We introduce an Anosov action on the bundle of Weyl chambers of a riemannian symmetric space of non-compact type, which for rank one spaces coincides with the geodesic flo

Hof, Hans-Christoph Im

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Ergod. Th. & Dynam. Sys. (1985), 5, 587-593Printed in Great BritainAn Anosov action on the bundle ofWeyl chambersHANS-CHRISTOPH IM HOFMathematisches Institut der Universitdt Basel,Rheinsprung 21, CH-4051 Basel, Switzerland(Received 15 October 1984 and revised 1 March 1985)Abstract. We introduce an Anosov action on the bundle of Weyl chambers of ariemannian symmetric space of non-compact type, which for rank one spacescoincides with the geodesic flow.IntroductionThe geodesic flow of a riemmanian symmetric space of non-compact type is anAnosov flow if and only if the rank of the space is one. In this note we shall introducean action, which for spaces of rank one coincides with the geodesic flow, but whichis Anosov for all riemannian symmetric spaces of non-compact type. The basic ideain defining this action is to view geodesies not as particular curves of the space, butas totally geodesic flat subspaces. So we are led to consider flats, i.e. maximal totallygeodesic flat subspaces, and Weyl chambers as particular subsets of flats. The bundleof all Weyl chambers will be the phase space of the action. The geodesic flow isgiven by moving geodesic rays along their supporting geodesies. Similarly, the actionto be denned here consists of parallel translating Weyl chambers within theirsupporting flats. If the rank of the space is one, then flats are geodesies, Weylchambers are geodesic rays, and our action coincides with the geodesic flow. Forspaces of rank higher than one, i.e. spaces with higher-dimensional flats, this actionis no longer a flow, but an action of a higher-dimensional abelian Lie group.Let us recall the definition of an Anosov action (cf. [5]; for the ergodicity ofAnosov actions, see [8]). An action <p of a Lie group L on a riemannian manifoldM is said to be Anosov if the orbits under <p foliate M, and if there exists at leastone Anosov element in L, i.e. an element a s L such that the tangent bundle T(M)admits a d<pa- invariant splittinginto three sub-bundles: the central bundle Ec tangent to the orbits under <p, thestrong stable bundle Es, on which the differential d<pa contracts more than on Ec,and the strong unstable bundle E", on which d<pa expands more than on Ec.Given such a splitting, the stable and unstable bundles are defined to be the sumsE'®EC and E"®EC, respectively. Finally, the foliations of T(M) tangent to Es,Eu, ES@EC, and EU®EC are called respectively the strong stable, strong unstable,stable, and unstable foliation induced by <pa.at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700003175Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 11:03:40, subject to the Cambridge Core terms of use, available588 H.-C. Im HofThe first part of this note is purely group-theoretical. There we shall define ouraction and prove its Anosov character. Geometrical interpretations will be given inthe second part, where we shall introduce the bundle of Weyl chambers and relateour action to the geodesic flow. Finally the stable and unstable foliations will findan interpretation in terms of horocycles.The content of this note is part of the author's Habilitationsschrift [6]. I wish tothank Ernst Ruh and Mikhael Gromov for their encouragement and criticism. Forfinancial support I am indebted to the Sonderforschungsbereich 'TheoretischeMathematik' at the University of Bonn, the Stiftung Volkswagenwerk, and the IHESin Bures-sur-Yvette, France.1. The action of A on G/MHere we state and prove our main result in group-theoretical terms. Let G be a realsemisimple Lie group of non-compact type and finite centre. A maximal compactsubgroup K <= G gives rise to a Cartan decomposition p = A® ji, where g and Adenote the respective Lie algebras of G and K, and Ji is the orthogonal complementto A with respect to the Killing form on g. Let a <= ji be a maximal abelian subalgebraand A = exp a<^ G the corresponding subgroup. By log: A -> a we shall denote theinverse of exp: a -» A. The pair (#, a) gives rise to a root system A c Horn {a, U) anda root space decompositionAn element a e A is called regular if A (log a) ^ 0 for all k e A, otherwise it is calledsingular. Now consider M = ZK(A), the centralizer of A in K. By definingcp(a, gM) = (pa(gM) = gaM