Multiplicative ergodic theorem on ‡ag bundles for ‡ows on principal bundles of reductive Lie groups Luciana A. Alves yand Luiz A. B. San Martinzx Abstract Let Q ! X be a principal bundle having as structural group G a reductive Lie group in the Harish-Chandra class or in particular G is semi-simple with …nite center. A semi‡ow t of endomorphisms of Q induces a semi‡ow t on the associated bundle E = Q G F, where F is the maximal ‡ag bundle of G. The A-component of the Iwasawa decomposition G = KAN yields an additive vector valued cocycle a (t; ), 2 E, over t with values in the Lie algebra a of A. We prove an analogous of the Multiplicative Ergodic Theorem of Oseledets for this cocycle: If is a probability measure invariant by the semi‡ow on X then the a-Lyapunov exponent ( ) = lim 1t a (t; ) exists for every on the …bers above a set of full -measure. The level sets of ( ) on the …bers are described in algebraic terms. When t is a ‡ow the description of the level sets is sharpened. We relate the cocycle a (t; ) with the Lyapunov exponents of a linear ‡ow on a vector bundle and other growth rates.
AMS 2010 subject classi…cation: Primary: 37H15, 22E46, 37B55. Key words and phrases: Lyapunov exponents, Multiplicative Ergodic Theorem, Semi-simple Lie groups, Reductive Lie groups, Flag Manifolds. Supported by FAPESP grant no 06/60031-3 Address: Faculdade de Matemática - Universidade Federal de Uberlândia. Campus Santa Mônica, Av. João Naves de Avila - 2121 38.408-100 Uberlândia - MG - Brasil. e-mail:
[email protected] z Supported by CNPq grant no 305513/2003-6 and FAPESP grant no 07/06896-5 x Address: Imecc - Unicamp, Departamento de Matemática. Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz. 13083-859 Campinas São Paulo, Brasil. e-mail:
[email protected] y
1
1
Introduction
The purpose of this paper is to prove an analogous of the Multiplicative Ergodic Theorem of Oseledets that holds for ‡ows and semi‡ows evolving on a principal bundle Q ! X of a reductive or a semi-simple Lie group G, where the ‡ow on the base space X preserves an invariant probability measure . An Iwasawa decomposition G = KAN of G yields a similar decomposition of the principal bundle Q. Based on this decomposition the logarithm of the projection onto the A component de…nes a vector valued cocycle a (t; ) over the ‡ow on the associated (maximal) ‡ag bundle E = Q G F. Here t is time (t 2 N or Z) and 2 E. Our Multiplicative Ergodic Theorem proves that the limit 1 a (t; ) t!+1 t
( ) = lim
exists almost surely and describes algebraically the level sets of ( ) on the ‡ag bundle E. It modi…es and extends Oseledets Theorem by considering general reductive and semi-simple Lie groups instead of Gl (n; R) (or C), that is the relevant group for ‡ows on vector bundles. Also to deal with the vector valued cocycle a (t; ) we are forced work with ‡ows on the maximal ‡ag bundle E = Q G F, because the cocycle cannot be de…ned over partial ‡ag bundles. This is a distinguished feature even when G = Gl (n; R), requiring another algebraic and geometric techniques. We call the limit ( ) = limt!+1 1t a (t; ) the a-Lyapunov exponent in the direction of 2 E. Here a = log A is the Lie algebra of the abelian group A, where a (t; ) takes its values. The cocycle a (t; ) were already considered in [14]. The interest in the A-component of the Iwasawa decomposition G = KAN is that it measures the variation of a lengths or volumes under the action of an element g 2 G in several geometric situations. This measuring is extensively used in Lie theory. Hence for (semi) ‡ows on principal bundles the a-Lyapunov exponents provide the exponential growth rate of di¤erent geometric quantities, that are obtained by composing the cocycle a (t; ) it with suitable linear maps de…ned on a. Our approach gives a uni…ed treatment for all these growths rates. We present some examples of these growth rates in Sections 7 and 8 below. In particular we discuss the Lyapunov exponents limt!+1 1t log jj t vjj of a linear ‡ow t on a vector bundle and make 2
a detailed derivation of Oseledets Theorem from the Multiplicative Ergodic Theorem for the a-Lyapunov exponents. The multiplicative ergodic theorem consists of two pieces. The …rst one ensures the existence of the limits ( ) = limt!+1 1t a (t; ) for every above a set of -total measure. The second part is the analogous of Oseledets …ltration or decomposition, and describe the level sets ( ) along a …xed …ber over x 2 . To prove the existence of limits we rely on the characterization of regular sequences given by Khaimanovich [9], ensuring the existence of limits of sequences in G (actually, sequences in the corresponding symmetric space G=K or rather KnG). Then an application of the subadditive ergodic theorem shows that -almost all points x 2 X are regular, much like in the proof of Oseledets theorem of Raghunathan [11] (see also Ruelle [12]). As to the second part, there is an analogous of the Oseledets …ltration that holds for semi‡ows of endomorphisms. The …ltration is described algebraically on the …ber over x 2 as the stable sets for the action of exp D (x), where D (x) is an element of the Lie algebra g of G. In the case of ‡ows (invertible maps) we exploit further the algebraic and geometric structure of ‡ag manifolds to get an analogous of Oseledets decomposition. It is given, for x 2 , as the set of …xed points (instead of stable sets) of exp H (x), where again H (x) 2 g. This is done by comparing the …ltrations for the forward and backward ‡ows and using the concepts of dual ‡ag manifolds and transversality. Thus, for instance in case is ergodic, we get H as measurable section of an associated bundle whose …ber is an adjoint orbit of G. We adopt throughout the more geometric point of view provided by the formalism of principal bundles. This is not usual in the literature about the multiplicative ergodic theorem, where the language of cocycles (skew product ‡ow) is more commonly used. These objects are the same when the bundle is trivial. Also it is part of the folklore of the theory that under mild conditions a principal bundle is measurably trivial. Since this triviality does not hold for continuous (or di¤erentiable) bundles it is worth to have the Multiplicative Ergodic Theorem written in the geometric language of principal bundles as well. Now we give a more detailed description of the paper. In the preliminary Section 2.1 we mainly state the scope of our results by specifying the class of reductive Lie groups we work with. At this regard we follow the book by Knapp [8]. We also describe some basic aspects of 3
these groups. The additive cocycle a (t; ) is de…ned in Section 3, where the Iwasawa decomposition of the group G is carried out to a decomposition of the bundle Q, under the (mild) assumption that Q admits a reduction to a K-subbundle R, where K is the maximal compact subgroup of G. (This assumption is in the same level of generality as the assumption that a vector bundle admits a Riemannian metric.) Then the cocycle a (t; ) is de…ned as the logarithm of the projection onto the A-component. We show that this cocycle factors to a cocycle (also denoted by a (t; )) to the associated ‡ag bundle E = Q G F, where F is the maximal ‡ag manifold of G. In Section 3 we also construct the Cartan and polar decompositions of Q, which are basic technical tools. The latter yields another map (denoted by a+ ) onto a, which is not a cocycle, but becomes a subadditive cocycle when composed with suitable linear maps. The asymptotics of a+ is called the polar exponent of the (semi-) ‡ow. We reserve Section 4 for the statement of the Multiplicative Ergodic Theorem. Its proof is presented in Section 5, which is divided in several subsections, where we discuss the regular sequences of Khaimanovich [9], apply the subadditive ergodic theorem and construct the analogous of the Oseledets …ltration. Section 6 is dedicated to ‡ows (invertible maps). We recall the concepts of dual ‡ag manifolds and transversality between ‡ags to prove that the …ltrations for the ‡ow and the backward ‡ow are transversal to each other. This yields the Oseledets decomposition via a section of an associated bundle Q G Ad (G) having the adjoint orbit Ad (G) , 2 g, as typical …ber. In the last two sections we describe new cocycles that are obtained by composing the a-cocycle a (t; ) with linear maps 2 a . First in Section 7 cocycles over partial ‡ag bundles are constructed. The cocycle a (t; ) itself does not factor to a cocycle above the ‡ow on a partial ‡ag bundle E . We consider instead cocycles of the form a (t; ) = (a (t; )), 2 a . We characterize those 2 a such that a (t; ) factors to a partial ‡ag bundle E . Several examples of such cocycles are included. Among them the cocycle log k t vk, which is a cocycle over the projective bundle P, viewed as a partial ‡ag bundle for the groups G = Gl (n; R) or Sl (n; R). Finally in Section 8 we consider a homomorphism : G ! H, where H is another reductive group and get an H-bundle by the usual construction of an associated bundle. The induced (semi-) ‡ow on the new bundle has his own a-Lyapunov exponent that turns out to be the image under of the a-Lyapunov exponent of the original ‡ow. This construction includes the 4
linear ‡ows on the vector bundles obtained via representations of G.
