ is finite, then dim ν = δ Γ . The first step in the proof of topological transitivity is the proof that the closureof the set of U -orbits intersecting Ω is E . The analogue here is to show that fora U -invariant set E , it is sufficient to show that µ ( E ) = 0 or µ ( E ) = 1 to deducethat λ ( E ) = 0 or λ ( E c ) = 0 respectively. Marstrand’s projection Theorem andthe hypothesis δ Γ + dim( U ) > d − λ isin fact equivalent to the ergodicity of µ (Proposition 4.10). Although it is highlyunusual to study the ergodicity of non-quasi-invariant measures, it turns out hereto be easier, thanks to finiteness of µ .For the second step, we know thanks to Winter [32] that the A -action on (Ω , µ ⊗ µ ) is mixing. So we can find couples ( x , y ) ∈ Ω , which are typical in the sensethat they satisfy Birkhoff ergodic Theorem for the diagonal action of A for negativetimes and continuous test-functions. By the same alignment argument than in thetopological part, one can find such typical couples in the same U -orbit.Unfortunately, from the point of view of measures, existence of one individualorbit with some specified properties is meaningless. To circumvent this difficulty,we have to consider plenty of such typical couples on the same U -orbit. More pre-cisely, we consider a measure η on Ω such that almost surely, a couple ( x , y ) pickedat random using η is in the same U -orbit, and is typical for the diagonal A -action.For this to make sense when comparing with the measure µ , we also require thatboth marginal laws of η on Ω are absolutely continuous with respect to µ . We checkin section 5.2 that the existence of such a measure η is sufficient to prove Theorem1.3. This measure η is a kind of self-joining of the dynamical system (Ω , µ ), butinstead of being invariant by a diagonal action, we ask that it reflects both thestructure of U -orbits, and the mixing property of A .It remains to show that such a measure η actually exists. In dimension d = 3,we can construct it (at least locally on F H ) as the direct image of µ ⊗ µ by the NIPOTENT FRAME FLOWS 5 alignment map, so we present the simpler 3-dimensional case separately in sec-tion 5.4. The fact that η is supported by typical couples on the same U -orbitis tautological from the chosen construction. The difficult part is to show that itsmarginal laws are absolutely continuous. This is a consequence of the following fact: If two compactly supported, probability measures on the plane ν , ν have finite -energy, then for ν -almost every x , the radial projection of ν on the unit circlearound x is absolutely continuous with respect to the Lebesgue measure on the circle. Although probably unsurprising to the specialists, as there exists many relatedstatements in the literature (see e.g. [20],[19]), we were unable to find a reference.We prove this implicitly in our situation, using the L -regularity of the orthogonalprojection in Marstrand’s Theorem, and the maximal inequality of Hardy and Lit-tlewood.In dimension d ≥ 4, the construction of η , done in section 5.5, is a bit moreinvolved since there is not a unique couple aligned on the same U -orbit, espe-cially if dim( U ) ≥ 2, so we have to choose randomly amongst them, using smoothmeasures on Grassmannian manifolds. Again, the absolute continuity follows fromMastrand’s projection Theorem and the maximal inequality.1.4. Organization of the paper. Section 2 is devoted to introductory material.In section 3, we prove our results on topological dynamics. In section 4, we introducethe measures µ and λ , establish the dimensional properties that we need, and proveTheorem 4.6 and the fact that U -ergodicity of µ and λ are equivalent. Finally, weprove Theorem 1.3 in section 5.2. Setup and Notations Lie groups, Iwasawa decomposition. Let d ≥ 2, and G = SO o ( d, d + 1 , R ) preserving the quadratic form q ( x , .., x d +1 ) = x + x + .. − x d +1 . It is the group of direct isometries of the hyperbolic n -space H d = { x ∈ R d +1 , q ( x ) = − , x d +1 > } . Define K < G as K = (cid:26)(cid:18) k 00 1 (cid:19) : k ∈ SO( d ) (cid:27) . It is a maximal compact subgroup of G , and it is the stabilizer of the origin x =(0 , ... , ∈ H d .We choose the one-dimensional Cartan subgroup A , defined by A = a t = I d − 00 cosh( t ) sinh( t )sinh( t ) cosh( t ) : t ∈ R . It commutes with the following subgroup M , which can be identified with SO( d − M = (cid:26)(cid:18) m I (cid:19) : m ∈ SO( d − (cid:27) . In other words, the group M is the centralizer of A in K . The stabilizer of anyvector v ∈ T H d identifies with a conjugate of M , so that T H d = SO o ( d, /M . FRANC¸ OIS MAUCOURANT, BARBARA SCHAPIRA Let n ⊂ so ( d, 1) be the eigenspace of Ad ( a t ) with eigenvalue e − t . Let N = exp( n ) . It is an abelian, maximal unipotent subgroup, normalized by A . The group G is diffeomorphic to the product K × A × N . This decomposition is the Iwasawadecomposition of the group G .The subgroup N is normalized by M , and M (cid:110) N is a closed subgroup isomorphicto the orientation-preserving affine isometry group of an d − U is any closed, connected unipotent subgroup of G , it is conjugated to asubgroup of N (see for example [3]). Therefore, it is isomorphic to R k , for some d ∈ { , .., d − } . Through the article, we will always assume that k ≥ A, N, U on the space Γ \ G .2.2. Geometry. Fundamental group, critical exponent, limit set. Let Γ ⊂ G = Isom + ( H d ) be adiscrete group. Let M = Γ \ H d be the corresponding hyperbolic manifold. Thelimit set Λ Γ is the set of accumulation points in ∂ H n (cid:39) S d − of any orbit Γ o ,where o ∈ H d . We will always assume that the group Γ is nonelementary, that is Γ = + ∞ .The critical exponent δ of the group Γ is the infimum of the s > P Γ ( s ) = (cid:88) γ ∈ Γ e − sd ( o,γo ) , is finite, where o is the choice of a fixed point in H d . In the convex-cocompact case,the critical exponent δ equals the Hausdorff dimension of the limit set Λ Γ . Since Γis non-elementary, we have 0 < δ ≤ d − Frames. The space of orthonormal, positively oriented frames over H d (resp. M )will be denoted by F H d (resp. FM ). As G acts simply transitively on F H d , F H d (resp. FM ) can be identified with G (resp. Γ \ G ) by the map g (cid:55)→ g. x , where x is a fixed reference frame. Note that F H d is a M -principal bundle over T H d ,and so is FM over T M . Denote by π : FM → T M (resp. F H d → T H d ) theprojection of a frame onto its first vector.As said above, we are interested in the properties of the right actions of A, N, U on FM .Given a subset E ⊂ M (resp. T M , FM ), we will write ˜ E for its lift to H d (resp. T H d , F H d ).Denote by F S d − the set of (positively oriented) frames over ∂ H d = S d − . Wewill write F Λ Γ for the subset of frames which are based at Λ Γ . NIPOTENT FRAME FLOWS 7 x .n ∂ H x x .u x .a x .a x .n x x .u Figure 1. Right actions of A , N , U Generalised Hopf coordinates. Choose o to be the point (0 , . . . , , ∈ H d . Recallthat the Busemann cocycle is defined on S d − × H d × H d by β ξ ( x, y ) = lim z → ξ d ( x, z ) − d ( y, z )By abuse of notation, if x , x (cid:48) are frames (or v, v (cid:48) vectors) with basepoints x, x (cid:48) ∈ H d ,we will write β ξ ( x , x (cid:48) ) or β ξ ( v, v (cid:48) ) for β ξ ( x, x (cid:48) ).We will use the following extension of the classical Hopf coordinates to describeframes. To a frame x ∈ F H d , we associate F H d → ( F S d − × ∆ S d − ) × R , x = ( v , . . . , v d ) (cid:55)→ ( x + , x − , t x ) , where x − (resp. x + ) is the negative (resp. positive) endpoint in S d − of the geodesic x A , t x = β x + ( o, x ), and x + ∈ F S d − is the frame over x + obtained for exampleby parallel transport along x A of the ( d − v , . . . , v n ). Thesubscript ∆ in ( F S d − × ∆ S d − ) indicates that this is the product set, minus thediagonal, i.e. the set of ( x + , x − ) where x + is based at x − . x x − x + t x o Figure 2. Hopf frame coordinatesDefine the following subsets of frames in Hopf coordinates (cid:101) Ω = ( F Λ Γ × ∆ Λ Γ ) × R , FRANC¸ OIS MAUCOURANT, BARBARA SCHAPIRA and ˜ E = ( F Λ Γ × ∆ ∂ H n ) × R . Consider their quotients Ω = Γ \ (cid:101) Ω and E = Γ \ ˜ E . These are closed invariant subsetsof FM for the dynamics of M × A and ( M × A ) (cid:110) N respectively, where all thedynamic happens. Let us state it more precisely.The non-wandering set of the action of N (resp. U ) on Γ \ G is the set of frames x ∈ FM such that given any neighbourhood O of x there exists a sequence n k ∈ N (resp. u k ∈ U ) going to ∞ such that n k O ∩ O (cid:54) = ∅ . As a consequence of Theorem1.2, the following result holds. Proposition 2.1. The set E is the nonwandering set of N and of any unipotentsubgroup { } (cid:54) = U < N . Topological dynamics of geodesic and unipotent frame flows Dense leaves and periodic vectors. For the proof of Theorem 1.1, we willneed the following intermediate result, of independent interest . Proposition 3.1. Let Γ be a Zariski-dense subgroup of SO o ( d, . Let x ∈ Ω bea frame such that π ( x ) is a periodic orbit of the geodesic flow on T M . Then its N -orbit x N is dense in E .Proof. First, observe that if v = π ( x ) ∈ T M is a periodic vector for the geodesicflow, then its strong stable manifold W ss ( v ) is dense in π ( E ) [5, Proposition B].Therefore, π − ( W ss ( π ( x ))) = x N M = x M N is dense in E . Thus it is enoughto prove that x M ⊂ x N . The crucial tool is a Theorem of Guivarc’h and Raugi [13, Theorem 2]. We will useit in two different ways depending if G = SO o (3 , 1) or G = SO o ( d, d ≥ M = SO( d − 1) is abelian in the case d = 3.Choose ˜ x a lift of x to (cid:101) Ω. As π ( x ) is periodic, say of period l > 0, but x itselfhas no reason to be periodic, there exists γ ∈ Γ and m ∈ M such that˜ x a l m = γ ˜ x . First assume d = 3, so both M and M A are abelian groups. Let C be theconnected compact abelian group C = M A/ (cid:104) a l m (cid:105) . Let ρ be the homomorphismfrom M AN to C defined by ρ ( man ) = ma mod (cid:104) a l m (cid:105) . Define X ρ = G × C/ ∼ ,where ( g, c ) ∼ ( gman, ρ ( man ) − .c ). The set X ρ is a fiber bundle over G/M AN = ∂ H n , whose fibers are isomorphic to C . In other terms, it is an extension of theboundary containing additional information on how g is positioned along AM ,modulo a l m . Let Λ ρ Γ be the preimage of Λ Γ ⊂ ∂ H n inside X ρ . Now, since C isconnected, [13, Theorem 2] asserts that the action of Γ on Λ ρ Γ is minimal. Denoteby [ g, m ] the class of ( g, m ) in X ρ .Let us deduce that x M ⊂ x N . Choose some m ∈ M . As Γ acts minimally onΛ ρ Γ , there exists a sequence ( γ k ) k ≥ of elements of Γ, such that γ k [˜ x , e ] converges to[˜ x m, e ]. It means that there exist sequences ( m k ) k ∈ M N , ( a k ) k ∈ A N , ( n k ) k ∈ N N ,such that γ k ˜ x m k a k