we obtain an action <p of A on the homogeneous spaceG/M.THEOREM. The action <p:Ax G/ M -* G/ M defined by <p(a, gM) = gaM is an Anosovaction.Proof. The orbits under <p coincide with the fibres of the natural projection G/M-*G/ MA, therefore they are the leaves of a foliation. Now we shall prove that allregular elements of A are Anosov elements for <p. At first we fix an arbitrary a e A.We have to compute the differential of<pa:G/M->G/Mat a point gM e G/M. Let m denote the Lie algebra of M. Since all tangent spacesof G/M are identified with g/m. by left translation, it suffices to computeLet Ra:G->G denote right translation by a. The projection of Ra onto G/M isjust <pa. The differential of Ra is Ad (a"1):^-*^, and this in turn projects onto p/m.Let us denote this projection again by Ad (a"1), then we haved<pa= Ad (a^y.p/m^p/m.In order to compute Ad(a~') we use the root space decomposition of # with respectat https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700003175Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 11:03:40, subject to the Cambridge Core terms of use, availableAn Anosov action 589to a. Observing ^0 = *»©<*, we haveY. 9K,AeAthereforep pAeAAccording to this isomorphism an element gep/m will be written asf = &+ I L, (1)AeAwith £0 e a and £A e ^ v The equationAd (a""1) = exp (ad (-log a))in GL {p) impliesAd(a-1)(£) = £0+ I «r*<to«">& (2)AeAfor £ as given by (1).Assume now that a is regular. Then A splits into the two subsetsA+ = {AeA; A(k>ga)>0}, A~ = {A e A; A log a)<0}.Defining«+= I #X> *~= I ^A,AeA AeAwe obtainp/m = a®n+@n~. (3)We claim that this decomposition, extended to all tangent spaces of G/M by lefttranslation, induces the desired splitting of T(G/M). Indeed, a is tangent to theorbits of <p, and by (2) we getfor f 6 a, f = IA6A+ ^ G «+, I = IAEA- IA G *~, respectively. Thus (3) is invariantunder d<pa.In order to establish the necessary estimates we need a metric on G/M. Invertingthe sign of the Killing form on I we get a positive definite scalar product on g whichwe restrict onto p/m = a®Y.),eAp\- By left translation this induces a riemannianmetric on G/M. Since root spaces are orthogonal to each other, we haveII£| |2=I|£>H2+I II6J2,AeAfor £ as given by (1). Therefore our expressions for d(pa(i;) imply\\d<pM)\\ = M\\ forfe*;where k - min {A(log a); A € A+} is a positive constant depending only on a. Dat https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700003175Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 11:03:40, subject to the Cambridge Core terms of use, available590 H.-C. Im HofRemark. The same arguments applied to a singular element aeA lead to thedecompositionA(]oga)=0 / A(loga)>0 A(logfl)<0which exhibits a component transverse to a, where d<pa is neither contractingnor expanding. Therefore singular elements of A are not Anosov elements for theaction <p.2. Geometric interpretationsHere we give geometric interpretations of the action <p: A x G/ M -» G/ M and ofthe stable and unstable foliations induced by this action. Let X be a riemanniansymmetric space of non-compact type. The maximal totally geodesic flat subspacesof X are called flats. The common dimension of all flats of X is called the rank ofX. Each geodesic is contained in at least one flat. A geodesic is called regular(singular), if it is contained in exactly one flat (in more than one flat). We fix apoint x € X and a flat F c X with xe F. The singular geodesies in F through x, arecontained in the union of finitely many hyperplanes in F through x, and conversely,any geodesic lying in one of these hyperplanes is singular. The connected componentsof F after removing all singular geodesies through x are called the Weyl chambersof F based at x. Admitting all possible flats Fa X and all possible base pointsx e F, we obtain the set CX of all Weyl chambers of X. Let n: CX -* X denote themap which assigns to each Weyl chamber its base point. Following Mostow ([7,§ 4]) we call two Weyl chambers asymptotic, if each one lies in a tubular neighbour-hood of the other. The set of classes of asymptotic Weyl chambers is called theboundary of X. Let T denote the map which assigns to each Weyl chamber itsasymptoticity class.A flat with a distinguished class of asymptotic Weyl chambers is called an orientedflat. Let p denote the map which assigns to a Weyl chamber c its supporting flattogether with the orientation induced by c.Finally we shall sketch a geometric definition of horocycles (for more