2
Results and notation about Lie groups and ‡ag manifolds
In this section, we explain our notation and background results about semisimple Lie groups, reductive Lie groups and their ‡ag manifolds. We refer to Knapp [8], Duistermat-Kolk-Varadarajan [4] and Warner [15]. In order to make the paper understandable to readers without acquaintance with Lie Theory we adopted the strategy of de…ning the notation by writing explicitely their meanings for the matrix group Sl (d; R) and its Lie algebra sl (d; R) (cf. Zimmer [16]). We hope that the reader with expertise in semi-simple theory will recognize the notation for the general objects (e.g. k is a maximal compact embedded subalgebra, etc.) Let g be a semi-simple non-compact Lie algebra. At the Lie algebra level the Cartan decomposition reads g = k s, where k = so (d) is the subalgebra of skew-symmetric matrices and s is the space of symmetric matrices. The Iwasawa decomposition of the Lie algebra is g = k a n where a is the subalgebra of diagonal matrices and n is the subalgebra of upper triangular matrices with zeros on the diagonal. The set of roots is denoted by . These are linear maps ij 2 a , i 6= j, de…ned by ij = i j , where i (diagfa1 ; : : : ; ad g) = ai . If H 2 a then the set (H), 2 , are the nonzero eigenvalues of ad (H). The set of positive roots is + = f ij : i < jg and the set of simple roots is = f ij : j = i+1g. The root space is g (g ij is spanned by the basic matrix Eij ) and g=m
a
X
g
2
where m = zk (a) = z (a) \ k is the centralizer of a in k (m = f0g in sl (d; R)). The basic (positive) Weyl chamber is denoted by a+ = fH 2 a :
(H) > 0;
2 g
(cone of diagonal matrices diagfa1 ; : : : ; ad g satisfying a1 > > ad ). We + write cla for this closure (formed by diagonal matrices with decreasing eigenvalues). 5
At the Lie group level the Cartan decomposition reads G = KS, K = exp k and S = exp s (K is the group SO (d) and S the space of positive de…nite symmetric matrices in Sl (d; R)). The Iwasawa decomposition is G = KAN , A = exp a, N = exp n. The Cartan decomposition splits further into the polar decomposition G = K (clA+ ) K, A+ = exp a+ . M = CentK (a) is the centralizer of a in K (diagonal matrices with entries 1), M = NormK (a) is the normalizer of a in K (signed permutation matrices) and W = M =M is the Weyl group (for Sl (d; R) is the group of permutations in d letters, that acts in a by permuting the entries of a diagonal matrix). The (standard) minimal parabolic subalgebra is p = m a n (= upper triangular matrices), and a general standard parabolic subalgebra p is de…ned by a subset as X X p =m a g g +
2
2h i+
where h i is the set of positive roots spanned (overPZ) by and h i+ = . h i \ + . That is, p = p n ( ), where n ( ) = 2h i+ g + Alternatively, given , take H 2 cla such that (H ) = 0, 2 , if and only if 2 . Such H exists and we call it a characteristic element of . Then p is the sum of eigenspaces of ad (H ) having eigenvalues 0. Thus p is the subalgebra of matrices that are upper triangular in blocks, whose sizes are the multiplicities of the eigenvalues of ad (H ). When is empty, p; boils down to the minimal parabolic subalgebra p. In sl (d; R), H = diag (a1 ; : : : ; ad ) with a1 ad , where the multiplicities of the eigenvalues is prescribed by ai = ai+1 if i;i+1 2 . This yields a partition of f1; : : : ; dg and a coorreponding block decomposition of the matrices. Then p is the subalgebra of the matrices upper triangular in blocks. The Levi component of p is the centralizer z = zH of H in g (which becomes diagonal matrices in blocks). There are the decompositions p =k
a
n+
and
z =k
a
n+ ( );
where k = Cent Pk (H ) (block diagonal matrices with skew-symmetric blocks) + and n ( ) = 2h i+ g (blocks diagonal matrices with upper triangular blocks). 6
The parabolic subgroup P , associated to , is de…ned as the normalizer of p in G (as a group of matrices it has the same block structure as p ). It decomposes as P = K AN , where K = CentK (H ) is the centralizer of H in K. We usually omit the subscript when = ; and P = P; is the minimal parabolic subgroup. The ‡ag manifold associated to is the homogeneous space F = G=P (just F when = ;). If 1 2 then the corresponding parabolic subgroups 1 satisfy P 1 P 2 , so that there is a canonical …bration : F 1 ! F 2, 2 given by gP 1 7! gP 2 (just 2 if 1 = ;). For the matrix group the ‡ag manifold F identi…es with the manifold of ‡ags of subspaces V1 Vk where the di¤erences dim Vi+1 dim Vi are the sizes of the blocks de…ned by (or rather the diagonal matrix H ). The projection 12 : F 1 ! F 2 is de…ned by “forgetting subspaces”. The concept of dual ‡ag manifold is de…ned as follows: Let w0 be the principal involution of W, that is, the only element of W such that w0 a+ = a+ , and put = w0 . Then ( ) = and for write = ( ). Then F is called the ‡ag manifold dual fo F . For the matrix group the vector subspaces of the ‡ags in F have complementary dimensions to those in F (for instance the dual to a Grassmannian Grk (d) is the Grasmmannian Grd k (d)). We conclude this section by stating some results on the dynamics of the action on the ‡ag manifolds of the one-parameter group exp tH, H 2 cla+ , t 2 R (which is the same as the discrete dynamics given by iterations of h = exp H). The ‡ow de…ned by exp tH is gradient in whatsoever ‡ag manifold F . To write down its …xed point set let b = P the origin of F = G=P and put ZH = fg 2 G : Ad(g)H = Hg and KH = ZH \ K, for the centralizers of H in G and K respectively. For w 2 W we write also w for any one of its representatives in M . Then the orbits ZH wb = KH wb
w 2 W;
are the connected components of the …xed point set of exp tH, t 2 R. We write x (H; w) = ZH wb and refer to it as the set of …xed points of type w. In addition, x (H; 1) is the single attractor and x (H; w0 ) is the single repeller. Here w0 the principal involution of W. For the stable and unstable sets of x (H; w) let = f 2 : (H) = 0g 7
and consider the nilpotent subalgebras X n+ = g n = 2
+ nh
i
2
X
+ nh
g i
and the connected subgroups N = exp n . Put st (H; w) = un (H; w) =
NH KH wb ; NH+ KH wb :
Then st (H; w) and un (H; w) are the stable and unstable sets of x (H; w), respectively.