n k → ˜ x m in G , whereas ρ ( m k a k n k ) → e in C , which means thatthere exists some sequence j k of integers, such that d k := ( m k a k ) − ( a l m ) j k → e in M A . NIPOTENT FRAME FLOWS 9 Now observe that the sequence γ k ˜ x ( a l m ) j k ( d − k n k d k ) = ( γ k γ j k )˜ x ( d − k n k d k ) ∈ Γ˜ x N has the same limit as the sequence γ k ˜ x ( a l m ) j k d − k n k = γ k ˜ x m k a k n k , which by construction converges to ˜ x m . On FM = Γ \ G , it proves precisely that x m ∈ x N . As m was arbitrary, it concludes the proof in the case n = 3.In dimension d ≥ (cid:104) a l m (cid:105) is not always a normal subgroup of M A anymore,so we have to modify the argument as follows.Denote by M x the set M x = { m ∈ M, x m ∈ x N } . This is a closed subgroup of M ; indeed, if m , m ∈ M x , then x m ∈ x N , so x m m ∈ x N m = x m N since m normalises N . Since x m ∈ x N , we have x m N ⊂ x N . So x m m ∈ x m N ⊂ x N . Thus M x is a subsemigroup, non-emptysince it contains e , and closed. Since M is a compact group, such a closed semi-group is automatically a group.We aim to show that the group M x is necessarily equal to M .Let C = M A/ (cid:104) a l (cid:105) . It is a compact connected group. Consider ρ ( man ) = ma mod (cid:104) a l (cid:105) , and the associated boundary X ρ = G × C/ ∼ . Choose some m ∈ M . Asabove, [13, Theorem 2] asserts that the action of Γ on Λ ρ Γ is minimal. Therefore,there exists a sequence ( γ k ) k ≥ of elements of Γ, such that γ k [˜ x , e ] converges to[˜ x m, e ]. As above, consider sequences ( m k ) k ∈ M N , ( a k ) k ∈ A N , ( n k ) k ∈ N N ,such that γ k ˜ x m k a k n k → ˜ x m in G , whereas ρ ( m k a k n k ) → e in C , which withthis new group C means that there exists some sequence j k of integers, such that d k := ( m k a k ) − ( a l ) j k → e in M A .Similarly to the 3-dimension case, we can write γ k ˜ x m k a k n k d k = γ k ˜ x a j k l ( d − k n k d k ) = ( γ k γ j k )˜ x m − j k ( d − k n k d k )The above argument shows that some sequence of frames in x (cid:104) m (cid:105) N = x N (cid:104) m (cid:105) converges to x m . This implies that the set of products M x . (cid:104) m (cid:105) is equal to M .We use a dimension argument to conclude the proof. The group (cid:104) m (cid:105) is a torusinside M = SO( d − d − . The group M hasdimension ( d − d − , so that M x . (cid:104) m (cid:105) = M implies that dim M x ≥ ( d − d − .By [24, lemma 4], the dimension of any proper closed subgroup of M = SO( d − d − 2) = ( d − d − . Therefore, M x cannot be a propersubgroup of M , so that M x = M . (cid:3) The following corollary is a generalization to FM of a well-known result on T M , due to Eberlein. A vector v ∈ T M is said quasi-minimizing if there existsa constant C > t ≥ d ( g t v, v ) ≥ t − C . In other terms, thegeodesic ( g t v ) goes to infinity at maximal speed. We will say that a frame x ∈ FM is quasi-minimizing if its first vector π ( x ) is quasi-minimizing. Corollary 3.2. Let Γ be a Zariski dense subgroup of G = SO o ( d, . A frame x ∈ Ω is not quasi-minimizing if and only if x N is dense in E .Proof. First, observe that when x ∈ Ω is quasi-minimizing, then the strong stablemanifold W ss ( π ( x )) of its first vector is not dense in π (Ω). Therefore, x N ⊂ π − ( W ss ( π ( x )) cannot be dense in Ω.Now, let x ∈ Ω be a non quasi-minimizing vector. Then W ss ( π ( x )) is dense in π (Ω), so that x N M = x M N = π − ( W ss ( π ( x )) is dense in Ω, and therefore in E = Ω N . Choose some y ∈ Ω such that π ( y ) is a periodic orbit of the geodesicflow. By the above proposition, y N is dense in E . As x N M is dense in E ⊃ Ω, wehave y M ⊂ x N M = x N M (this last equality following from the compactness of M ), so that there exists m ∈ M with y m ∈ x N . But π ( y m ) = π ( y ) is periodic,so that y mN is dense in E and x N ⊃ y mN ⊃ E . (cid:3) Topological Mixing of the geodesic frame flow. Recall that the continu-ous flow ( φ t ) t ∈ R (or a continuous transformation ( φ k ) k ∈ Z ) on the topological space X is topologically mixing if for any two non-empty open sets U , V ⊂ X , there exists T > t > T , φ − t U ∩ V (cid:54) = ∅ . Let us now prove Theorem 1.1, by a refinement of an argument of Shub also usedby Dal’bo [5, p988]. Proof. We will proceed by contradiction and assume that the action of A is notmixing. Thus there exist U , V two non-empty open sets in Ω, and a sequence t k → + ∞ , such that U .a t k ∩ V = ∅ . Choose x ∈ V such that π ( x ) is periodic forthe geodesic flow - this is possible by density of periodic orbits in π (Ω) [9, Theorem3.10]. Let l > , m ∈ M be such that x a l m = x .In particular, we have x a jl = x m − j ∈ x G ε for all j ∈ J . We can find integers( j k ) k (the integer parts of t k /l ) and real numbers ( s k ) k such that: t k = j k l + s k , with 0 ≤ s k < l . Without loss of generality, we can assume that the sequence ( s k ) k ≥ convergesto some s ∞ ∈ [0 , l ], and that m j k converge in the compact group M to some m ∞ ∈ M . By Proposition 3.1, the N -orbit x a − s ∞ m ∞ N is dense in E . Notice that U N is an open subset of E ; therefore one can choose a point w = x a − s ∞ m ∞ n ∈ U ,for some n ∈ N . U V w w a tk x Figure 3. The frame flow is mixing NIPOTENT FRAME FLOWS 11 We have w a t k = x a − s ∞ m ∞ na t k = x ( a l m ) j k ( m − j k m ∞ )( a s k − s ∞ )( a t k na − t k )= x ( m − j k m ∞ )( a s k − s ∞ )( a t k na − t k ) . Observe that, as N -orbits are strong stable manifolds for the A -action, solim k a t k na − t k = e. By definition of m ∞ and s ∞ , lim k m − j k m ∞ = e and lim k a s k − s ∞ = e . Therefore, w a t k tends to the frame x in the open set V . Thus, we found a frame w ∈ U , with w a t k ∈ V for all k large enough. Contradiction. (cid:3) Dense orbits for the diagonal frame flow on Ω . Recall that a continuousflow ( φ t ) t ∈ R (or a continuous transformation ( φ k ) k ∈ Z ) on the topological space X is said to be topologically transitive if any nonempty invariant open set is dense.In the case of a continuous transformation on a complete separable metric spacewithout isolated points, topological transitivity is equivalent to the existence ofa dense positive orbit, or equivalently, that the set of dense positive orbits is a G δ -dense set (see for example [6]).It is clear that topological mixing implies topological transitivity. Moreover, asis easily checked, topological mixing of ( X, φ t ) implies topological mixing for thediagonal action on the product ( X × X, ( φ t , φ t )).A couple ( x , y ) ∈ Ω will be said generic if the negative diagonal, discrete-timeorbit ( x a − k , y a − k ) k ≥ is dense in Ω . Theorem 1.1 about topological mixing ofthe A -action on Ω has the following corollary, which will be useful in the proof ofTheorem 1.2. Corollary 3.3. If Γ ⊂ G = SO o ( d, is a Zariski-dense discrete subgroup, thenthere exists a generic couple ( x , y ) ∈ Ω .Proof. By Theorem 1.1, the geodesic frame flow is topologically mixing. Therefore,so is the diagonal flow action of A on Ω . This implies that the transformation( a − , a − ) on Ω is also topologically mixing, hence topologically transitive, i.e.has a dense positive orbit. (cid:3) Existence of a generic couple on the same U -orbit.Lemma 3.4. There exists a generic couple of the form ( x , x u ) , with x ∈ Ω and u ∈ U .Proof. By Corollary 3.3, there exists a generic couple.Let ( y , z ) ∈ ( F H d ) be the lift of a generic couple. Notice that, since the actionsof A and M commute with A , the set of generic couples is invariant under theaction of ( A × M ) × ( A × M ). This means that in Hopf coordinates, being the liftof a generic couple does not depend on the orientation of the frame y + , z + , norof the times t y , t z . Moreover, since being generic is defined as density for negative times, one can also freely change the base-points of y + , z + because the new negativeorbit will be exponentially close to the old one. In short, being the lift of a genericcouple (or not) depends only on the past endpoints ( y − , z − ), or equivalently, is (( M × A ) (cid:110) N − ) -invariant. Obviously, y − (cid:54) = z − since generic couple cannot be onthe diagonal.Up to conjugation by elements of M , we can freely assume that U contains thesubgroup corresponding to following the direction given by the second vector of aframe. Pick a third point ξ ∈ Λ Γ distinct from y − and z − , and choose a frame x + ∈ F Λ Γ based at ξ , whose first vector is tangent to the circle determined by( ξ, y − , z − ). Therefore, the two frames of Hopf coordinates x = ( x + , y − , 0) and( x + , z − , 0) lie in the same U -orbit, thus ( x + , z − , 0) = x u for some u ∈ U . Byconstruction, the couple ( x , x u ) is the lift of a generic couple. (cid:3) Minimality of Γ on F Λ Γ . We recall the following known fact. Proposition 3.5. Let Γ be a Zariski-dense subgroup of SO o ( d, . Then the actionof Γ on F Λ Γ is minimal. In dimension d = 3, this is due to Ferte [11, Corollaire E]. In general, this is againa consequence of Guivarc’h-Raugi [13, Theorem 2], applied with G = SO o ( d, C = M . The set F H d identifies with G × M/ ∼ where ( g, m ) ∼ ( gm (cid:48) an, m (cid:48)− m ).[13, Theorem 2] asserts that the Γ-action on F H d = G × M/ ∼ has a unique minimalset, which is necessarily F Λ Γ .3.6. Density of the orbit of Ω .Proposition 3.6. The U -orbit of Ω is dense in E .Proof. Up to conjugation by an element of M , it is sufficient to prove the propositionin the case where U contains the subgroup corresponding to shifting in the directionof the first vector of the frame x + .Consider the subset E of F Λ Γ defined by ( ξ, R ) ∈ E if ξ ∈ Λ Γ and there existsa sequence ( ξ n ) n ≥ ⊂ Λ Γ \ { ξ } such that ξ n → ξ tangentially to the direction ofthe first vector of R , in the sense that the direction of the geodesic (on the sphere ∂ H n ) from ξ to ξ n converges to the direction of the first vector of R . Clearly, E isa non-empty, Γ-invariant set. By Proposition 3.5, it is dense in F Λ Γ .Let x be a frame in ˜ E , we wish to find a frame arbitrarily close to x , which isin the U -orbit of (cid:101) Ω. Let x = ( x + , x − , t x ) be its Hopf coordinates, by assumption x + ∈ F Λ Γ . Pick ( ξ, R ) ∈ E very close to x + . By definition of E , there exist ξ (cid:48) ∈ Λ,very close to ξ such that the direction ( ξξ (cid:48) ) is close to the first vector of the frame R . We can find a frame y + ∈ F Λ Γ , based at ξ , close to x + , whose first vector istangent to the circle going through ( ξ, ξ (cid:48) , x − ).By construction, the two frames y = ( y + , x − , t x ) and z = ( y + , ξ (cid:48) , t x ) belong tothe same U -orbit; notice that z ∈ (cid:101) Ω, so we have y ∈ (cid:101) Ω U . Since y + and x + arearbitrarily close, so are x and y . (cid:3) Proof of Theorem 1.2. Let O , O (cid:48) ⊂ E be non-empty