details see[6]). Each geodesic ray in X determines a horosphere passing through the initialpoint of the ray and orthogonal to it (cf. [1, § 3]). For a fixed Weyl chamber cbased at x, the horocycle associated with c is defined as the intersection of allhorospheres determined by the (regular) geodesic rays emanating from x andcontained in c. This intersection is a submanifold whose codimension equals therank of X. Let a denote the map which assigns to each Weyl chamber its associatedhorocycle.Now we shall translate the geometrical picture introduced so far into the group-theoretical language. Let G be the connected component of the isometry group ofX; it is a real semisimple Lie group of non-compact type and trivial centre. Choosea point x0 e X and let K be the isotropy group of x0. Then K is a maximal compactsubgroup of G, and X is identified with the homogeneous space G/ K. Furthermore,the choice of x0 gives rise to a Cartan decomposition g = A®ji, where the vectorpart /i is identified with the tangent space of X at x0.at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700003175Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 11:03:40, subject to the Cambridge Core terms of use, availableAn Anosov action 591Now consider a flat F through x0. Its tangent space at x0 is a maximal abeliansubalgebra a <=• jt, and conversely, each such subalgebra corresponds to a flat throughx0. The group K operates transitively on the set of all flats through x0 (cf. [3, ChapterV, Lemma 6.3]).We now fix a flat Fo through x0. Let a denote the corresponding maximal abeliansubalgebra of yd, and set A = expa<^G. The group A acts on Fo as its group oftranslations. The root system A with respect to the pair (#, a) defines Weyl chambersin a and, via the exponential map, in A. These correspond to the Weyl chambersin Fo based at x0. The Weyl group W of X is defined as the quotient M'/ M, whereM = ZK{A) is the centralizer and M' = NK(A) the normalizer of A in K. It actssimply transitively on the set of Weyl chambers in Fo based at x0 (for spaces ofcompact type, see [3, Chapter VII, Theorem 2.12]). Let c0 be one of the Weylchambers in Fo based at x0. The choice of c0 determines a set A+ <= A of positiveroots. Define « = XAeA+#A ana" N = exp«<= G. Finally define P = MAN.The Weyl chamber c0 and the groups derived from this choice shall be fixed forthe rest of this paper.PROPOSITION. Let X be a riemannian symmetric space of non-compact type. Then theset of Weyl chambers of X is identified with the homogeneous space G/ M, the set oforiented flats with G/ MA, the set ofhorocycles with G/ MN, and the boundary ofXwith G/ P. Furthermore, the maps it, p, cr, and T introduced above coincide with thenatural projections from G/ M to G/ K, G/ MA, Gj MN, and G/P, respectively.Proof. We shall give a proof for the case of G/ M. First we have to show that Goperates transitively on CX. Let c be any Weyl chamber, F the flat containing c,and x the base point of c. An isometry g e G takes the flat F into a flat gF throughthe point gx, and it preserves the configuration of regular and singular geodesies.Therefore the Weyl chamber c is mapped into a Weyl chamber gc contained in gFand based at gx.Now we shall find an isometry which takes c into c0. There is a g e G such thatgx = x0, thus x0 e gF. An element k e K maps gF into Fo, hence kgF = Fo. Now kgcand c0 are two Weyl chambers in Fo based at x0, therefore there is an elementm'eM' such that m'kgc = c0.Finally we shall determine the isotropy group of c0 in G. Suppose gc0 = c0. Thencertainly gx0 = x0, hence ge K. Moreover gF0 = Fo, thus g e M'. Since W = M'/ Macts simply transitively on the set of Weyl chambers in Fo based at x0, it followsthat ge M. Conversely, we clearly have mc0 = c0 for me M.The proof for G/ MA is similar to the one for G/ M, so we shall omit it.The case of G/ MN is treated in [4] in purely group-theoretical terms. There, thehorocycle associated with c = gc0 is defined as the orbit of gx0 under the groupgNg'1. The relation with our geometric notion of horocycles is established in [6].The identification of the boundary G/P with the set of classes of asymptoticWeyl chambers is proved in [7, § 4].The statements about the natural projections are evident. •at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700003175Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 11:03:40, subject to the Cambridge Core terms of use, available592 H.