2.1
Reductive groups
To get our results on ‡ag manifolds we work mainly with the structure of semi-simple Lie groups, which can be extended to reductive Lie groups, that are supposed to have all the essential structure-theoretic properties of semisimple groups. We reproduce here the precise de…nition of reductive Lie of Knapp [8], Section VII.2. A reductive Lie group is a 4-tuple (G; K; ; B) consisting of a Lie group G, a compact subgroup K of G, a involution of the Lie algebra g de G, and a nondegenerate, Ad(G) invariant, invariant, bilinear form B on g such that (i) g is a reductive lie algebra, i.e., g = Zg of g and [g; g] is semi-simple. (ii) the decomposition of g into +1 and where k is the Lie algebra of K;
[g; g], where Zg is the center
1 eigenspaces under
is g = k s,
(iii) k and s are orthogonal under B, and B is positive de…nite on s and negative de…nite on k; (iv) multiplication, as a map from K onto, and
exp s into G, is a di¤eomorphism
(v) every automorphism Ad(g) of gC , the complexi…cation of g, is inner for g 2 G, i.e., is given by some x 2 Int(g). 8
In this case, will be called the Cartan involution and g = k s will be called the Cartan decomposition of g. A reductive Lie group is called a Harish-Chandra group if its semi-simple component has …nite center. The notion of an Iwasawa decomposition extends to reductive Lie groups. Let a reductive Lie group G be given, and write its Lie algebra as g = Zg [g; g]. Let a be a maximal abelian subespace de s. In this case, a = (s \ Zg )
(a \ [g; g]) = aa
as ;
where a \ [g; g] is a maximal abelian subspace of s \ [g; g]. Relative to a, a root of (g; a) is a nonzero 2 a such that the space g = fX 2 g; ad(H)(X) = (H)X; 8 H 2 ag
is nonzero. Clearly, a root is obtained by taking a root for [g; g] and extending it from a \ [g; g] to a by making it be 0 on s \ Zg . Take a Weyl chamber a+ a and denote by + the corresponding set of positive roots and the set of simple roots. De…ne X X n+ = g and n = g 2
2
the nilpotent Lie subalgebras de g. Thus, we have the Iwasawa decomposition of g g = k a n+ :
Denote by A = exp a and N + = n+ the connected subgroups with corresponding Lie algebras. The Iwasawa decomposition of reductive Lie group G is G = KAN + . Analogously, G = K (clA+ ) K is a polar decomposition of G. A concrete example of a Harish-Chandra group is Gl (d; R), whose semisimple component is Sl (d; R). A Cartan involution is (A) = AT , while a Cartan decomposition is gl (d; R) = so (d; R) s, where s is the subspace of symmetric matrices.
3
Decompositions and cocycles
We assume throughout the paper that G is a reductive group in the HarishChandra class. In particular G has …nite if it is semi-simple. In this section we recall the decompositions of a principal bundle Q ! X with reductive structural group G that yield the vector valued spectra. We refer to [14], Section 4, for details. 9
3.1
Iwasawa decomposition and the a-cocycle
Let Q ! X be a principal bundle whose structural group G is a reductive Lie group. The right action of G on Q is denoted by q 7! q g, q 2 Q, g 2 G. We endow G with an Iwasawa decomposition G = KAN , that is kept …xed throughout. The bundle Q has a reduction to a subbundle R Q which is a principal bundle with structural group K in such a way that Q = R AN . This means that any q 2 Q decomposes uniquely as q = r hn;
r 2 R;
hn 2 AN:
We let R : Q ! R;
A : Q ! A;
q 7! r;
q 7! h;
be the associated projections. They satisfy the following properties: 1. R(r) = r, A(r) = 1, when r 2 R, 2. R(q g) = R(q)m, A(q g) = A(q)h, when q 2 Q, g = mhn 2 P = M AN . In particular, A(r g) = h. In what follows we write for q 2 Q, a (q) = log A (q) 2 a: We use the same notation for the Iwasawa decomposition of g 2 G, namely, a (g) = log h if g = uhn 2 KAN . By the second of the above properties we have a (q p) = a (q) + a (p) ; Now let
p 2 P:
(1)
be an endomorphism of Q. It induces the map R
: r 2 R 7! R( (r)) 2 R;
R which satis…es R =( )R if is another endomorphism. Let t be the continuous ‡ow or semi‡ow of endomorphisms of Q, t 2 T = Z or N. Then R t de…nes a continuous ‡ow in R. We write a also for the map a : T R ! a; a(t; r) = a( t (r)):
This is a cocycle over
R t ,
that is,
a(t + s; r) = a(t;
R s (r))
+ a(s; r);
where t; s 2 T, r 2 R. In what follows we write simply 10
t
instead of
R t .
Proposition 3.1 The additive cocycle a(t; r) over R factors to a cocycle over the maximal ‡ag bundle E = Q G F = R G K=M , also denoted by a(t; ). Proof: In fact, the associated bundle R G K=M R=M can be viewed as the total space of the bundle R=M ! E over the ‡ag bundle, whose …ber is M . For m 2 M and q 2 R we have A(q m) = A(q) by the second property listed above. Hence, a(t; r m) = a(t; r), showing that a(t; r) factors to R=M .
Now let t , t 2 T (T = Z or T = Z+ ) be a ‡ow (or a semi‡ow) on Q. It induces ‡ows on both E and on E A also denoted by t . The cross section de…nes a continuous a-valued cocycle a : T E ! a by a(t; ) = log at where t ( ( )) = ( t ( )) at , 2 FQ. This is essentially the cocycle over R de…ned from the Iwasawa decomposition (see [14]). In fact, if r 2 R and 2 E are related by = r b0 , then t (r) = rt at nt and t ( ( )) = ( t ( )) at . Finally the a-Lyapunov exponent of t in the direction of 2 E is de…ned by 1 ( ) = lim a (t; ) 2 a; t!+1 t when the limit exists.