open sets. We wish toprove that O (cid:48) U ∩ O U (cid:54) = ∅ . By Proposition 3.6, O ∩ Ω U (cid:54) = ∅ , therefore O U ∩ Ω isan open nonempty subset of Ω. Similarly, O (cid:48) U ∩ Ω (cid:54) = ∅ .Let ( x , x u ) a generic couple given by Lemma 3.4. By density, there exists a k ≥ x a − k , x ua − k ) ∈ ( O U ∩ Ω) × ( O (cid:48) U ∩ Ω). But since A normalizes U , x ua − k ∈ x a − k U ⊂ O U . Therefore x ua − k ∈ O (cid:48) U ∩ O U , which is thus non-empty,as required. NIPOTENT FRAME FLOWS 13 ξ (cid:48) x − z ξ x + yx Figure 4. Mesurable dynamics Measures. Let us introduce the measures that will play a role here.The Patterson-Sullivan measure on the limit set is a measure ν on the boundary,whose support is Λ Γ , which is quasi-invariant under the action of Γ, and moreprecisely satisfies for all γ ∈ Γ and ν -almost every ξ ∈ Λ Γ , dγ ∗ νdν ( ξ ) = e − δβ ξ ( o,γo ) . When Γ is convex-cocompact, this measure is proportional to the Hausdorff mea-sure of the limit set [31], it is the intuition to keep in mind here.On the unit tangent bundle T H d , let us define a Γ-invariant measure by d ˜ m BM ( v ) = e δβ v − ( o,v )+ δβ v + ( o,v ) dν ( v − ) dν ( v + ) dt , . By construction, this measure is invariant under the geodesic flow and induceson the quotient on T M the so-called Bowen-Margulis-Sullivan measure m BMS .When finite, it is the unique measure of maximal entropy of the geodesic flow, andis ergodic and mixing.On the frame bundle F H d (resp. FM ), there is a unique way to define a M -invariant lift of the Bowen-Margulis measure, that we will denote by ˜ µ (resp. µ ).We still call it the Bowen-Margulis-Sullivan measure . On FM , this measure hassupport Ω. When it is finite, it is ergodic and mixing [32]. The key point in ourproofs will be that it is mixing, and that it is locally equivalent to the product dν ( x − ) dν ( x + ) dt dm x , where dm x denotes the Haar measure on the fiber of π ( x ),for the fiber bundle FM → T M . This measure is M A -invariant, but not N -(or U )-invariant, nor even quasi-invariant.The Burger-Roblin measure is defined locally on T H d as d ˜ m BR ( v ) = e ( d − β v − ( o,v )+ δβ v + ( o,v ) d L ( v − ) dν ( v + ) dt , where L denotes the Lebesgue measure on the boundary S d − = ∂ H d , invariantunder the stabiliser K (cid:39) SO( d ) of o . We denote its M -invariant extension to F H d (resp. FM ), still called the Burger-Roblin measure , by ˜ λ (resp. λ ). This measureis infinite, A -quasi-invariant, N -invariant. It is N -ergodic as soon as µ is finite.This has been proven by Winter [32]. See also [27] for a short proof that it is theunique N -invariant measure supported in E rad .In some proofs, we will need to use the properties of the conditional measures of µ on the strong stable leaves of the A -orbits, that is the N -orbits. These conditionalmeasures can easily be expressed as dµ x N ( x n ) = e δβ ( x n ) − ( x , x n ) dν (( x n ) − ) , and the quantity e δβ ( x n ) − ( x, x n ) is equivalent to | n | δ when | n | → + ∞ .Observe also that by construction, the measure µ x N has full support in the set { y ∈ x N, y − ∈ Λ Γ } .Another useful fact is that µ x N does not depend really on x in the sense that itcomes from a measure on ∂ H n \ { x + } . In other terms, if m ∈ M and y ∈ x mN ,and z ∈ x N is a frame with π ( z ) = π ( y ), one has dµ x mN ( y ) = dµ x N ( z ).4.2. Dimension properties on the measure ν . Most results in this paper relyon certain dimension properties on ν , allowing to use projection theorems due toMarstrand [18], and explained in the books of Falconer [10] and Mattila [19]. Theseproperties are easier to check in the convex-cocompact case, relatively easy in thegeometrically finite, and more subtle in general, under the sole assumption that µ is finite.Define the dimension of ν , like in [17], bydim ν = infess lim inf r → log ν ( B ( ξ, r ))log r . Denote by g t the geodesic flow on T M . For v ∈ T M , let d ( v, t ) be the distancebetween the base point of g t v and the point Γ .o .Proposition 1.5 in the introduction has been established by Ledrappier [16] when µ is finite. It is also an immediate consequence of Proposition 4.1 and Lemma 4.2below, as it is well known that when the measure µ is finite, it is ergodic andconservative. Proposition 4.1. If µ -almost surely, we have d ( v,t ) t → , then dim ν ≥ δ Γ .If µ is ergodic and conservative, then dim ν ≤ δ Γ .Proof. We will come back to the original proof of the Shadow Lemma, of Sulli-van, and adapt it (the proof, not the statement) to our purpose. The Shadow O o ( B ( x, R )) of the ball B ( x, R ) viewed from o is the set { ξ ∈ ∂ H d , [ oξ ) ∩ B ( x, R ) (cid:54) = ∅} . Denote by ξ ( t ) the point at distance t of o on the geodesic [ oξ ). It is well knownthat for the usual spherical distance, a ball B ( ξ, r ) in the boundary is comparableto a shadow O o ( B ( ξ ( − log r ) , R )). More precisely, there exists a universal constant t > ξ ∈ ∂ H d and 0 < r < 1, one has O o ( B ( ξ ( − log r + t ) , ⊂ B ( ξ, r ) ⊂ O o ( B ( ξ ( − log r − t ) , d ( ξ, t ) the distance d ( ξ ( t ) , Γ .o ). By assumption (in the application thiswill be given by Lemma 4.2), for ν -almost all ξ ∈ ∂ H d and 0 < r < d ( ξ, − log r ± t ) ≤ t + d ( ξ, − log r ) is negligible compared to t = − log r . Let γ ∈ Γ be an element minimizing this distance d ( ξ, t ). It satisfies NIPOTENT FRAME FLOWS 15 obviously | d ( o, γo ) − t | ≤ d ( ξ, t ). Observe that, by a very naive inclusion, using just1 ≤ C + 1) d ( ξ, t ), O o ( B ( ξ ( t − t ) , ⊂ O o ( B ( γ.o, d ( ξ, t − t ))Now, using the Γ-invariance properties of the probability measure ν , and the factthat for η ∈ O o ( B ( γ.o, d ( ξ, t − t )), the quantity |− β η ( o, γo )+ d ( o, γ.o ) − d ( ξ, t ) | is bounded by some universal constant c , one can compute ν ( B ( ξ, r )) ≤ ν ( O o ( B ( γ.o, d ( ξ, t − t ))))= (cid:90) O o ( B ( γ.o, d ( ξ,t − t ))) e − δ Γ β η ( o,γo ) dγ ∗ ν ( η ) ≤ e δ Γ c e − δ Γ d ( o,γo )+2 δ Γ d ( ξ,t ) γ ∗ ν ( O o ( B ( γ.o, d ( ξ, t − t )))) ≤ e δ Γ c e − δ Γ t +2 δ Γ d ( ξ,t ) Recall that t = − log r . Up to some universal constants, we deduce that ν ( B ( ξ, r )) ≤ r δ Γ e δ Γ d ( ξ, log r ) (1)It follows immediately that dim ν ≥ δ Γ .The other inequality follows easily from the classical version of Sullivan’s ShadowLemma, or from the well known fact that δ Γ is the Hausdorff dimension of the radiallimit set, which has full ν -measure. (cid:3) Lemma 4.2. The following assertions are equivalent, and hold when µ is finite. • for µ -a.e. x ∈ F M , one has lim t → + ∞ d ( x , x a t ) t = 0 . • for λ -a.e. x ∈ F M , one has lim t → + ∞ d ( x , x a t ) t = 0 . • for m BM or m BR a.e. v ∈ T M , one has lim t → + ∞ d ( v, g t v ) t = lim t → + ∞ d ( v, t ) t = 0 . • ν -almost surely, lim t → + ∞ d ( ξ ( t ) , Γ .o ) t = lim t → + ∞ d ( ξ, t ) t = 0 . When Γ is geometrically finite, a much better estimate is known thanks to Sul-livan’s logarithm law (see [30], [28], [15, Theorem 5.6]), since the distance growstypically in a logarithmic fashion. However, this may not hold for geometricallyinfinite manifolds with finite µ . In any case, the above sublinear growth is sufficientfor our purposes. Proof. First, observe that all statements are equivalent. Indeed, first, as m BR and m BM differ only by their conditionals on stable leaves, and the limit d ( v, g t v ) /t when t → + ∞ depends only on the stable leaf W ss ( v ), this property holds (or not)equivalently for m BR and m BM .Moreover, as F M is a compact extension of T M , this property holds (or not)equivalently for λ on F M and m BR on T M or µ on F M and m BM on T M . As this limit depends only on the endpoint v + of the geodesic, and not really on v , the product structure of m BR implies that this property holds true equivalentlyfor m BM -a.e. v ∈ T M and ν almost surely on the boundary.Let us prove that all these equivalent properties indeed hold when µ is finite.Let f ( v ) = d ( v, − d ( v, µ -integrable. Thus, S n fn converges a.s. to (cid:82) f dµ , andtherefore d ( v, t ) /t → (cid:82) f dµ , µ -a.s.It is now enough to show that this integral is 0. This would be obvious if weknew that the distance d ( v, 0) is µ -integrable.Divide Ω in annuli K n = { v ∈ T M , d ( π ( v ) , o ) ∈ ( n, n + 1) } , and set B n = T B ( o, n + 1). If a n = µ ( K n ), we have (cid:80) n a n = 1.Observe that (cid:82) f dµ = lim n →∞ (cid:82) B n f dµ .It is enough to find a sequence n k → + ∞ such that these integrals are arbitrarilysmall. Observe that (cid:90) B N f ( x ) dµ ( x ) = (cid:90) g ( B N ) d ( v, dµ − (cid:90) B N d ( v, dµ But now, the symmetric difference between g B N and B N is included in K N ∪ K N +1 .As d ( v, ≤ N + 2 in this union, we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) B N f ( x ) dµ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( N + 2)( a N + a N +1 ) . As (cid:80) a n = 1, there exists a subsequence n k → + ∞ , such that ( n k + 2)( a n k + a n k +1 ) → 0. This proves the lemma. (cid:3) Energy of the measure ν . The t -energy of ν is defined as I t ( ν ) = (cid:90) (cid:90) Λ | ξ − η | t dν ( ξ ) d ν ( η ) . The finiteness of a t -energy is sufficient to get the absolute continuity of the pro-jection of ν on almost every k -plane of dimension k < t . However, a weaker formof finiteness of energy will be sufficient for our purposes, namely Lemma 4.3. For all t < dim ν , there exists an increasing sequence ( A k ) k ≥ suchthat I t ( ν | A k ) < ∞ , and ν ( ∪ k A k ) = 1 .Proof. When t < dim ν , choose some t < t (cid:48) < dim ν . One has, for ν -almost all x ,and r small enough, ν ( B ( x, r )) ≤ Cst.r t (cid:48) . It implies the convergence of the integral (cid:90) Λ | ξ − η | t d ν ( η ) = t (cid:90) ∞ ν ( B ( ξ, r )) r t − dr < ∞ Therefore, the sequence of sets A M = { x ∈ ∂ H n , (cid:82) Λ Γ | ξ − η | t d ν ( η ) ≤ M } is anincreasing sequence whose union has full measure. And of course, I t ( ν | A M ) < ∞ . (cid:3) It is interesting to know when the following stronger assumption of finiteness ofenergy is satisfied. In [23], when dim N = 2 and dim U = 1, Mohammadi and Ohused the following: Lemma 4.4. If Γ is convex-cocompact and δ > d − − dim U then I d − − dim U ( ν ) < ∞ . NIPOTENT FRAME FLOWS 17 Proof. For ξ ∈ Λ Γ , define A k = { η ∈ ∂ H d , | ξ − η | ∈ ]2 − k − , − k ] } , and compute (cid:90) Λ Γ | ξ − η | dim N − dim U d ν ( η ) (cid:39) (cid:88) k ∈ N ∗ k (dim N − dim U ) ν ( A k )Denote by ξ k log 2 the point at distance k log 2 of o on the geodesic ray [ oξ ). AsΓ is convex-cocompact, Ω is compact, so that ξ k log 2 is at bounded distance fromΓ o . Sullivan’ Shadow lemma implies that, up to some multiplicative constant, ν ( A k ) ≤ ν ( B ( ξ, − k )) ≤ Cst. − kδ . We deduce that, up to multiplicative constants, (cid:90) Λ Γ | ξ − η | dim N − dim U d ν ( η ) ≤ (cid:88) k k (dim N − dim U − (1 − ε ) kδ ) If δ > dim N − dim U , for ε > ξ ∈ Λ Γ , so that the integral (cid:82) (cid:82) Λ | ξ − η | dim N − dim U d ν ( η ) d ν ( ξ ) is finite, and theLemma is proven. (cid:3) As mentioned before, the reason we have to be interested in these energies is thefollowing version of Marstrand’s projection theorem, see for example [19, thm 9.7]. Theorem 4.5. Let ν be a finite measure with compact support in R m , such that I t ( ν ) < ∞ , for some < t < m . For all integer k < t , and almost all k -planes P of R m , the orthogonal projection (Π P ) ∗ ν of ν on P is absolutely continuous w.r.t.the k -dimensional Lebesgue measure of P . Moreover, its Radon-Nikodym derivativesatisfies the following inequality (cid:90) G mk (cid:90) P (cid:18) d (Π P ) ∗ ν d L P (cid:19) d L P dσ mk < c.I k ( ν ) where σ mk is the natural measure on the Grassmannian G mk , invariant by isometry,and c some constant depending only on k and m . Conservativity/ Dissipativity of λ . In this section, we aim to prove The-orem 1.4.The measure λ is N -invariant (and N -ergodic), therefore, U -invariant for allunipotent subgroups U < N .It is U -conservative iff for all sets E ⊂ FM with positive measure, and λ -almostall frames x ∈ FM , the integral (cid:82) ∞ E ( x u ) du diverges, where du is the Haarmeasure of U . In other words, it is conservative when it satisfies the conclusion ofPoincar´e recurrence theorem (always true for a finite measure).It is U -dissipative iff for all sets E ⊂ FM with positive finite measure, and λ -almost all frames x ∈ FM , the integral (cid:82) ∞ E ( x u ) du converges.A measure supported by a single orbit can be both ergodic and dissipative.In other cases, ergodicity implies conservativity [1]. Therefore, Theorem 1.3 im-plies that when the Bowen-Margulis-Sullivan measure is finite, and δ Γ > dim N − dim U = d − − dim U , the Burger-Roblin measure λ is U -conservative.In the case δ Γ < dim N − dim U , we prove below (Theorem 4.6) that the measure λ is U -dissipative. Unfortunately, our method does not work in the case δ Γ =dim N − dim U . We refer to works of Dufloux [7] and [8] for the proof that • When µ is finite and Γ Zariski dense, the measure µ is U -dissipative iff δ Γ ≤ dim N − dim U • When moreover Γ is convex-cocompact, if δ Γ = dim N − dim U , then λ is U -conservative. Theorem 4.6. Let Γ be a discrete Zariski dense subgroup of G = SO o ( d, groupand U < G a unipotent subgroup. If δ < d − − dim U , then for all compact sets K ⊂ FM and λ -almost all x ∈ F M the time spent by x U in K is finite. Let d = dim U . Let r > 0. Let N r ⊂ N (resp. U r ⊂ U ) be the closed ball ofradius r > N (resp. in U ). Let K r = K.N r be the r -neighbourhoodof K along the N -direction.Let µ x N be the conditional measure on W ss ( x ) = x N of the Bowen-Margulismeasure. x UN r x x .U Λ Γ x .N Γ . (cid:101) K r Figure 5. Intersection of a U -orbit with the Γ-orbit of a compactset (cid:101) K r Lemma 4.7. For all compact sets K ⊂ Ω, and all x ∈ E , if K r = K.N r , for all r > 0, there exists c = c ( x, r, K ) > (cid:90) U K r ( x u ) du ≤ c µ x N ( x U N r ) . Proof. Let r > 0. First, the map x ∈ Ω (cid:55)→ µ x N ( x N r ) is continuous. It is animmediate consequence of [12, Cor. 1.4], where they establish that µ x N ( ∂ x N r ) = 0for all x ∈ Γ \ G and r > 0. In this reference, they assume at the beginning Γ to beconvex-cocompact, but they use in the proof of corollary 1.4 only the finiteness of µ . The above map is also positive, and therefore bounded away from 0 and + ∞ onany compact set. Let 0 < c r = inf z ∈ K r µ z N ( z N r ) ≤ C r = sup z ∈ K r µ z N ( z N r ) < ∞ .Let us work now on G and not on Γ \ G . Fix a frame x ∈ (cid:101) E ⊂ F H d . Forall y ∈ x U ∩ Γ KN r , choose some z ∈ y N r ∩ Γ K and consider the ball z N r .Choose among them a maximal (countable) family of balls z i N r ⊂ x U N r whichare pairwise disjoint. By maximality, the family of balls z i N r cover x U N r ∩ Γ KN r .We deduce on the one hand (cid:90) U K r ( x u ) du ≤ (cid:88) i µ x N ( z i N r ) ≤ C r | I | . NIPOTENT FRAME FLOWS 19 On the other hand, as the balls z i N r are disjoint, µ x N ( x U N r ) ≥ (cid:88) i µ x N ( z i N r ) ≥ c r | I | . This proves the lemma. (cid:3) To prove Theorem 4.6, it is therefore sufficient to prove the following lemma. Lemma 4.8. Assume that δ Γ < dim N − dim U . Then for all x ∈ E such that d ( x , x a t ) t → when t → + ∞ , we have (cid:90) M µ x mN ( x mU N r ) dm < ∞ . Indeed, Lemma 4.2 ensures that the assumption of Lemma 4.8 is satisfied λ -almost surely. And by Lemma 4.7, its conclusion implies that for λ -a.e. x ∈ E andalmost all m ∈ M , the orbit x mU does not return infinitely often in a compact set K . As λ is by construction the lift to F M of m BR on T M , with the Haar measureof M on the fibers, this implies that for λ -almost all x , the orbit x mU does notreturn infinitely often in a compact set K . This implies the dissipativity of λ w.r.t.the action of U , so that Theorem 4.6 is proved. Proof. Recall first that for n ∈ N not too small, one has dµ x N ( x n ) (cid:39) | n | δ dν (( x n ) − ).We want to estimate the integral (cid:82) M µ x mN ( x mU N r ) dm .First, observe that the measure µ x N on x N does not depend really on the orbit x N , in the sense that it is the lift of a measure on W ss ( π ( x )) through the inverseof the canonical projection y ∈ x N → π ( y ) from x N to W ss ( π ( x )). Therefore,one has µ x mN ( x mU N r ) = µ x N ( x mU m − N r ).Thus, by Fubini Theorem, one can compute : F ( x ) = (cid:90) M µ x mN ( x mU N r ) dm = (cid:90) M µ x N ( x mU m − N r ) dm = (cid:90) M × N m ∈ M,mUm − ∩ nN r (cid:54) = ∅ ( m ) dmdµ x N ( n ) (cid:39) (cid:90) N r dim N − dim U | n | dim U − dim N dµ x N ( n ) (cid:39) (cid:90) N r dim N − dim U | n | dim U − dim N +2 δ dν (( x n ) − )where N t = { n ∈ N ; | n | ≥ t } .The estimate comes from the probability that a point in the sphere of dimension k − r/ | n | -neigborhood of a fixed subsphere of dimension d − 1, see forexample [19, chapter 3].Therefore, we get F ( x ) ≤ (cid:88) l ≥ l (dim U − dim N +2 δ ) ν (( x N l ) − ) Now, observe that ( x N l ) − is comparable to the ball of center x + and radius 2 − l on the boundary. By Inequality (1), we deduce that ν (( x N l ) − ) ≤ − δl e δ Γ d ( x a l log 2 , Γ o ) For all ε > 0, there exists l ≥ 0, such that d ( x a l log 2 ) ≤ εl log 2 for l ≥ l . Thus,up to the l first terms of the series, we get the following upper bound for F ( x ). F ( x ) ≤ l − (cid:88) l =0 · · · + (cid:88) l ≥ l (dim U − dim N + δ ) e δ Γ d ( x a l log 2 , Γ o ) ≤ l − (cid:88) l =0 · · · + (cid:88) l ≥ l l (dim U − dim N + δ + εδ ) Thus, if δ < dim N − dim U , we can choose ε > U − dim N + δ + εδ < 0, and F ( x ) is finite. (cid:3) Remark 4.9. Observe that the above argument, in the case δ + dim U = dim N ,would lead to the fact that (cid:90) M µ x mN ( x mU N r ) dm = ∞ , which is not enough to conclude to the conservativity, that is that almost surely, µ x mN ( x mU N r ) = + ∞ . We refer to the works of Dufloux for a finer analysis in thiscase.4.5. Equivalence of the Bowen-Margulis-Sullivan measure and the Burger-Roblin measure for invariants sets. As claimed in the introduction, we reducethe study of ergodicity of the Burger-Roblin measure λ to the ergodicity of theBowen-Margulis-Sullivan measure µ . The rest of the section is devoted to the proofof the following Proposition: Proposition 4.10. Assume that Γ is Zariski-dense. If µ finite and δ Γ + dim( U ) >d − , then for any U -invariant Borel set E , we have λ ( E ) > if and only if µ ( E ) > . We denote by B the Borel σ -algebra of E , and I U ⊂ B the sub- σ -algebra of U -invariant sets. The first part of the proof of Proposition 4.10 is the following. Lemma 4.11. Assume that Γ is Zarisi-dense in SO o ( d, and that µ is finite. If δ > dim N − dim U and E is a Borel U -invariant set such that µ ( E ) > , then λ ( E ) > .Proof. Let E be a Borel U -invariant set with µ ( E ) > 0. It is sufficient to showthat ˜ λ ( ˜ E ) > 0. Let x = ( x +0 , x − , t x ) be a frame in the support of the (non-zero)measure 1 ˜ E ˜ µ , and F be a small neighbourhood of x . Denote by H ( x + , t x ) thehorosphere passing through the base-point of the frame x . The measure ˜ µ ( ˜ E ∩ F )can be written˜ µ ( ˜ E ∩ F ) = (cid:90) F Λ Γ × R (cid:32)(cid:90) H ( x + ,t ) ˜ E ∩ F ( x + , x − , t ) .gdν ( x − ) (cid:33) d ˜ ν ( x + ) dt x , where g is a positive continuous function, namely the exponential of some Busemannfunctions, and ˜ ν the M -invariant lift of ν to F Λ Γ . The main point is that it is NIPOTENT FRAME FLOWS 21 positive, so for a set J ⊂ F Λ Γ × R of positive ˜ ν ⊗ dt measure, for any ( x + , t x ) ∈ J ,the set E F x + ,t = { x − : ( x + , x − , t x ) ∈ ˜ E ∩ F, } , has positive ν -measure. ∂ H d \ { x + } x .N supp ( µ x .N ) x (cid:101) E ∩ x V x .V (cid:101) E ∩ x N Figure 6. Since similarly,˜ µ ( ˜ E ) = (cid:90) F Λ Γ × R (cid:32)(cid:90) H ( x + ,t x ) g (cid:48) . ˜ E ( x + , x − , t x ) d L ( x − ) (cid:33) d ˜ ν ( x + ) dt x , with g (cid:48) > 0, it is sufficient to show that for a subset of ( x + , t x ) ∈ J of positivemeasure, the set E x + ,t = { x − : ( x + , x − , t x ) ∈ ˜ E, } has positive Lebesgue L -measure.On each horosphere H ( x + , t x ), we wish to use Marstrand’s projection Theorem,and therefore to use an identification of the horosphere with R d − . A naive waywould be to say that H ( x + , t x ) is diffeomorphic to x N , and therefore to N (cid:39) R d − .However, it will be more convenient to use an identification of these horosphereswith N (cid:39) R d − which does not depend on a frame x in π − ( H ( x + , t x )).In order to obtain these convenient coordinates, we fix a smooth section s froma neighbourhood of x +0 to F ∂ H d . If x ∈ F , the horosphere H ( x + , t x ) can be iden-tified (in a non-canonical way) with N the following way: let n ∈ N , we associateto it the base-point of ( s ( x + ) , x − , t x ) n . This way, the identification does dependonly on the M N -orbit of x , that is depends on the horosphere only.For x + ∈ F Λ Γ , define m = m ( x + ) ∈ M by the relation x + = s ( x + ) m . If x ∈ ˜ E , then so does x u = ( s ( x + ) m, x − , t x ) u = ( s ( x + ) , x − , t x ) mu , which has thesame base-point as ( s ( x + ) , x − , t x ) mum − . This means that the set E x + ,t , viewedas a subset of N , is invariant by translations by the subspace mU m − in thesecoordinates. From now on, E x + ,t will always be seen as a subset of N . Let V be the orthogonal complement of U in N , and Π mV m − : N → mV m − be the orthogonal projection onto mV m − . What we saw is that the set E x + ,t is a product of mU m − and Π mV m − ( E x + ,t ). Clearly, it contains the product of mU m − and Π mV m − ( E F x + ,t ), so it is of positive Lebesgue V -measure as soon asΠ mV m − ( E F x + ,t ) has positive Lebesgue measure in mV m − .The strategy is now to use the projection Theorem 4.5 on each horosphere to de-duce that Π mV m − ( E F x + ,t ) is of positive Lebesgue measure for almost every m ∈ M .Unfortunately, we cannot apply it to the measure 1 E F x + ,t ν directly, since the set E F x + ,t depends on the orientation m of the frame x + = s ( x + ) m (and not only onthe Horosphere H ( x + , t x )), so it depends on M .By Lemma 4.3, we can find a subset L ⊂ Λ Γ , such that ν | L has finite dim( N ) − dim( U )-energy, and E F x + ,t ∩ L has positive ν -measure for any ( x + , t ) ∈ J (cid:48) , where J (cid:48) ⊂ J is of positive ˜ ν ⊗ dt - measure.One can moreover assume that for every horosphere H ( x + , t x ) with x ∈ F , L lies in a fixed compact set of N using both identifications of the horosphere with ∂ H d and N . Notice that these identifications are smooth maps, so the finiteness ofthe energy of ν | L does not depend on the model metric space chosen.By Theorem 4.5, applied on each horosphere H ( x + , t ) (cid:39) N = U ⊕ V , the or-thogonal projection (Π mV m − ) ∗ ν | L is m -almost surely absolutely continuous withrespect to the Lebesgue measure on mV m − . But since 1 E F x + ,t ν | L (cid:28) ν | L , we havefor almost every m (Π mV m − ) ∗ (1 E F x + ,t ν | L ) (cid:28) (Π mV m − ) ∗ ν | L (cid:28) L mV m − . This forces the projection set Π mV m − E F x + ,t to be of positive L mV m − -measure m -almost surely, for those m such that ( s ( x + ) m, t ) ∈ J (cid:48) . (cid:3) The second step of the proof is the following. Lemma 4.12. Assume that Γ is Zariski-dense in SO o ( d, , that µ is ergodic andconservative, and δ > dim N − dim U . If E is a Borel U -invariant set such that µ ( E ) = 1 , then λ ( E (cid:52)E ) = 0 .Proof. First, pick some element a ∈ A whose adjoint action has eigenvalue log( λ a ) > n such that ana − = λ a v for all n ∈ N .Replacing E by ∩ k ∈ Z E.a k (another set of full µ -measure), we can freely assumethat E is a -invariant.By Lemma 4.11, we already know that λ ( E ) > 0. As above also, let V ⊂ N bea supplementary of U in N . As λ ( E ) > 0, we know that for λ -almost all x ∈ E ,the set V ( x , E ) = { v ∈ V, x v ∈ E } has positive V -Lebesgue (Haar) measure dv .The Lebesgue density points of V ( x , E ) have full dv -measure. Recall that V t isthe ball of radius t in V .Let (cid:15) ∈ (0 , x ∈ E (not only x ∈ E ) F ε,E ( x ) = sup (cid:26) T > ∀ t ∈ (0 , T ) , (cid:90) V x V t ∩ E dv ≥ (1 − ε ) | V t | (cid:27) , NIPOTENT FRAME FLOWS 23 with the convention that it is zero if no such T exists; it may take the value + ∞ .Observe that F ε,E is a U -invariant map, because E is U -invariant.Since the Lebesgue density points of V ( x , E ) have full dv -measure, then for λ -almost all x ∈ E , and dv -almost all v ∈ V ( x , E ), F ε,E ( x v ) > 0. Moreover, thisstatement stay valid for other U -invariant sets E (cid:48) of positive λ -measure.We claim that for µ -almost every x ∈ E , F ε,E ( x ) > 0. Assuming the contrary, E (cid:48) = F − ε,E (0) ∩ E is a U -invariant set of positive µ -measure, so by Lemma 4.11, it isalso of positive λ -measure. As E (cid:48) ⊂ E , F ε,E (cid:48) ≤ F ε,E , so that the function F ε,E (cid:48) isidentically zero on E (cid:48) . But there exists x ∈ E (cid:48) and v ∈ V ( x , E (cid:48) ) such that x v ∈ E (cid:48) (by definition of V ( x , E (cid:48) )) and F ε,E (cid:48) ( x v ) > 0, by the previous consideration ofLebesgue density points, leading to an absurdity.We will now show that F ε,E is in fact infinite, µ -almost surely. First, the clas-sical commutation relations between A and N (and therefore A and V ⊂ N ) give aV T a − = V λ a T . Observe also that,by a -invariance of E , V ( x a, E ) = { v, x av ∈ E } = { v ∈ V, x ava − = x . ( λ a .v ) ∈ E } = λ − a V ( x , E ) . Therefore, F ε,E ( x a ) = λ a F ε,E ( x ), i.e. it is a function increasing along the dynamicof an ergodic and conservative measure-preserving system. This situation is con-strained by the conservativity of µ . Indeed, assume there exists t < t such that µ ( F − ε,E ( t , t )) > 0. Then for all k large enough (namely s.t. λ ka > t /t ), we have (cid:16) F − ε,E ( t , t ) (cid:17) a k ∩ (cid:16) F − ε,E ( t , t ) (cid:17) = ∅ , in contradiction to the conservativity of µ w.r.t. the action of a .This shows that F ε,E ( x ) = + ∞ for µ -almost all x ∈ E .Define now I E = ∩ j ∈ N ∗ F − /j,E (+ ∞ ). It is a U -invariant set of full µ -measure asa countable intersection of sets of full µ -measure. Therefore λ ( I ) > F ε,E , I consists of the frames x such that V ( x , E ) is of fullmeasure in V , a property that is V -invariant. Hence I is N -invariant of positive λ -measure, so by ergodicity of ( N, λ ), it is of full λ -measure.Unfortunately, we know that E ⊂ I E but I E does not have to be a subset of E . To be able to conclude the proof (i.e. show that λ ( E c ) = 0), we consider thecomplement set E (cid:48) = E c , and assume it to be of positive λ -measure. For any x ∈ I E and v ∈ V , by definition of I E , F ε,E c ( x v ) = 0. So the intersection of I E and E c is of zero measure, and thus λ ( E c ) = 0. (cid:3) Let us now conclude the proof of Proposition 4.10. Let E be a U -invariant set.We already know that µ ( E ) > λ ( E ) > 0. For the other direction, assumethat µ ( E ) = 0, so that µ ( E c ) = 1. The above Lemma applied to E c thereforewould imply E c = E λ -almost surely, so that λ ( E ) = 0. Thus, λ ( E ) > µ ( E ) > Ergodicity of the Bowen-Margulis-Sullivan measure Typical couples for the negative geodesic flow. Let us say that a couple( x , y ) ∈ Ω is typical (for µ ⊗ µ ) if for every compactly supported continuous function f ∈ C c ( E ), the conclusion of the Birkhoff ergodic Theorem holds for the couple( x , y ) in negative discrete time for the action of a , more precisely:lim N → + ∞ N N − (cid:88) k =0 f ( x a − k , y a − k ) = µ ⊗ µ ( f ) . Write T for the set of typical couples, which is a subset of the set of generic couples.Let us explain briefly why this is a set of full µ ⊗ µ -measure. Since the actionof A on (Ω , µ ) is mixing, so is the action of a − . A fortiori, the action of a − on(Ω , µ ) is weak-mixing, so the diagonal action of a − on (Ω , µ ⊗ µ ) is ergodic. Itfollows from the Birkhoff ergodic Theorem applied to a countable dense subset ofthe separable space ( C c ( E ) , (cid:107) . (cid:107) ∞ ) that µ ⊗ µ -almost every couple is typical.As the set of generic couples used in the topological part of the article (seesection 3), the subset of typical couples enjoys the same nice invariance propertiesby (( M × A ) (cid:110) N − ) . That is, ( x , y ) ∈ ( F H d ) being the lift of a typical coupleonly depends on ( x − , y − ) in Hopf coordinates. This follows from the fact that M × A acts isometrically on C c ( E ) and commutes with a − , so T is ( M × A ) -invariant,and the fact that, since elements of C c ( E ) are uniformly continuous, two orbits inthe same strong unstable leaf have the same limit for their ergodic averages.5.2. Plenty of typical couples on the same U -orbit. We will say that there are plenty of typical couples on the same U -orbit if there exists a probability measure η on Ω such that the three following conditions are satisfied:(1) Typical couples are of full η -measure, that is η ( T ) = 1.(2) Let p ( x , y ) = x , p ( x , y ) = y be the coordinates projections. We assumethat, for i = 1 , 2, ( p i ) ∗ η is absolutely continuous with respect to µ . Wedenote by D , D their respective Radon-Nikodym derivatives, so that( p i ) ∗ η = D i µ . We assume moreover that D ∈ L ( µ ).(3) Let η x and η y be the measures on Ω obtained by disintegration of η alongthe maps p i , i = 1 , f ∈ L ( η ), (cid:90) Ω f dη = (cid:90) Ω (cid:18)(cid:90) Ω f ( x , y ) dη x ( y ) (cid:19) dµ ( x ) = (cid:90) Ω (cid:18)(cid:90) Ω f ( x , y ) dη y ( x ) (cid:19) dµ ( y ) . Note that η x (resp η y ) have total mass D ( x ) (resp. D ( y )). Whenever thismakes sense, define the operator Φ which to a function f on Ω associatesthe following function on Ω:Φ( f )( x ) = (cid:90) Ω f ( y ) dη x ( y ) . The condition (3) here is that if f is a bounded, measurable U -invariantfunction, then Φ( f )( x ) = f ( x ) D ( x )for µ -almost every x ∈ Ω. Note that even if f is bounded, Φ( f ) may notbe defined everywhere. NIPOTENT FRAME FLOWS 25 Remark 5.1. Observe that we do not require any invariance of the measure η .Condition (1) replaces the A -invariance, whereas Condition (3) establish a linkbetween the structure of U -orbits and η . Remark 5.2. Let us comment a little bit on condition (3): it is obviously satisfiedif, for example, η x is supported on x U for almost every x , that is, η is supportedon couples of the form ( x , x u ) with u ∈ U . It will be the case for the measures η we will construct in section 5.4 and 5.5 in dimension 3 and higher respectively.A good example of a measure η satisfying (2) and (3) is the following: let ( µ x ) x ∈ Ω be the conditional measures of µ with respect to the σ -algebra of U -invariant sets,and define η as the measure on Ω such that η x = µ x by the above disintegrationalong p . However, its seems difficult to prove directly that it also satisfies (1). Thisexample also highlights that condition (3) is in fact weaker than requiring that η x is supported on x U . Remark 5.3. The condition that the Radon-Nikodym derivatives D i be in L is notrestrictive. Indeed , we will construct a measure η (cid:48) satisfying all above conditionsexcept this L -condition. The Radon-Nikodym derivatives D i are integrable, sothat they are bounded on a set of large measure. We will simply restrict η (cid:48) to thissubset, and normalize it, to get the desired probability measure η .The interest we have in finding plenty of typical couples on the same U -orbit isdue to the following key observation. Lemma 5.4. To prove Theorem 1.3, it is sufficient to prove that there are plenty oftypical couples on the same U -orbit, that is that there exists a probability measure η satisfying (1),(2) and (3).The next section is devoted to the proof of this observation. The idea is thefollowing: suppose g is a bounded U -invariant function. We aim to prove that g is constant µ -almost everywhere. Consider the integral of the ergodic averages forthe function g ⊗ g on Ω with respect to η , J N = (cid:90) Ω N N − (cid:88) k =0 g ⊗ g ( x a − k , y a − k ) dη ( x , y ) . If η is supported only on couples on the same U -orbit, then since g is constant on U -orbits, g ( x a − k ) = g ( y a − k ) for η -almost every ( x , y ), so J N = (cid:90) Ω N N − (cid:88) k =0 g ( x a − k ) dη ( x , y )= (cid:90) Ω N N − (cid:88) k =0 g ( x a − k ) D ( x ) dµ ( x ) , = (cid:90) Ω g ( x ) (cid:32) N N − (cid:88) k =0 D ( x a k ) (cid:33) dµ ( x ) , so J N → (cid:82) Ω g dµ by the Birkhoff ergodic Theorem applied to D . Observe thatProperty (3) is used in the first equality, and Property (2) in the second.For the sake of the argument, assume that g is moreover continuous with com-pact support . Then by Condition (1) on typical couples, since g ⊗ g is continuouswith compact support, the same sequence J N tends to (cid:82) Ω g ⊗ gdµ = ( (cid:82) Ω gdµ ) . Hence g has zero variance, so is constant. Unfortunately, one cannot assume g to be continuous, nor approximate it by continuous functions in L ∞ ( µ ). The reg-ularity Condition (2) that D ∈ L will nevertheless allow us to use continuousapproximations in L ( µ ).5.3. Proof of Lemma 5.4. We first need to collect some facts about the operatorΦ, and its behaviour in relationship with ergodic averages for the negative-timegeodesic flow a − . Lemma 5.5. The operator Φ is a continuous linear operator from L ( µ ) to L ( µ ) . As we will see, Property (2) of the measure η is crucial here. Proof. Let f ∈ L ( µ ), we compute (cid:107) Φ( f ) (cid:107) L ( µ ) = (cid:90) Ω | Φ( f )( x ) | dµ ( x ) ≤ (cid:90) Ω (cid:18)(cid:90) Ω | f ( y ) | dη x ( y ) (cid:19) dµ ( x ) , ≤ (cid:90) Ω | f ( y ) | dη ( x , y ) ≤ (cid:90) Ω | f ( y ) | (cid:18)(cid:90) Ω dη y ( x ) (cid:19) dµ ( y ) , ≤ (cid:90) Ω | f ( y ) | D ( y ) dµ ( y ) ≤ (cid:107) f (cid:107) L ( µ ) (cid:107) D (cid:107) L ( µ ) . (cid:3) Given f, g two functions on Ω, write f ⊗ g for the function f ⊗ g ( x , y ) = f ( x ) g ( y )on Ω . Denote by (cid:104) f, g (cid:105) µ = (cid:82) Ω f.g dµ the usual scalar product on L ( µ ). For f ∈ L ∞ ( µ ) and g ∈ L ( µ ), a simple calculation gives (cid:90) Ω f ⊗ g dη = (cid:104) f, Φ( g ) (cid:105) µ . Let Ψ be the Koopman operator associated to a , that is Ψ( f )( x ) = f ( x a ).The Ergodic average of a tensor product can be written in terms of Φ and Ψ thefollowing way: (cid:90) Ω N N − (cid:88) k =0 f ⊗ g ( x a − k , y a − k ) dη ( x , y ) = 1 N N − (cid:88) k =0 (cid:104) Ψ − k ( f ) , Φ(Ψ − k ( g )) (cid:105) µ , = (cid:104) f, N N − (cid:88) k =0 Ψ k ◦ Φ ◦ Ψ − k ( g ) (cid:105) µ = (cid:104) f, Ξ N ( g ) (cid:105) µ , where Ξ N is the operator Ξ N = N (cid:80) N − k =0 Ψ k ◦ Φ ◦ Ψ − k . Since the Koopman operatoris an isometry from L q ( µ ) to L q ( µ ) for both q = 1 and q = 2, the operator Ξ N from L ( µ ) to L ( µ ) has norm at most (cid:107) Ξ N (cid:107) L → L ≤ (cid:107) Φ (cid:107) L → L . Notice also that if f, g are continuous with compact support, the above ergodicaverage converges toward (cid:104) f, (cid:105) µ (cid:104) g, (cid:105) µ for η -almost every x, y , by Condition (1).By the Lebesgue dominated convergence Theorem, we also have(2) lim N →∞ (cid:104) f, Ξ N ( g ) (cid:105) µ = (cid:104) f, (cid:105) µ (cid:104) g, (cid:105) µ . NIPOTENT FRAME FLOWS 27 Let g be a bounded measurable, U -invariant function. Since Ψ − k ( g ) is alsobounded and U -invariant, by property (3), we haveΦ(Ψ − k ( g ))( x ) = g ( x a − k ) D ( x ) . Therefore, Ξ N ( g )( x ) = g ( x ) (cid:32) N N − (cid:88) k =0 D ( x a k ) (cid:33) . By the Birkhoff L -ergodic Theorem and boundedness of g , it follows that Ξ N ( g )tends to g in L ( µ )-topology.Our aim is to show that g has variance zero. Let ( g n ) n ≥ be a sequence ofuniformly bounded continuous functions with compact support converging to g in L ( µ ) (and hence also in L ( µ )). Let D > (cid:107) g n (cid:107) ∞ ≤ D for all n . For n, N positive integers, we have (cid:104) g, g (cid:105) µ − (cid:104) g, (cid:105) µ = (cid:104) g − g n , g (cid:105) µ + (cid:104) g n , g − Ξ N ( g ) (cid:105) µ + (cid:104) g n , Ξ N ( g − g n ) (cid:105) µ + (cid:0) (cid:104) g n , Ξ N ( g n ) (cid:105) µ − (cid:104) g n , (cid:105) µ (cid:1) + (cid:0) (cid:104) g n , (cid:105) µ − (cid:104) g, (cid:105) µ (cid:1) . Therefore, (cid:12)(cid:12) (cid:104) g, g (cid:105) µ − (cid:104) g, (cid:105) µ (cid:12)(cid:12) ≤(cid:107) g − g n (cid:107) (cid:107) g (cid:107) ∞ + D (cid:107) g − Ξ N ( g ) (cid:107) + D (cid:107) Ξ N (cid:107) L → L (cid:107) g − g n (cid:107) + (cid:12)(cid:12) (cid:104) g n , Ξ N ( g n ) (cid:105) µ − (cid:104) g n , (cid:105) µ (cid:12)(cid:12) + (cid:107) g − g n (cid:107) (cid:107) g + g n (cid:107) . First fix n and let N goes to infinity. By what precedes, Ξ N ( g ) converges to g in L so that the second term vanishes. Since g n is continuous, by (2), the last butone term of the upper bound vanishes. We obtain (cid:12)(cid:12) (cid:104) g, g (cid:105) µ − (cid:104) g, (cid:105) µ (cid:12)(cid:12) ≤ (cid:107) g − g n (cid:107) (cid:107) g (cid:107) ∞ + D (cid:107) Φ (cid:107) L → L (cid:107) g − g n (cid:107) + 2 D (cid:107) g − g n (cid:107) . We now let n go to infinity, and we get (cid:104) g, g (cid:105) µ − (cid:104) g, (cid:105) µ = 0Therefore, g has variance zero, so is constant.5.4. Constructing plenty of typical couples : the dimension 3 case. The candidate to be the measure η , in dimension . First, recall that N is identifiedwith R d − = R . Fix also an isomorphism U (cid:39) R , so that the set U + of positiveelements is well defined.Consider the map (cid:101) R : (cid:101) Ω → (cid:101) Ω defined as follows. The image ( x (cid:48) , y (cid:48) ) of ( x , y )is the unique couple such that x (cid:48) + = x + = y (cid:48) +, x (cid:48)− = x − , y (cid:48)− = y − , t x (cid:48) = t x = t y ,and x + , y + are the unique frames such that there exists u ∈ U + with x (cid:48) u = y (cid:48) .Consider the restriction of this map to couples ( x , y ) inside some fundamentaldomain for the action of Γ on (cid:101) Ω, so that we get a well defined map R : Ω → Ω .Define η as the image η := R ∗ ( µ ⊗ µ ).Observe that condition (1) in 5.2 is automatic, as being typical depends only on x − and y − . Remark 5.2 shows that condition (3) is also automatic. By Remark5.3, we only need to show that its projections ( p ) ∗ η and ( p ) ∗ η are absolutelycontinuous w.r.t. µ . That is the crucial part of the proof. We do it in the nextsections. x − x + y + y − y (cid:48) x (cid:48) x − x y x + y + y − The map R Figure 7. The alignement map R The key assumption will of course be our dimension assumption on δ Γ > dim N − dim U . Then, we will try to follow the classical strategy of Marstrand, Falconer,Mattila. However, a new technical difficulty will appear, because we will need to doradial projections on circles instead of orthogonal projections on lines. The lengthof the proof below is due to this technical obstacle. Projections. First of all, by lemma 4.3, we can restrict the measure ν to some sub-set A ⊂ Λ Γ of measure as close to 1 as we want, with I ( ν | A ) < ∞ . In the sequel,we denote by ν A the measure restricted to A and normalized to be a probabilitymeasure. Fix four disjoint compact subsets X + , X − , Y + , Y − of A ⊂ Λ Γ , each of pos-itive ν -measure, and write ν X + , ν X − , ν Y + , ν Y − for the Patterson measures restrictedto each of these sets, normalized to be probability measures. Therefore, all theirenergies I ( ν X ± ) and I ( ν Y ± ) are finite.In fact, the definition of the measure η will be slightly different than said above.First, ˜ η will be the image by the projection map (cid:101) R defined above of the restrictionof ˜ µ ⊗ ˜ µ to the set of couples ( x , y ) ∈ (cid:101) Ω , such that x ± ∈ X ± and y ± ∈ Y ± , t x ∈ [0 , t y ∈ [0 , η will be defined on Ω as the image of ˜ η .Pick two distinct points outside X + , called ’zero’ and ’one’. For any x + ∈ X + ,we identify ∂ H \ { x + } to the complex plane C by the unique homography, say h x + : ∂ H → C ∪ {∞} , sending x + to + ∞ , zero to 0 and one to 1. We get a welldefined parametrization of angles, as soon as x + is fixed. Remark 5.6. Observe that when x + varies in the compact set X + , as and donot belong to X + , all the quantities defined geometrically (projections, intersectionsof circles, ...) vary analytically in x + . In particular, if x ∈ Ω is a frame, the frame x + in the boundary determines aunique half-circle from x + to x − in ∂ H , which is tangent to the first direction of x + at x + , and therefore, a unique half-line originating from x − in C (cid:39) ∂ H \ { x + } .We use therefore an angular coordinate θ x ∈ [0 , π ) instead of x + .Let (cid:126)u θ be the unit vector e i ( θ + π/ in the complex plane. Define the projection π x + θ in the direction θ from ∂ H \ { x + } to itself as π x + θ ( z ) = z.(cid:126)u θ . Observe that theline R (cid:126)u θ in C , orthogonal to θ , has a canonical parametrization, and a Lebesguemeasure, denoted by (cid:96) x + θ . NIPOTENT FRAME FLOWS 29 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) C (cid:39) ∂ H \ { x + } Y − θ = θ x x − X − π x + θ (cid:126)u θ Figure 8. Angular parameter on C (cid:39) ∂ H \ { x + } Once again, the variations of x + (cid:55)→ π x + θ and x + (cid:55)→ (cid:96) x + θ are as regular as possible.For measures, it means that the Lebesgue measures (cid:96) x + θ are equivalent one to an-other when x + varies, with analytic Radon-Nikodym derivatives in x + in restrictionto any compact set of ∂ H d which does not contain x + .Observe also that when x + varies in X + , the distances d x + induced by thecomplex metric on C (cid:39) ∂ H \ { x + } , when restricted to the compact set X − ∪ Y − ,are uniformly equivalent to the usual metric on ∂ H . In particular, if we denote by I x + the energy of a measure relatively to the distance d x + , there exists a constant c = c ( X + , X − , Y + ) such that for all x + ∈ X + ,1 c I ( ν A ) ≤ I x + ( ν X − ) ≤ cI ( ν A ) and 1 c I ( ν A ) ≤ I x + ( ν Y − ) ≤ cI ( ν A )(3)Rephrasing Marstrand’s projection Theorem in dimension 2, we have: Theorem 5.7. (Falconer, [10, p82] , Mattila [20, th 4.5] ) Assume that I ( ν A ) < ∞ .Then for all fixed x + ∈ X + , and almost all θ ∈ [0 , π ) , the projection ( π x + θ ) ∗ ν Y − (resp. ( π x + θ ) ∗ ν X − ) is absolutely continuous w.r.t (cid:96) x + θ . Moreover, the map H x + defined as H x + : ( θ, ξ ) ∈ [0 , π ) × R (cid:55)→ d ( π x + θ ) ∗ ν Y − d(cid:96) x + θ ( ξ ) belongs to L ([0 , π ) × R ) , and we have (cid:107) H x + (cid:107) L ([0 ,π ) × R ) ≤ CI ( ν A ) , with C auniversal constant which does not depend on x + ∈ X + .In particular, as the variation in x + is analytic and X + compact, the map ( x + , θ, ξ ) → H x + θ ( ξ ) belongs to L ( X + × [0 , π ] × R ) , with L -norm bounded bythe same upper bound CI ( ν A ) .The same result is true when replacing Y − with X − .Proof. Thanks to the comparison (3) between the different notions of energy, wecan replace I x + ( ν X + ) by I ( ν A ), and get the desired result. (cid:3) Hardy-Littlewood Maximal Inequality. Let H x + θ be the map H x + θ : ξ ∈ R .