-C. Im HofIn addition to the obvious left action of G on the spaces G/M, G/ MA, G/MN,and G/P, there is a right action of the Weyl group W on these spaces. Let mwdenote a representative for an element we W. Then the effect of w on a Weylchamber gM is given by gM <-> gmwM. The action of W on the other spaces isdenned similarly, and so all these W-actions are compatible with natural projections.Now recall the action tp:AxG/M^-G/M given by <p(a, gM) = gaM. The orbitof a Weyl chamber gM e G/M under <p is {gaM e G/M; a e A}, it coincides withthe fibre p~1(gMA) of the projection p: G/M-» G/MA. In other words, the effectof <p consists of translating a given Weyl chamber along its supporting flat to anotherWeyl chamber of the same flat and asymptotic to the given one. In the rank onecase, Weyl chambers coincide with geodesic rays, and these in turn may be identifiedwith unit tangent vectors. Moreover, since the group A is one-dimensional it admitsa parametrization A = {a(t)e G; teU}. Therefore the action <p may be viewed asan action of U on the unit tangent bundle of X. The effect of <p consists in this caseof translating a given geodesic ray along its supporting geodesic to another ray ofthe same geodesic and pointing to the same direction. This is exactly the effect ofthe geodesic flow. (For a group-theoretical description of the geodesic flow, see [2].)Finally we shall determine the leaves of the stable and strong stable foliationsinduced by <p. First we shall consider a special case. Let a0 be a regular element ofA with A (log a0) > 0 for all A e A+. For such an element, the contracting subspaceof ?/m = a@'ZK^A gK is n = ZA6A+ 9\- Therefore the leaf of the strong stable foliationthrough co = M eG/M equals the orbit{nMeG/M;neN}of c0 under N. This orbit coincides with the fibre <x~l{MN) of the projectionG/MN, i.e. it consists of all Weyl chambers asymptotic to c0 and basedat the points of the horocycle Zo = Nx0 = MN e G/ MN.The stable foliation is tangent to a®m. Its leaf through c0 equals the orbit{anM eG/M;aeA,neN)of c0 under AN, it coincides with the fibre T~\P) of the projection x: G/M^G/P,i.e. it consists of all Weyl chambers asymptotic to c0.Now consider the case of an arbitrary regular element a e A. There is a uniqueWeyl group element we W such that a = mwaom~^, where mweM' represents w,and a0 satisfies A (log a0) > 0 for all A e A+. The contracting subspace with respectto the action of a isI ^A=Ad(mJ*.A(loga)>0Therefore the leaf of the strong stable foliation through c0 equals the orbit{mM,«ffi~'Me G/M; ne N}of c0 under mwNm^,1. This is no longer a set of asymptotic Weyl chambers, it isobtained from the set {mwnM e G/ M; n e N} of Weyl chambers asymptotic to mwc0and based at the points of the horocycle mJZo by applying w"1 from the right.at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700003175Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 11:03:40, subject to the Cambridge Core terms of use, availableAn Anosov action 593The leaf of the stable foliation through c0 equals the orbit{mwanm^xM e G/ M; a e A, ne N}of c0 under m^AMw"1 = AmBNm~'. It consists of the right w~'-images of all Weylchambers asymptotic to mwc0.The unstable foliations with respect to a coincide with the stable foliations withrespect to a"1, so they need no further attention.REFERENCES[1] P. Eberlein & B. O'Neill. Visibility manifolds. Pacific J. Math. 46 (1973), 45-109.[2] I. M. Gel'fand & S. V. Fomin. Geodesic flows on manifolds of constant negative curvature. UspehiMat. Nauk (N.S.) 7 (1952), 118-137; Amer. Math. Soc. Translations (2) 1 (1955), 49-65.[3] S. Helgason. Differential Geometry and Symmetric Spaces. Academic Press: New York 1962.[4] S. Helgason. Duality and Radon transform for symmetric spaces. Amer. J. Math. 85 (1963), 667-692.[5] M. W. Hirsch. Foliations and non-compact transformation groups. Bull. Amer. Math. Soc. 76 (1970),1020-1023.[6] H. C. Im Hof. Die Geometrie der Weylkammern in symmetrischen Raumen vom nichtkompaktenTyp. Habilitationsschrift, Bonn, 1979.[7] G. D. Mostow. Strong Rigidity of Locally Symmetric Spaces. Princeton University Press: Princeton,1973.[8] C. Pugh & M. Shub. Ergodicity of Ansov actions. Invent. Math. 15 (1972), 1-23.at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700003175Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 11:03:40, subject to the Cambridge Core terms of use, available

An Anosov action on the bundle of Weyl chambers

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