3.2
Polar decomposition
So far we used the Iwasawa decomposition of G to get the cocycles over R and E. If we consider instead a Cartan G = KS and a polar decomposition G = K(clA+ )K we get subaditive cocycle a+ , which factors to the base space and has values in cla+ , the closure of a Weyl chamber. (See [14], Section 4 for details.) These two cocycles are di¤erent but closely related. Again we …x a K-reduction R Q. Then the decomposition G = KS induces a decomposition Q R S, where any q 2 Q can be written uniquely as q = r s with r 2 R and s 2 S. Let RC : Q ! R; q 7! r; S : Q ! S; q 7! s be the associated projections. (We note that RC is not equal to the projection R against the Iwasawa decomposition.) These projections satisfy the following properties: 11
1. RC (q k) = RC (q) k and S(q k) = k 1 S(q)k if q 2 Q and k 2 K. (This is a consequence of the fact that kSk 1 = S.) 2. RC (r) = r and S(r g) = S(g) if r 2 R, g 2 G, where we denote also by S(g) the S-component of g 2 G = K S. 3. S(q g) = S(S(q)g) if q 2 Q and g 2 G. (In fact, q g = R(q) S(q)g so that if S(q)g = kt is the Cartan decomposition of S(q)g in G where t = S(S(q)g) then q g = R(q) kt. Hence S(q g) = t = S(S(q)g).) Now …x a Weyl chamber A+ sitting inside a maximal abelian A S and consider the polar decomposition G = K(clA+ )K. If g = ks 2 KS then g = uhv where u = kv 1 2 K and s = v 1 hv, h 2 clA+ . Combining this polar decomposition with the Cartan decomposition of Q we can write every q 2 Q as q = r hv; r 2 R; h 2 clA+ ; v 2 K: Here the component h 2 clA+ is uniquely de…ned (although r and v are not). Thus we have a well de…ned map A+ : Q ! clA+ ;
q 7! h
which satis…es A+ (q k) = A+ (q) if q 2 Q, k 2 K. In the sequel we denote with the corresponding lower case letters the logarithms of the above maps: a+ (q) = log A+ (q) 2 cla+ :
s(q) = log S(q) 2 s Now take again a (semi-) ‡ow a+ : T
t,
R ! cla+ ;
t 2 T, on Q and de…ne the map a+ (t; r) = a+ ( t (r)):
Accordingly the polar exponent of +
t
is de…ned to be
1 + a (t; r) 2 cla+ t!+1 t
(r) = lim
when the limit exists. Next we state two basic properties saying that a+ (t; r) and constant along the …bers of R.
12
+
(r) are
Proposition 3.2 a+ (t; r) and + (r) are invariant by the right action of K: a+ (t; r k) = a+ (t; r) and + (r k) = + (r), r 2 R and k 2 K, is constant on the …bers of R ! X. Proof: In fact, a+ (t; r k) = a+ ( t (r k)) = a+ ( t (r) k) = a+ ( t (r)) because a+ ( ) is invariant. The invariance of + is an immediate consequence of the invariance of a+ (t; r). In view of this proposition we write a+ (t; x) = a+ (t; r) and + (x) = + (r) where r 2 R is any element in the …ber above x. Contrary to a (t; ) de…ned by the Iwasawa decomposition, the map a+ (t; x) coming from the polar decomposition is not an additive cocycle. However, when composed with suitable linear maps in a subadditive cocycles are obtained. For 2 a we write a+ (t; x) =
a+ (t; x) :
Next we use representation theory to prove that a+ (t; x) is subadditive when is a dominant weight of a …nite dimensional complex representation of G. If gC is the complexi…cation of the Lie algebra g then the …nite dimensional complex representations of gC are parametrized by the dominant weights. Namely, if h is a Cartan subalgebra of gC and is a simple system of roots then 2 h is dominant if 2hh ;; ii is a positive integer for any 2 . Up to conjugacy de…nes a unique irreducible representation of gC on a vector space V , which restricts to a representation of g. We say that is a dominant weight for G in case the representation of g extends to a representation of G. By a general construction we can take the Cartan subalgebra h gC so that a h and a dominant weight assume real values in a. The set of dominant weights for G is denoted G . It can be proved that if H 2 a and (H) = 0 for every 2 G then H 2 az , that is, the restrictions to as of the dominant weights span its dual as . Now if is a dominant weight for G then via the representation we have a linear action of G on V from which we can construct the vector bundle V = Q G V . The ‡ow t on Q induces a linear ‡ow on V , also denoted by t . The next lemma gives the operator norm of t in terms of a+ (t; x) (see also [6], Lemma 4.22).
13
Lemma 3.3 Let be a dominant weight for G. Then V can be endowed with a norm jj jj such that a+ (t; x) = log jj( t )x jj where jj( t )x jj is the operator norm of ( t )x : (V )x ! (V )t x . Proof: It is known that there is an inner product h ; i on V such that (k) is an isometry for k 2 K. This inner product can be plugged …berwise to V by hr v; r wi = hv; wi r 2 R; v; w 2 V : Using the polar decomposition of Q we write t (r) = rt ht ut , rt 2 R, ht 2 clA+ and ut 2 K. Then jj t (r v)jj = rt (ht ut ) v , v 2 V . But rt (ht ut ) v = (ht ut ) v because rt is an isometry between V and the …ber of V above x = (r). Hence we have equality jj( t )x jj =
(ht ut )
between operator norms. Since (ut ) is an isometry, it follows that (ht ut ) = (ht ) (ut ) = (ht ) , that is, jj( t )x jj = (ht ) . Now, (ht ) is symmetric w.r.t. the K-invariant inner product h ; i on V so that (ht ) is the highest eigenvalue of (ht ) which is exp (log ht ). Therefore jj( t )x jj = exp (log ht ) = exp a+ (t; x) as claimed. This lemma implies easily the subadditivity of a+ (t; x). Proposition 3.4 Let be a dominant weight for G. Then a+ (t; x) is subadditive with respect to the ‡ow on the base X, that is, a+ (t + s; x)
a+ (t; s x) + a+ (s; x)
where (s; x) 7! s x denotes the semi‡ow on the base space. Proof: In fact, by the lemma a+ (t + s; x) = log
t+s x
14
log jj( t )s x jj jj( s )x jj
and by the lemma again the last term is a+ (t; s x) + a+ (s; x).
Remark: It is known that the dominant weights for G span h and hence their restrictions to a span the dual a of a. This enables to prove existence of a limit + (r) = limt!+1 1t a+ (t; r) by proving the existence of the limits limt!+1 1t a+ (t; r) with a dominant weight. The existence of the latter limit can be ensured by the subadditive ergodic theorem. 0 because Remark: If is a dominant weight for G then a+ (t; x) + + a (t; x) 2 cla and a dominant weight is positive on the Weyl chamber a+ .