(cid:126)u θ (cid:55)→ d ( π x + θ ) ∗ ν Y − d(cid:96) x + θ ( ξ )Its maximal function is defined as M H x + θ ( t ) = sup ε> ε (cid:90) t + εt − ε d ( π x + θ ) ∗ ν Y − d(cid:96) x + θ ( ξ ) dξ = sup ε> ε ν Y − ( { y ∈ Y − , π x + θ ( y ) ∈ [ t − ε, t + ε ] } ) . The strong maximal inequality of Hardy-Littlewood [14] with p = 2 on R (ofdimension 1) asserts that there exists C = C , independent of θ such that for all θ ∈ [0 , π ), (cid:107) M H x + θ (cid:107) L ( R ) ≤ C , (cid:107) H x + θ (cid:107) L ( R ) We deduce that (cid:107) M H x + (cid:107) L ([0 ,π ) × R ) ≤ (cid:90) π C , (cid:107) H x + θ (cid:107) L ( R ) dθ = C , (cid:107) H x + θ (cid:107) L ([0 ,π ) × R ) < + ∞ The above also holds for the map G x + defined by G x + θ : ξ ∈ R .(cid:126)u θ (cid:55)→ d ( π x + θ ) ∗ ν X − d(cid:96) x + θ ( ξ ) , with the same constants. A geometric inequality. We want to show that the projections ( p i ) ∗ η on Ω areabsolutely continuous w.r.t. µ . We will first prove it for p , and then observe thatfor p , the situation is completely symmetric, when reversing the role of x − and y − .Given a Borel set P = E + × E − × E t × E θ ⊂ X + × X − × [0 , × [0 , π ), observethat( p ) ∗ η ( P ) = ˜ µ ⊗ ˜ µ ( { ( x , y ) ∈ (cid:101) Ω , x + ∈ E + , x − ∈ E − , t x ∈ E t , y − ∈ C x + ( x − , E θ ) } where C x + ( x − , E θ ) is the cone of center x − with angles in E θ in the complex plane C (cid:39) ∂ H \ { x + } .Similarly,( p ) ∗ η ( P ) = ˜ µ ⊗ ˜ µ ( { ( x , y ) ∈ (cid:101) Ω , x + ∈ E + , y − ∈ E − , t x ∈ E t , x − ∈ C x + ( y − , E θ ) } . Lemma 5.8. To prove that ( p ) ∗ η (resp. ( p ) ∗ η ) is absolutely continuous w.r.t. µ ,it is enough to show that there exists a nonnegative measurable map F (resp. F )such that for all rectangles P = E + × E − × E t × E θ ∈ X + × X − × [0 , × M (resp. P = E + × E − × E t × E θ ∈ X + × Y − × [0 , × M ) we have ( p ) ∗ η ( P ) ≤ (cid:90) P F ( x + , x − , θ ) dν X + ( x + ) dν X − ( x − ) dtdθ and ( p ) ∗ η ( P ) ≤ (cid:90) P F ( x + , y − , θ ) dν X + ( x + ) dν Y − ( y − ) dtdθ with F ∈ L ( ν X + × ν X − × [0 , π ]) , and F ∈ L ( ν X + × ν Y − × [0 , π )) Proof. It is clear that µ ( P ) = 0 will imply ( p ) ∗ η ( P ) = 0 for all rectangles. Asthey generate the σ -algebra of (cid:101) Ω ∩ ( X + × X − × [0 , × [0 , π ) it implies that ( p ) ∗ η is absolutely continuous w.r.t. µ . The proof is the same with p . (cid:3) Let us show that such integrable maps F and F exist.In fact, we will prove that for all given x + , F i ( x + , . ) is integrable. And the factthat, as usual, the variation of all involved quantities in x + is analytic will implythat (cid:107) F i ( x + , . ) (cid:107) is integrable also in x + .As said above, for P = E + × E − × E t × E θ we have( p ) ∗ η ( P ) = (cid:90) E + × E − × E t (cid:90) Y − C x + ( x − ,E θ ) ( y − ) dν Y − ( y − ) dν X − ( x − ) dν X + ( x + ) dt NIPOTENT FRAME FLOWS 31 Now, we wish to study the quantity ν Y − ( C x + ( x − , E θ )) in order to prove that, x + being fixed, the radial projection of ν Y − on the circle of directions around x − isabsolutely continuous w.r.t the Lebesgue measure dθ , and control the norm of theRadon-Nikodym derivative, which a priori depends on, and needs to be integrablein the variable x + .It seems now appropriate to use Theorem 5.7 to conclude. Unfortunately, wehave to prove that a radial projection is absolutely continuous, whereas Theorem5.7 deals with orthogonal projection in a certain direction. The Hardy-Littlewoodmaximal L -inequality will allow us to overcome this difficulty.Denote by Θ x + ( x − , y − ) the angle in ∂ H \ { x + } (cid:39) C at x − of the half-line from x − to y − .First, as the distance from X − to Y − is uniformly bounded from below, the cone C x + ( x − , [ θ − ε, θ + ε ]) intersected with Y − is uniformly included in a rectangleof the form { y − ∈ Y − , | π x + θ ( y − ) − π x + θ ( x − ) | ≤ c ε } , for some uniform constantdepending only on the sets X ± and Y ± , and not on ε, x ± , y ± . In particular, thefollowing result holds. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) C = ∂ H \ { x + } x − π x + θ R .(cid:126)u θ C x + ( x − , [ θ − ε, θ + ε ]) C x + ( x − , E θ ) Y − { y − ∈ Y − , | π x + θ ( y − ) − π x + θ ( x − ) | ≤ c ε } Figure 9. Radial versus orthogonal projections of ν Y − Lemma 5.9. There exists a geometric constant c > depending only on the sizesand respective distances of the sets X ± and Y ± , such that ν Y − ( C x + ( x − , [ θ − ε, θ + ε ]) ∩ Y − ) ≤ c εM H x + θ ( π x + θ ( x − )) Conclusion of the argument. The above inequality does not allow directly to con-clude. Let us integrate it in θ , to recover the L -norm of the maximal Hardy-Littlewood function. The first inequality follows from the inclusion [ θ − ε, θ + ε ] ⊂ [ θ − ε, θ + 2 ε ] for θ in the first interval, the second inequality from Lemma 5.9. ν Y − ( C x + ( x − , [ θ − ε, θ + ε ]) ∩ Y − ) ≤ (cid:90) θ + εθ − ε ν Y − ( { y − ∈ Y − , Θ x + ( x − , y − ) ∈ [ θ − ε, θ + 2 ε ] } ) dθ ε ≤ c ε (cid:90) θ + εθ − ε M H x + θ ( π x + θ ( x − )) dθ ε = 2 c (cid:90) θ + εθ − ε M H x + θ ( π x + θ ( x − )) dθ Define F ( x + , x − , θ ) as F ( x + , x − , θ ) = 2 c M H x + θ (cid:16) π x + θ ( x − ) (cid:17) = 2 c sup (cid:15)> ε (cid:90) π x + θ ( x − )+ επ x + θ ( x − ) − ε H x + θ ( t ) dt . The absolute continuity of π x + θ w.r.t (cid:96) θ , the Cauchy-Schwartz inequality and theHardy-Littlewood maximal inequality imply that (cid:107) F ( x + , ., . ) (cid:107) L ([ X − × [0 ,π ]) = 2 c (cid:90) X − (cid:90) π M H x + θ ( π x + θ ( x − )) dν X − ( x − ) dθ = (cid:90) R (cid:90) π M H x + θ ( ξ ) d ( π x + θ ) ∗ ν X − d(cid:96) x + θ ( ξ ) dξdθ ≤ (cid:107) M H x + (cid:107) L ( R × [0 ,π ]) × (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d ( π x + θ ) ∗ ν X − d(cid:96) x + θ ( ξ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R × [0 ,π ]) ≤ C (cid:107) H x + (cid:107) L ( R × [0 ,π ]) × (cid:107) G x + (cid:107) L ( R × [0 ,π ]) which is, by Projection Theorem 4.5, bounded from above by C I ( ν A ) < ∞ .The uniformity of the bound in x + ∈ X + allows to integrate once again theabove quantities and deduce that F ∈ L ( X − × X + × [0 , π ]).5.5. The higher dimensional case. In higher dimension, the strategy of theproof is similar. We want to build a measure η on Ω which gives positive measureto plenty of couples on the same U -orbit .We will build η from the measure µ ⊗ µ , to obtain a measure defined on (a subsetof) { ( x , y ) ∈ Ω , x U = y U } , which gives full measure to typical couples ( x , y )(whose negative orbit satisfies Birkhoff ergodic theorem for the diagonal action of a − , and whose projections ( p ) ∗ η and ( p ) ∗ η on Ω are absolutely continuous w.r.t µ . Contrarily to the dimension 3 case, we will not define any ”alignment map”.Indeed, given a typical couple ( x , y ), one can begin as in dimension 3, and try tofind a frame x (cid:48) ∈ x M and a frame y (cid:48) ∈ x (cid:48) U (or in other words y (cid:48) U = x (cid:48) U ), so thatin particular y (cid:48) + = x (cid:48) + = x + , with the same past as y (that is, y (cid:48)− = y − ). However,there is no canonical choice of such x (cid:48) , y (cid:48) , due to the fact that the dimension and/orthe codimension of U in N will be greater than one.Therefore, we will directly define the new measure η , by a kind of averagingprocedure of all good choices of couples ( x (cid:48) , y (cid:48) ). NIPOTENT FRAME FLOWS 33 Identify the horosphere x N M = x M N in T H d with a d − x (cid:48) and y (cid:48) have their first vectorson x N M , that x (cid:48) belongs to the fiber x M of the vector π ( x ), and y (cid:48)− = y − ,so that y (cid:48) belongs to the fiber y (cid:48) M (with an abuse of notation, as y (cid:48) is not welldefined) of the well defined vector v y = ( y − , x + , t x ) of x M N .These vectors x M and y (cid:48) M are well defined, so that the line from x M to y (cid:48) M in the affine space x N M is also well defined.Now, given any two frames x (cid:48) and y (cid:48) in the respective fibers of x M and y (cid:48) M ,such that x (cid:48) U = y (cid:48) U , the k -dimensional oriented linear space P = x (cid:48) U M containsthe line from x M to y (cid:48) M . The set of such P can be identified with SO ( d − / ( SO ( k − × SO ( d − k − P using the SO ( d − P , the set of frames x (cid:48) such that the direction of the affinesubspace x (cid:48) U M is P can be identified with SO ( k ) × SO ( d − k − x (cid:48) randomly using the Haar measure of this group. This determines the element u ∈ U such that x (cid:48) uM = y (cid:48) M , so it determines y (cid:48) = x (cid:48) u completely.As in dimension 3, the non-trivial part is to show that the measure obtainedby this construction has absolutely continuous marginals. We first describe moreprecisely the construction to fix notations.5.5.1. Restriction of the support of µ ⊗ µ . Recall that the lift ˜ µ of the measure µ on (cid:101) Ω can be written locally as d ˜ µ ( x ) = dν ( x − ) dν ( x + ) dt x dm , where dm denotes the Haar measure on the fiber x M over π ( x ). Remember that aframe x with first vector π ( x ) induces (by parallel transport until infinity) a frameat infinity in T x + ∂ H d , or T x − ∂ H d , so that dm can also be seen as the Haar measureon the set of frames based at x − inside T x − ∂ H d .As in dimension 3, consider a subset A ⊂ Λ Γ of positive ν -measure such that I ( ν | A ) < ∞ . Choose four compact sets X ± , Y ± inside A , pairwise disjoint, andrestrict ˜ µ ⊗ ˜ µ to the couples ( x , y ) ∈ (cid:101) Ω such that x ± ∈ X ± and y ± ∈ Y ± , and t x , t y ∈ [0 , Coordinates on ∂ H d . For the purpose of contructing η , it will be convenientto have a family of identifications of horospheres, or here the complement of a point x + in ∂ H d , with the vector space R d − . Let ( e i ) ≤ i ≤ d − be the canonical basis of R d − . Choose three different points x +0 ∈ X + , x − ∈ X − and y ∈ Y − , in thesupport of ν | X + , ν | X − and ν | Y − respectively.Now we want to get a unique homography h x + from ∂ H d \ { x + } to R d − sending x to 0, y to e , and x + to infinity, with a smooth dependence in x + .To do so, choose successively d − q , ... q d − in ∂ H d , in such away that, uniformly in x + ∈ X + , none of the points x + , x , y , q , . . . , q d − belongsto a circle containing three other points. Now, it is elementary to check that thereis a unique conformal map h x + sending x + to infinity, x to 0, y to e , q insidethe half-plane R .e + R + e , q inside the half-space R e + R e + R + e , and so onup to q d − . This is the desired map.Up to decreasing the size of X + , X − and Y − using neighbourhoods of x +0 , x − , y − respectively, we can moreover assume that for all these conformal maps uniformly in x + ∈ X + , x − ∈ X − , y − ∈ Y − , the first coordinate of the vector −−−−−−−−−−−−→ h x + ( x − ) h x + ( y − )belongs to [ , h x + on ∂ H d .5.5.3. A nice bundle. We will construct a measure ˜ η on the set S η = { ( x , y ) ∈ Ω , : x + = y + ∈ X + , x − ∈ X − , y − ∈ Y − , x U = y U } , and prove that it satisfies assumptions (1),(2),(3) of Lemma 5.4, so that Theorem1.3 follows. Observe that this space S η is a fiber bundle over some subset P ⊂ X + × X − × Y − × G d − k , whose projection is simply( x , y ) ∈ S η → ( x + , x − , y − , V ect ( x , . . . , x k )) ∈ P , where V ect ( x , . . . , x k ) is the oriented k -linear space spanned by the k first vectorsof the frame x + at infinity with orientation x ∧ ... ∧ x k , or equivalently the k -planespanned by these k vectors viewed around x − at infinity, i.e. inside R d − identifiedwith ∂ H d \ { x + } using the map h x + .Moreover, observe that it is a principal bundle, whose fibers are isomorphicto SO ( k ) × SO ( d − − k ) × A . Indeed, given a couple ( x , y ) in the fiber of( x + , x − , y − , P ), after maybe let A act diagonally so that both couples are basedon the horosphere passing through the origin o ∈ H d , any other couple differs from( x , y ) only by changing ( x , . . . , x k ) into another orthonormal basis of V ect ( x , . . . , x k ),and ( x k +1 , . . . , x d − ) into another orthonormal basis of V ect ( x k +1 , . . . , x d − ), pre-serving the orientation.5.5.4. Defining the measure. Given x + ∈ X + , we first define a measure ¯ η x + sup-ported on the set P x + = { ( x − , y − , P ) : x − ∈ X − , y − ∈ Y − , P ∈ G d − k , s.t. −−−−−−−−−−−−→ h x + ( x − ) h x + ( y − ) ∈ P } . (a subset of X − × Y − × G d − k ) as follows.Observe that, thanks to our choice of coordinates, the vector −−−−−−−−−−−−→ h x + ( x − ) h x + ( y − )has always a nonzero coordinate along e . Therefore, any k -plane P containing −−−−−−−−−−−−→ h x + ( x − ) h x + ( y − ) is uniquely determined by its k − P ∩ e ⊥ with e ⊥ .Thus, we have a well defined measure on P x + : d ¯ η x + ( x − , y − , P ) = dν X − ( x − ) dν Y − ( y − ) dσ d − k − ( P ∩ e ⊥ ) , where σ d − k − is the SO ( d − k − e ⊥ .Now, P is a bundle over X + with fibers P x + . Define ¯ η on P as the measurewhich disintegrates as ν X + on the basis X + and ¯ η x + in the fibers.Pick (cid:15) small enough, and lift ¯ η to (cid:101) η on (cid:101) Ω , or more precisely on its subset (cid:101) S η = { ( x , y ) ∈ (cid:101) Ω , x U = y U, t x = t y ∈ [0 , (cid:15) ] } by endowing the fibers with the Haar measure of SO ( k ) × SO ( d − − k ) times theuniform probability measure on the interval [0 , (cid:15) ]. NIPOTENT FRAME FLOWS 35 If X ± , Y ± and (cid:15) are small enough, we can assume that the support of (cid:101) η isincluded inside the product of two single fundamental domains of the action of Γon SO o ( d, η on the quotient.By construction, it is supported on couples ( x , y ) in the same U -orbit, and as indimension 3, it gives full measure to couples ( x , y ) which are typical in the past,because this property of being typical depends only on x − , y − , and ν | X − ⊗ ν | Y − gives full measure to the pairs ( x − , y − ) which are negative endpoints of typicalcouples ( x , y ).The main point to check is that ( p ) ∗ η and ( p ) ∗ η are absolutely continuousw.r.t. µ .5.5.5. Absolute continuity. Let us reduce the abolute continuity of ( p i ) ∗ η to anotherabsolute continuity property, by a succession of elementary observations.First, to prove that ( p ) ∗ η and ( p ) ∗ η are absolutely continuous w.r.t. µ , it issufficient to prove that (˜ p ) ∗ (cid:101) η and (˜ p ) ∗ (cid:101) η , where ˜ p i : (cid:101) Ω → (cid:101) Ω are the coordinatesmaps, are both absolutely continuous with respect to ˜ µ .Both measures are defined on the compact set T = { x ∈ (cid:101) Ω : t x ∈ [0 , (cid:15) ] , x + ∈ X + , x − ∈ ( X − ∪ Y − ) } . This set T is fibered over X + × ( X − ∪ Y − ) × G d − k , with projection map x → ( x + , x − , x M U ) and fiber isomorphic to SO ( k ) × SO ( d − k − × A . (cid:101) S η ⊂ (cid:101) Ω (cid:15) (cid:15) ˜ p i (cid:47) (cid:47) T ⊂ (cid:101) Ω (cid:15) (cid:15) P ⊂ X + × X − × Y − × G d − k (cid:15) (cid:15) ¯ p i (cid:47) (cid:47) X + × ( X − ∪ Y − ) × G d − k (cid:15) (cid:15) X + X + On the upper left part of this diagram, observe that the measure ˜ η disintegratesover P , with the Haar measure of SO ( k ) × SO ( d − − k ) × A in the fibers, and ¯ η on P .Similarly, on the upper right of the diagram, the measure ˜ µ restricted to T disintegrates over X + × ( X − ∪ Y − ) × G d − k , with measure ν X + ⊗ ν X − ∪ Y − × σ d − k on the basis, and Haar measure of SO ( k ) × SO ( d − − k ) × A in the fibers.Therefore, to prove that (˜ p i ) ∗ ˜ η is absolutely continuous w.r.t. ˜ µ , it is enough toprove that (¯ p i ) ∗ ¯ η is absolutely continuous w.r.t. ν X + ⊗ ν X − ∪ Y − × σ d − k .Look at the lower part of the diagramm now. The measure ¯ η itself disintegratesover X + , with ν X + on the base and ¯ η x + on each fiber P x + , whereas the measure(¯ p i ) ∗ ¯ η disintegrates also over ν X + , with measure ν X − ∪ Y − × σ d − k on each fiber.Thus, it is in fact enough to prove that for ν X + -almost every x + , the image ofthe measure ¯ η x + under the natural projection map P x + → { x + } × X − ∪ Y − × G d − k is absolutely continuous w.r.t. ν X − ∪ Y − ⊗ σ d − k . The precise statement that we will prove is Lemma 5.10. By the above discus-sion, it implies that ( p i ) ∗ η is absolutely continuous w.r.t. µ , and therefore, as indimension 3, Theorem 1.3 follows from Lemma 5.4.5.5.6. Absolute continuity of conditional measures. We discuss now the absolutecontinuity of the marginals laws of ¯ η x + .In order to do so, it is necessary to say a few words about the distance on theGrassmannian manifolds of oriented subspaces that we shall use. As we are onlyinterested in the local properties of the distance, we will (abusively) define it onlyon the Grassmannian manifold of unoriented subspaces.If P is a l -dimensional subspace of a Euclidean space of dimension n , we write Π P for the orthogonal projection on P . If P, P (cid:48) ∈ G nl are two l -dimensional subspaces,a distance between P and P (cid:48) can be defined as the operator norm of Π P − Π P (cid:48) (which is also the operator norm of Π P ⊥ − Π ( P (cid:48) ) ⊥ ).We will use the following facts.(1) The above distance is Lipschitz-equivalent to any Riemannian metric on G nl ,and σ nl is a smooth measure. In particular, up to multiplicative constants,the measure of a ball of sufficiently small radius r around a point P is σ nl ( B G nl ( P, r )) (cid:39) r l ( n − l ) . (2) Identify e ⊥ with R d − . Define( G d − k ) (cid:48) = { P ∈ G d − k : P (cid:54)⊂ e ⊥ } . The map P ∈ ( G d − k ) (cid:48) (cid:55)→ P ∩ e ⊥ ∈ G d − k − is well-defined and smooth, so thatits restriction to any compact set is Lipschitz.(3) Let P, P be two k -dimensional subspaces of R d − . If v ∈ P , (cid:107) v (cid:107) ≤ d G d − k ( P, P ) ≤ r , then (cid:107) Π P ⊥ ( v ) (cid:107) ≤ r. Lemma 5.10. There exist two functions F x + , ∈ L ( ν X − ⊗ σ d − k ) , F x + , ∈ L ( ν Y − ⊗ σ d − k ) such that for any E ⊂ ( X − ∪ Y − ) , any ball B = B ( P , r ) ⊂ G d − k of suffi-ciently small radius r around some P ∈ G d − k , and any x + ∈ X + , ¯ η x + ( { ( x − , y − , P ) ∈ P x + : ( x − , P ) ∈ E × B } ) ≤ (cid:90) E × B F x + , dν X − ⊗ σ d − k , and ¯ η x + ( { ( x − , y − , P ) ∈ P x + : ( y − , P ) ∈ E × B } ) ≤ (cid:90) E × B F x + , dν Y − ⊗ σ d − k . Moreover, the L -norms of F x + ,i are uniformly bounded on X + .Proof. We prove only the second inequality, the first one is similar and only ex-changes the roles of x − and y − in the following.First choose some P ∈ B G d − k ( P , r ). If ( x − , y − , P ) ∈ P x + with P ∈ B G d − k ( P , r ),then, provided r is small enough, both P and P are in a fixed compact subset of( G dk ) (cid:48) . This implies that for some fixed c > Q = P ∩ e ⊥ ∈ B G d − k − ( P ∩ e ⊥ , c r ) . We also have d P ⊥ (Π P ⊥ ( x − ) , Π P ⊥ ( y − )) ≤ r. NIPOTENT FRAME FLOWS 37 Thus we have the inequalities¯ η x + ( { ( x − , y − , P ) ∈ P x + : ( y − , P ) ∈ E × B G d − k ( P , r ) } )= (cid:90) B G d − k ( P , r ) ( Q ⊕ −−−−−−−−−−−−→ h x + ( x − ) h x + ( y − )) dν X − ( x − ) dν Y − ( y − ) dσ d − k − ( Q ) , ≤ σ d − k − ( B G d − k − ( P ∩ e ⊥ , c r )) (cid:90) E (cid:90) X − B (Π P ⊥ ( y ) , r ) (Π P ⊥ ( x )) dν X − ( x − ) dν Y − ( y − ) , ≤ σ d − k − ( B G d − k − ( P ∩ e ⊥ , c r )) (cid:90) E (Π P ⊥ ) ∗ ν X − ( B (Π P ⊥ ( x ) , r )) dν Y − ( y − ) ≤ σ d − k − ( B G d − k − ( P ∩ e ⊥ , c r )) (cid:90) E (6 r ) d − k − M H x + ,P (Π P ⊥ ( y )) dν Y − ( y − ) , where M H x + ,P is the maximal function M H x + ,P ( v ) = sup ρ> ρ − ( d − k − (cid:90) B P ⊥ ( v,ρ ) d (Π P ⊥ ◦ h x + ) ∗ ν X − dw ( w ) dw . We now integrate this inequality over P ∈ B G d − k ( P , r ) using the uniform mea-sure and the fact that B G d − k ( P , r ) ⊂ B G d − k ( P , r ) . We obtain¯ η x + ( { ( x − , y − , P ) ∈ P x + : ( y − , P ) ∈ E × B G d − k ( P , r ) } ) ≤ (cid:90) B G d − k ( P ,r )) ¯ η x + ( { ( x − , y − , P ) ∈ P x + : ( y − , P ) ∈ E × B G d − k ( P , r ) } dσ d − k ( P ) σ d − k ( B G d − k ( P , r )) ≤ (cid:90) E × B G d − k ( P ,r )) σ d − k − ( B G d − k − ( P ∩ e ⊥ , c r ))(6 r ) d − k − σ d − k ( B G d − k ( P , r )) M H x + ,P (Π P ⊥ ( y − )) dν | Y − ( y − ) dσ d − k ( P ) . Now, the ratio σ d − k − ( B G d − k − ( P ∩ e ⊥ , c r ))(6 r ) d − k − σ d − k ( B G d − k ( P , r )) , is bounded by a uniform constant c > 0, since the dimension of the Grassmannianmanifolds G nr is r ( n − r ), so the above ratio is comparable, up to multiplicativeconstants, with r ( k − d − k − × r d − k − r k ( d − k − = 1.This proves an inequality of the desired form with the function F x + , ( y − , P ) = c M H x + ,P (Π P ⊥ ( h x + ( y − ))) . We still have to show that this function is in L ( ν Y − ⊗ σ d − k ). Let us compute itsnorm N = (cid:90) Y − ×G d − k − M H x + ,P (Π P ⊥ ( h x + ( y − ))) dν Y − ( y − ) dσ d − k ( P )= (cid:90) G d − k − (cid:18)(cid:90) P ⊥ M H x + ,P ( v ) d (Π P ⊥ ◦ h x + ) ∗ ν | Y − ( v ) (cid:19) dσ d − k ( P )= (cid:90) G d − k − (cid:18)(cid:90) P ⊥ M H x + ,P ( v ) d (Π P ⊥ ◦ h x + ) ∗ ν | Y − dv ( v ) dv (cid:19) dσ d − k ( P ) . By [19, Theorem 9.7], the two Radon-Nikodym derivatives d (Π P ⊥ ◦ h x + ) ∗ ν | Y − dv , d (Π P ⊥ ◦ h x + ) ∗ ν | X − dv , have the square of their L -norms bounded by a constant times the respectiveenergies I d − − k (( h x + ) ∗ ν | Y − ) , I d − − k (( h x + ) ∗ ν | X − ) . By the Hardy-Littlewood inequality [19, Theorem 2.19], this is also true for theirmaximal functions, with a different constant. By the choices of X + , X − , Y − and h x + , the family of maps ( h x + ) x + ∈ X + is uniformly bilispchitz when restricted to thecompact set X − ∪ Y − . 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