3.3
Central components of the cocycles
To study the cocycles a (t; r) as well as the map a+ (t; r) with values in a it is convenient to decompose them into the central and semi-simple components according to a = (a \ Zg ) (a \ [g; g]) = az as : We write az (t; r) and a+ z (t; r) for the components in the direction of az .
Proposition 3.5 az (t; r) is constant on the …bers and hence factors to an additive cocycle over the ‡ow on the base space X. a+ z (t; r) factors to a cocycle over X as well. Proof: We must show that az (t; r k) = az (t; r) for k 2 K and r 2 R. Thus R e n and take the decomposiwrite t (r) = R t (r) hn, t (r k) = t (r k) he tions h = h1 h2 , e h=e h1e h2 , with h1 = exp H1 , H1 2 as , h2 = exp H2 , H2 2 az , and the same notation for e h. Now, t (r k) = t (r) k = R t (r) h1 h2 nk. Since h2 is in the center of G we have h1 h2 nk = (h1 nk) h2 . Take the Iwasawa decomposition h1 nk = uh0 n0 2 KAN: Then t (r k) = R h0 h2 n0 . Hence by uniqueness of the Iwasawa t (r) u R 0 e decomposition of Q we have R e = n0 , t (r k) = t (r) u, h = h h2 and n so that e h2 = h2 . But by de…nition az (t; r) = log h2 and az (t; r k) = log e h2 , concluding the proof for az (t; r). 15
By Proposition 3.2, a+ z (t; r) is constant on the …bers. The proof that it is indeed a cocycle follows directly from the de…nitions: Ir r 2 R then t (r) = R 1 with k 2 K, h1 h2 = exp a+ (t; r) and h2 = exp a+ t (r) kh1 h2 k z (t; r). Since h2 is in the center of G we have t+s
(r) = =
s
(
R t+s
t
(r)) =
(r) kh1 k
1
h2
(r) kh1 h2 k h2
with obvious notation where h2 = a+ z s; composition we get the cocycle property + a+ z (t + s; r) = az s;
4
R t 1
s
R t
R t
(r) . By uniqueness of the de-
(r) + a+ z (t; r) :
Multiplicative ergodic theorem for semi‡ows
We state here the multiplicative ergodic theorem for a semi‡ow of endomorphisms t , t 2 N, of the principal bundle Q ! X, whose structural group G is a reductive or a semi-simple Lie group. Suppose that the semi‡ow on the base preserves a probability measure . Given a dominant weight 2 G we write f ( ) = a+ (1; ). Theorem 4.1 Suppose that f ( ) is -integrable, for every dominant weight 2 G. Then there exists a measurable invariant subset X, with ( ) = 1 such that the polar exponent + (x) = + (r) (r 2 R and x = (r)) exists for every x 2 . Put e = E 1 ( ). Then the a-Lyapunov exponent ( ) exists for every 2 e. e = 1( )\ As to the level sets of there exists a measurable map D : R R ! s such that +
1. D (r) = Ad (kr ) 2.
(r b) = w
1
+
(r), with kr 2 K and
(r) if b 2 st (D (r) ; w). 16
The last statement says that is the level sets are parametrized by the Weyl group W. Notation: For w 2 W and x 2 X we write st (x; w) = r st (D (r) ; w) where e is in the …ber over x. This set is independent of r in the …ber because r2R D (r k) = Ad (k 1 ) D (r), k 2 K. Also, for w 2 W we put [ st (w) = st (x; w) E; x2
so that by the last statement of the theorem
( )=w
+
(x) if
2 st (w).
Remark: The decomposition of e into the sets st (w) is the analogous of the Oseledets …ltration of a ‡ow on a vector bundle, and we refer to it by the same name. It will be proved below that the sets st (w) are invariant by the semi‡ow on E, although D itself is not invariant. Later on we will see that for invertible maps (‡ows) there exists an invariant measurable map with values in Ad (G) + (x) that de…ne alternatively the sets st (w). This map gives the analogous of the Oseledets decomposition of a ‡ow on a vector bundle. Remark: By item (2) of the theorem the a-Lyapunov exponent of the elements of st (1) (where 1 is the identity of W) lie in the cla+ . By the same formula st (1) is the only component having a-Lyapunov exponent in cla+ .
5
Proof of the multiplicative ergodic theorem
In this section we prove the multiplicative ergodic theorem for the a-Lyapunov exponents of a semi‡ow t , t 2 N, of endomorphisms of the principal bundle : Q ! X. We keep …xed the invariant measure on the base space X. As in the case of (semi) ‡ows of linear maps on vector bundles the multiplicative ergodic theorem contains two pieces: It …rst proves the almost surely existence of the limits and then constructs the decomposition of the bundle (the Oseledets …ltration or decomposition) into level sets of the Lyapunov exponents. Our approach is based on the concept of regular sequences in a semisimple Lie group developed by Khaimanovich [9]. We start by recalling the main results of [9] (see also [14]). 17
5.1
Regular sequences in G
Assume …rstly that G is semi-simple. For a sequence gk 2 G, k 2 Z+ , we write 1. the Lyapunov exponent in the direction of b 2 F as (gk ; b) = lim
k!+1
1 log a(gk u); k
where b = u b0 and u 2 G. This limit depends only on b because if b = u0 b0 , u0 2 G, then u0 = ug, g 2 P , so by (1) one has a(gk u0 ) = a(gk u) + a(g). In the limit the term k1 a(g) disappears. 2. The polar exponent of gk : +
1 log a+ (gk ); k!+1 k
(gk ) = lim
which depends on the sequence only, contrary to the Lyapunov exponents. Of course these limits may not exist. Next consider the left coset symmetric space KnG. Let d be the G-invariant distance in KnG, which is uniquely determined by d(o exp(X); o) = jXj ; X 2 s where o is the origin of KnG. Following [9] a sequence gk 2 G is said to be regular if there exists D 2 s such that d (o gk ; o exp kD) has sublinear growth as k ! +1, that is, 1 d (o gk ; o exp(kD)) ! 0: k
(2)
In this case we say that D is the asymptotic ray of gk . If u 2 K it follows that the sequence gk u is also regular with asymptotic ray Ad (u 1 ) D (see [14]). Write the polar decomposition of gk as gk = uk hk vk 2 K (clA+ ) K. Then one of the main results of [9] says that the sequence is regular if and only if it satis…es the following two conditions (see [9], Theorem 2.1): 1. The polar exponent
+
(gk ) = lim k1 log hk exists, and 18
2.
1 d(o k
gk ; o gk+1 ) ! 0, that is, o gk has sublinear growth in KnG.
Moreover, in this case the asymptotic ray of gk is given by D = Ad(u) + where u 2 K and + = + (gk ) 2 cla+ is the polar exponent of gk . The next statement gives the Lyapunov exponents of a regular sequence. Proposition 5.1 Suppose gk is a regular sequence in G with asymptotic ray D 2 s. Then the following statements hold. 1. Suppose D 2 cla+ and take y 2 PD . Then the sequence gk y is regular with the same asymptotic ray D. 2. Suppose D 2 a. Then limk!+1 k1 log a(gk ) = D. 3. Let + 2 cla+ be the polar exponent of gk . Then the Lyapunov exponent at b 2 st(D; w) is given by (gk ; b) = w 1 + . Proof: For the …rst statement we note that since the eigenvalues of ad (D) on pD are 0, it follows from the Iwasawa decomposition of y 2 PD that the sequence (exp kD) y (exp ( kD)) is bounded in PD so that d(o (exp kD) y; o (exp kD)) = d(o (exp kD) y (exp ( kD)) ; o) is bounded as a function of k (cf. [6], Proposition 3.9). But d(o gk y; o (exp kD))
d(o gk ; o (exp kD))+d(o (exp kD) ; o (exp kD) y 1 ):
So that k1 d(o gk y; o exp kD) ! 0, because gk is asymptotic to exp kD and the last term is bounded. For the second statement write gk = uk ak nk 2 KAN . We have jlog ak
kDj = d (o ak ; o (exp kD))
and by a well known inequality of horospherical coordinates (cf. Corollary 1.11 of [10]) the right hand side is bounded above by d (o ak nk ; o (exp kD)) = d (o gk ; o (exp kD)) : Therefore, k1 log ak ! D. Now, recall that st(D; w) = uPH + wb0 where u 2 K satis…es D = Ad(u) + and b0 2 F is the origin. Hence, for b 2 st(D; w) there exists y 2 PH + such 19
that b = uywb0 . In this case (gk ; b) = limk!+1 k1 log ak where gk uyw = u0k ak nk 2 KAN . Since u 2 K the sequence gk u is regular with asymptotic ray + , so by the …rst statement the same holds for gk uy. But w 2 K hence gk uyw is asymptotic to the ray Ad (w 1 ) + 2 a. Therefore, (gk ; b) = Ad (w 1 ) + , as follows from the second statement, concluding the proof. In [9] it is also given the following characterization of regularity in terms of the Lyapunov exponents (gk ; b). Proposition 5.2 The sequence gk is regular if and only if it has sublinear growth and the Lyapunov exponent (gk ; b) exists at some (and hence at all) b 2 F. Proof: See [9], Theorem 2.5, and its corollary where it is proved that gk is regular it it has sublinear growth and lim k1 log ak exists, where gk = u0k ak nk 2 KAN is the Iwasawa decomposition. But as mentioned above gk is regular if and only if gk u, u 2 K, is regular. Hence the existence of (gk ; b) at some b 2 F together with sublinear growth implies regularity. Regular sequences for a reductive Lie group G in the Harish Chandra class are treated the same way. To make some comments on them we follow the notation of Knapp [8]. Let Gss be its semi-simple component and put 0 G = KGss . Then 0 G = K exp (s \ [g; g]) and G is isomorphic to 0 G Zvec , where Zvec = exp (s \ z) is the split component of G (see [8], Proposition 7.27). By these decompositions it follows that KnG is di¤eomorphic to the product Zvec (K \ Gss ) nGss and the de…nition of a regular sequence in G is the same as in the semi-simple case. Here the asymptotic ray D decomposes as D = Dz +Ds with Dz 2 s\z = a\z and Ds 2 s\[g; g]. The two conditions of Khaimanovich still hold and the existence of polar exponent can be splitted into two limits, one for each component of log hk in the directions of a \ z and a \ [g; g], respectively.
5.2
Regular elements of the bundle
The Lyapunov exponents of sequences in G yield Lyapunov exponents of a ‡ow in the bundle by the following statement that follows directly from the de…nitions. 20
Proposition 5.3 Let r 2 R and write t (r) = rt gt , with rt 2 R, gt 2 G, t 2 N. If b0 2 F is the origin and b = ub0 , u 2 K, then 1 log at ; t!+1 t
(r b) = (r ub0 ) = lim
u2G
where gt u = u0t at nt 2 KAN is the Iwasawa decomposition. In other words any Lyapunov exponent of the semi‡ow is a Lyapunov exponent of a sequence in G. Let Q = R S ' R S be a Cartan decomposition of the bundle (see Section 3.2). As before we let S : Q ! S stand for the projection onto S ' KnG. For any initial condition q 2 Q we de…ne the sequence S( k (q)) 2 S, k 2 Z+ . In what follows we say that q 2 Q is a regular point for the ‡ow in case S( k (q)) is a regular sequence in S G in the sense of last subsection. We note that q is a regular point if and only if q g is regular for every g 2 G. That is, regularity depends only on the …ber of q. In this case we say that the base point x = (q) 2 X is regular for the ‡ow. Now we can prove that any Lyapunov exponent of the ‡ow on Q comes from a regular sequence in G. Proposition 5.4 Let r 2 R. Then the following statements are equivalent. 1. r is a regular point for the ‡ow. Denote by D 2 s the asymptotic ray and by + 2 cla+ the polar exponent of S( k (r)). 2. The Lyapunov exponent (r b) exists in one, and hence in any direction r b, b 2 F, along the …ber of r. In that case (r b) = w 1 + for any b 2 st(D; w). Proof: Take the Cartan decomposition k (r) = rk sk 2 R S. Then sk = S( k (r)) has sublinear growth. By Proposition 5.3 the Lyapunov exponent at r b, b 2 F, is the Lyapunov exponent at b of the sequence sk . Now the result follows by Proposition 5.2 which ensures that a sequence with sublinear growth is regular if and only if some (and hence all) Lyapunov exponent exists. It follows immediately that any Lyapunov exponent of the ‡ow is the Lyapunov exponent of a regular sequence. 21
Corollary 5.5 Take = r b 2 E = FQ and assume that its Lyapunov exponent ( ) under t exists. Then r is a regular point for the ‡ow and ( ) = (sk ; b) where sk = S( k (r)). By Proposition 5.1 the set of a-Lyapunov exponents (gk ; b), b 2 F, of a regular sequence gk is invariant under the Weyl group W. Therefore the above corollary has as a consequence the following symmetry of the Lyapunov spectrum. Corollary 5.6 Take = r b 2 FQ and assume that its Lyapunov exponent ( ) for the ‡ow t exists. Then for every w 2 W, w ( ) is also an aLyapunov exponent of the ‡ow.
5.3
Almost sure convergence
Theorem 5.7 Given the semi‡ow t , t 2 N, of endomorphisms of the principal bundle : Q ! X let be an invariant probability measure on the base space X. Assume 1. a+ z (1; x) is -integrable. 2. a+ (1; x) is -integrable for every dominant weight Then there exists a mensurable set ( ) = 1 such that x 2 is regular.
for G.
X invariant by the ‡ow with
For the proof we apply Khaimanovich criterion of regular sequences. Namely if write k (r) = rk hk vk with r; rk 2 R, hk 2 clA+ and vk 2 K, then the two conditions to be checked are the existence of the limit lim
1 1 log hk = lim a+ (k; x) k k k
x=
(r)
and the sublinear growth of sk = vk 1 hk vk . + Now, we split a+ (k; x) = a+ as . Since a+ z (k; x) + ax (k; x) 2 az z (1; x) + is an additive cocycle, by the integrability of fz (x) = az (1; x) we can apply the ergodic theorem of Birkho¤ to get the almost sure existence of the limit 1X 1 lim a+ (1; x) = lim fz ( k z k n=0 k 1
22
n
(x)) :
On the other hand if is a dominant weight for G then a+ (k; x) is a subadditive cocycle by Proposition 3.4. Hence the integrability assumption allows to apply the subadditive ergodic theorem to conclude that k1 a+ (k; x) converges for -almost all x. Since the set of dominant weights for G generates the dual as , it follows that k1 a+ s (k; x) also converges almost surely. 1 + Hence k a (k; x) converges almost surely. To get the sublinear growth 1 d (o sk ; o sk+1 ) ! 0 k of sequences sk we …rst do the following computation. Proposition 5.8 With the above notation we have d (o sk ; o sk+1 ) = a+ (1; k x)
x=
(r)
where o is the origin of KnG. Proof: Take k 2 N. Then k+1 (r) = rk+1 sk+1 and also k+1 (r) = 1 ( k (r)) = 1 (rk sk ) = 1 (rk ) sk . The Cartan decompsition of 1 (rk ) reads 1 (rk ) = r0 s with s = v exp a+ (1; k x) v
1
for some v 2 K. Hence k+1 (r) = r0 ssk . To relate ssk with sk+1 write the Cartan decomposition ssk = ut 2 KS. Then k+1 (r) = (r0 u) t and since k+1 (r) = rk+1 sk+1 it follows by the uniqueness of the decomposition that sk+1 = t. Hence o t = o sk+1 and o ssk = o sk+1 , so that d (o sk ; o sk+1 ) = d (o sk ; o ssk ) = d (o; o s) : Finally, d (o; o s) = ja+ (1; k x)j, concluding the proof. Now an application of Birkho¤ ergodic theorem to f (x) = a+ (1; x) and fz (x) = a+ z (1; x) shows that -almost surely the limit lim k
1 (f (x) + k
23
+ f (k x))
exists, where f (x) = a+ (1; x). This implies that lim k1 f (k x) = 0 hence 1 1 + d (o sk ; o sk+1 ) = a (1; k x) ! 0; k k showing the sublinear growth ratio of a sequence sk de…ned by (r) = rk sk with r ranging through a subset of R projecting onto a set of full -measure. This concludes the proof of Theorem 5.7, and hence the existence of the set with full -measure where + (x) exists, as in the statement of the multiplicative ergodic theorem.
5.4
Oseledets …ltration
Let be the set of full -measure ensured by Theorem 5.7. The elements of e = 1 ( ) \ R. For a regular element are regular. So are the elements of R e we write k (r) = rk sk and sk = vk hk v 1 with rk 2 R, sk 2 S, r 2 R k hk 2 clA+ and vk 2 K. Also put D (r) = lim
1 log sk k
+
(r) = lim
1 log hk ; k
where the limits exists by regularity. In addition, D(r) = Ad(u)
+
(r);
(3)
for some u 2 K. In fact, we can write sk = exp (Ad(vk )(log hk )) hence lim k1 log sk exists by the the convergence of vk ! u 2 K and k1 log hk . By Proposition 5.1 we have the Lyapunov exponent (r b) = w
1
+
(r)
b 2 st (D (r) ; w)
(4)
where w 2 W. Thus we have well de…ned maps D:e!s
and
+
: e ! cla+
that are combined to give the a-Lyapunov exponents as in (4). e Thus it By Proposition 3.2, the map + is constant on the …bers of R. + + de…nes a map : ! cla which is invariant by the semi‡ow, as ensured by the subadditive ergodic theorem. Also, + is measurable. 24
On the other hand, the map D satis…es D (ru) = Ad u as follows by
k
1
D (r)
(5)
e and u 2 K then (ru) = rk u u 1 sk u. Therefore if r 2 R r st (D (r) ; w) = ru st (D (ru) ; w)
because for E 2 s the stable set st (Ad (u 1 ) E; w) = u 1 (st (E; w)). In other words the set r st (D (r) ; w) E stays the same as r runs throughout a …xed …ber. e is in the …ber over x, We write st (x; w) = r st (D (r) ; w) where r 2 R and for w 2 W put [ st (w) = st (x; w) E: x2
De…nition 5.9 We say that st (w), w 2 W, are the Oseledets components S of the Oseledets decomposition e = w2W st (w) of e = E 1 ( ). By (4) the a-Lyapunov exponents are given by ( ) = w 1 + (x) if 2 st (w) where x = ( ). It follows that an Oseledets component st (w) is measurable, since st (w) = f 2 e : ( ) = w 1 + ( ( ))g. Moreover, e is the disjoint union of the components st (w) that are di¤erent. (We discuss below the possible equalities st (w1 ) = st (w2 ), w1 ; w2 2 W.)
Proposition 5.10 st (w) is invariant by the ‡ow in E. Proof: This is due to the fact that is invariant by the ‡ow on E in the sense that if ( ), 2 E, exists then ( s ( )), s 2 Z+ , also exists and they are equal. To see this use the cocycle property of a (t; ) to get 1 a (t; t
s
1 ) = a (t + s; ) t
1 a (s; ) : t
The last term ! 0 when t ! +1 and the …rst term in the right hand side satis…es 1 t+s 1 a (t + s; ) = a (t + s; ) ! ( ) : t t t+s This implies the invariance of . Now st (w) is a level set of and hence is also invariant. 25
Remark: Formula (5) means that D is an equivariant map. Therefore it gives rise to a section , above , of the associated bundle R K s, obtained by the adjoint action of K on s. The section : ! R K s is de…ned by (x) = r D (r)
(r) = x:
Remark: Despite the fact that the stable sets st (w) are invariant it is not true in general that the map D and the corresponding section are invariant by the semi‡ow. Below we describe invariant sections of suitable bundles.
5.5
Flag (parabolic) type
Given a regular point x 2 Ly
(x) =
we write Ly
(x; ) = f 2
:
+
(x) = 0g:
This subset of the simple system of roots de…ne a ‡ag manifold F Ly (x) . We call the ‡ag manifold F Ly (x) (or equivalently the subset Ly (x)) the ‡ag type of at x. We use the term parabolic type as a synonymous to ‡ag type. When the measure is ergodic + (x) = + and Ly (x) = Ly are constant. In this case we say simply that Ly ( ) is the ‡ag type of . Recall that the dual F of a ‡ag manifold F is given by = w0 , where w0 is the only element of the Weyl group mapping a+ into a+ . Proposition 5.11 Given a regular point x 2 , let E Ly (x) = Q G F Ly (x) dual to the ‡ag type at x. Write : E ! E Ly (x) for the standard map. Then (st (x; w0 )) reduces to a point denoted by (x), that is, (st (x; w0 )) = f (x)g. Proof: By de…nition st (x; w0 ) = r st (D (r) ; w0 ) for any r in the …ber over x. By (3) we have D (r) = Ad (u) + (x) for some u 2 K, so that st (D (r) ; w0 ) = u st
+
(x) ; w0 :
Now st ( + (x) ; w0 ) F is the repeller …xed point set x ( + (x) ; w0 ) of H + (x) by the choice of w0 , which projects to a point in the dual ‡ag manifold F Ly (x) . Since (st (x; w0 )) = r (st (D (r) ; w0 )), it follows that (st (x; w0 )) 26
is a singleton. The point (x) in (st (x; w0 )) given by this proposition associates to x 2 an element of the ‡ag bundle E Ly (x) . We call the Oseledets section of the semi‡ow. Note that it is not properly a section of a ‡ag bundle, because Ly (x) changes with x. However when Ly (x) = Ly is constant on (for instance when is ergodic) the map : ! E Ly is a section of the bundle E Ly ! X. This section is measurable because st (w0 ) is a measurable set.
6
Flows: invertible maps
In this section we consider a ‡ow n , n 2 Z, of automorphisms of the principal bundle : Q ! X. Again we let be a probability measure on the base space assuming now that it is invariant by both n and n = ( n ) 1 . Throughout this section we let be a set of full -measure such that every x 2 is regular for .
6.1
Backward ‡ow
The …rst step is to relate the polar exponent and the a-Lyapunov exponents of ( n ) 1 to those of n that is the Lyapunov exponents of n when n ! 1. As before we take a K-reduction R and denote by R n the ‡ow induced 1 R 1 R on R. Its inverse is n = n , the ‡ow induced by the inverse of . Taking the Iwsawa decomposition decomposition Q = R S we get two sequence of maps tn ; e tn : R ! AN given by n
(r) =
R n
(r) tn (r)
n
Proposition 6.1 b tn (r) = tn
(r) =
R n
(r)
R
n 1
(r) b tn (r)
r 2 R;
.
Proof: We have r = n n (r) = n Rn (r) b tn (r) = by right invariance of the ‡ow . Hence r=
Therefore tn
R n
R
n
R n
(r)
tn
R
n
(r) b tn (r) = r tn
(r) b tn (r) = 1, that is, b tn (r) = tn 27
n 2 N:
R
n R n
R n
n
(r)
(r) b tn (r) : (r)
1
b tn (r)
as claimed.
Now, write b tn (r) = b an (r) zbn
tn (r) = an (r) zn
with an (r) ; b an (r) 2 A and zn (r) ; zbn (r) 2 N , so that the a-cocycle is a (n; ) = log an (r), already factored to the ‡ag bundle E. We put b a (n; ) = log b an (r), for the backward cocycle and for its Lyapunov exponent.
b ( ) = lim 1 b a (n; ) n
a (n; ) = Proposition 6.2 b
a n;
Proposition 6.3 b ( ) =
( ) if
n
( ) ,
2 E, n 2 N.
1 1 Proof: We have b an (r) zbn = b tn (r) = tn Rn (r) = zn 1 an Rn (r) . 1 This last term can be writen as an Rn (r) with 2 N , after multi1 plying and dividing it on the left by an Rn (r) . By uniqueness of the 1 Iwasawa decomposition it follows that b an (r) = an Rn (r) . Hence the claim follows by taking logarithms (and factoring to the ‡ag bundle).
2 E.
Proof: The function gn ( ) = a (n; ) is additive with respect to , that is, gn+m ( ) = gn ( m ( )) + gm ( ). Put fn ( ) = b a (n; ). Then fn is also additive and by Proposition 6.2 we have fn ( ) = gn n ( ) . Hence f2n ( ) = fn ( ) + fn ( n ( )) = fn ( ) + gn ( ). Taking limits we get b ( ) = lim 1 f2n ( ) = lim 1 2n 2
which means that
b( ) =
1 1 fn ( ) + gn ( ) n n
( ).
=
1 2
b( )+ ( ) ;
We denote by b + (x) the polar exponent of the backward ‡ow, when x is a regular point. Corollary 6.4 b + (x) = such that w0 a+ = a+ .
w0
+
(x), where w0 is the only element of W 28
Proof: In fact, the Lyapunov exponents in the …ber above x are the Wtranslates w + (x), w 2 W, of the polar exponent, which in turn is the only Lyapunov exponent that lies on the positive chamber cla+ . Now w0 + (x) is a Lyapunov exponent of , hence by the above proposition w0 + (x) is a Lyapunov exponent of the reversed ‡ow. Since w0 + (x) 2 cla+ , the claim follows.
Corollary 6.5 The ‡ag type of the inverse ‡ow at x is the dual the ‡ag type Ly (x) of the forward ‡ow.
Ly
(x) of
Proof: The ‡ag type of is de…ned by the subset Ly (x; ) = f 2 : 1 + ( (x)) = 0g, while that of is the set of simple roots annihilating b + (x) = w0 + (x). Hence Ly x; 1 = w0 Ly (x; ), which by de…nition is Ly (x). This corollary combined with Proposition 5.11 yields immediately the following section on the bundle of the ‡ag type of . Corollary 6.6 Given the standard map : FQ ! F Ly(x) Q. Then, Ly(x) for each x 2 , the image of st (x; w0 ) by reduces to a single point Ly(x) F Ly(x) Q, which will be denoted by (x). That is, Ly(x)
(st (x; w0 )) = f (x)g:
Finally, by Proposition 5.10, the sets st(w) are the invariance (x) and (x). Corollary 6.7 For each x 2 , (x) and When E Ly and
6.2
t -invariant,
which implies
(x) are invariant.
is ergodic, we have two bona …de measurable sections : ! E Ly .
:
!
Transversality
By Corollary 6.5 the Oseledets section of the inverse ‡ow takes value in the ‡ag bundle E Ly (x) dual to E Ly (x) . We denote this section by , while is the Oseledets section of the forward ‡ow. We shall prove now that these two sections are transversal in the sense we de…ne now. 29
First let F and F be dual ‡ag manifolds. Let b be the origin of F = G=P = K=K . Then b1 2 F and b2 2 F are transversal in case the pair (b1 ; b2 ) belongs to the open orbit of the diagonal action of G on F F (see [13], [14]). This transversality concept can be plugged …berwise into the ‡ag bundles E and E . Precisely, a point q 2 Q de…ne bijections between F and F with the …bers of E and E , respectively. The bijections associate to b in a ‡ag manifold the element q b in the ‡ag bundle. Then we say that 1 = q b1 2 E and 2 = q b2 2 E are transversal if b1 2 F is transversal to b2 2 F . This is independent of q in its …ber because if q = pg, g 2 G, then q bi = p gbi , i = 1; 2, and b1 and b2 are transversal if and only if gb1 is transversal to gb2 (see [13], for further details). To prove that the Oseledets sections (x) 2 E Ly and (x) 2 E Ly are transversal we use the following lemma about transversality in ‡ag manifolds. Lemma 6.8 Let H be a characteristic element for , that is, = f 2 : (H) = 0g. Then ( x (H; w0 )) = fb g, and the set of elements transversal to b = ( x(H; w0 )) is the open Bruhat cell st (H; 1) F . Proof: The nilradical n of the parabolic subalgebra associated to b X g
is
