JORDAN NORMAL FORM FOR LINEAR COCYCLES

JORDAN NORMAL FORM FOR LINEAR COCYCLES LUDWIG ARNOLD, NGUYEN DINH CONG, AND VALERY IUSTINOVICH OSELEDETS Abstract. The paper is devoted to the problem of classi cation of linear cocycles up to cohomology. The main result is a theorem on the Jordan normal form saying that any linear cocycle is cohomologous to a block-triangular cocycle with irreducible block-conformal cocycles on the diagonal. Two invariants of cocycle cohomology, the algebraic hull and the set of invariant measures, and their interrelations are studied. We show that all random invariant measures of a cocycle are determined by the algebraic hull and, up to a cohomology, are deterministic. For orthogonal cocycles the two invariants are equivalent and they give a sub-relation of the equivalence relation of cocycle cohomology. A complete classi cation of the one- and two-dimensional linear cocycles is given. Our results are re nements of the multiplicative ergodic theorem of Oseledets, as we are able to describe the structure of a linear cocycle inside the invariant subspaces corresponding to dierent Lyapunov exponents. A by-product of our theory is a classi cation of amenable Lie subgroups of ( R). Gl d;

Contents

1. Introduction 2 2. Decomposition of linear cocycles 4 3. The algebraic hull and invariant measures of a cocycle 12 3.1. De nitions 12 3.2. Relation between algebraic hull and invariant measures 13 3.3. Structure of invariant measures 14 3.4. Criterion for orthogonal and conformal cocycles 16 4. Block-conformal cocycles 18 4.1. The linear cover of a subset of S d?1 18 4.2. Structure of a cocycle on the span of the support of an invariant measure. Block-conformal cocycles 25 5. The Jordan normal form 32 5.1. The Jordan form 32 5.2. An algorithm for constructing the Jordan normal form 35 6. Orthogonal cocycles 37 6.1. The algebraic hull of an orthogonal cocycle 37 6.2. Invariant measures of orthogonal cocycles 38 6.3. Equivalence of orthogonal cocycles 39 6.4. About classi cation of orthogonal cocycles 40 7. Classi cation of low-dimensional cocycles 41 7.1. Classi cation of one-dimensional linear cocycles 41 7.2. Classi cation of two-dimensional linear cocycles 41 1991 Mathematics Subject Classi cation. Primary 58F36, 58F35; Secondary 58F11, 28D05. Key words and phrases. Jordan normal form, cocycle, classi cation, cohomology, algebraic hull, invariant measure. 1

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8. Relation to the multiplicative ergodic theorem Acknowledgment References

44 51 51

1. Introduction Let ( ; F ; P) be a probability space, and an automorphism of ( ; F ; P) preserving the probability measure P. Throughout this paper, we assume that is ergodic. The non-ergodic case can be reduced to the ergodic one by using the ergodic decomposition of dynamical systems if available (see Cornfeld at al. [4]). Consider a linear random map A() : ! Gl(d; R), i.e. A is a measurable mapping from the probability space ( ; F ; P) to the Lie group Gl(d; R) (for short: Gl(d)) of linear nonsingular operators of Rd equipped with its Borel -algebra. It generates a linear cocycle over the dynamical system via 8 A(n?1 !) : : : A(!); n > 0; < n = 0; A (n; !) := : id ; A?1 (n !) : : : A?1 (?1 !); n < 0: Conversely, if we are given a linear cocycle over , then its time-one map is a linear random map. Therefore, the correspondence between A and A is one-to-one and we are free to choose one from them to work with. The above construction applies to any topological group G in place of Gl(d) (in particular, G can be a Lie subgroup of Gl(d)), and we shall speak of G-cocycles and random G-map in that case. We shall look at linear cocycles as linear operators of Rd and identify linear operators with their matrix representations in the standard Euclidean basis of Rd . This applies equally to G-cocycles for a Lie subgroup G Gl(d). The space of all Gl(d)-cocycles will be denoted by G (d). Since we deal with discrete-time cocycles we can always neglect sets of null measure, and we shall identify the random mappings which coincide P-almost surely. We often omit the \P-almost surely" in equations between random variables. De nition 1.1. Two G-cocycles A and B are called G-cohomologous if there exists a random G-map C such that for almost all ! 2

B (!) = C (!)?1 A(!) C (!): In this case C is called a G-cohomology and we write A B . The main result of this paper is the Jordan normal form for linear cocycles (Theorem 5.6): We show that any linear cocycle A is reduced by means of a cohomology to block-triangular form with block-conformal irreducible subcocycles on the diagonal; these subcocycles are uniquely, up to ordering, determined by A. Y. Guivarc'h and A. Raugi [13] considered cocycles which are products of independent and identically distributed random matrices satisfying the integrability conditions of the multiplicative ergodic theorem and an additional condition called \total irreducibility". They proved that such cocycles are Lyapunov cohomologous to block-diagonal cocycles with conformal blocks. We would like to mention here a closely related paper of R. Zimmer [33], where he proved that every cocycle is cohomologous to a cocycle taking values in an amenable subgroup. We note that in the proof of Theorem 5.5 of [33] Zimmer has

JORDAN NORMAL FORM FOR LINEAR COCYCLES

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used a conformal structure on the ags which is somehow related to our blockconformal structure, but he did not derive a block-triangular form with blockconformal subcocycles on the diagonal, not to say a Jordan form (we note that the normal form problem was not his aim). Our Jordan form theory gives another proof of Zimmer's result about amenability. As a by-product we obtain the Jordan normal form of amenable subgroups of Gl(d; R), hence a classi cation of them (Theorem 5.10). Another major result of this paper is the description of all invariant measures of a linear cocycle: in Section 3 we show that up to a cohomology all invariant measures are deterministic Lebesgue measures of algebraic manifolds. The key tools for our work are Furstenberg's lemma (Lemma 3.20), Zimmer's construction of linear cover (see Section 4) and techniques from representation theory (Section 2). This paper is organized as follows. In the remaining part of the introduction we prove a useful rst reduction of cocycles. In Section 2 we present the techniques of representation theory applied to cocycles and derive a preliminary version of the normal form, namely a block-triangular form with irreducible cocycles on the diagonal. It remains to show that irreducible cocycles are block-conformal. To do so we need to study the algebraic hull and invariant measures of a cocycle (Section 3). In Section 4 we derive the block-conformal form on the span of an invariant measure. In Section 5 we present the nal Jordan normal form and an algorithm for constructing it. Since the block-conformal form suggests to study conformal and orthogonal cocycles we devote Section 6 to the investigation of orthogonal cocycles for which a better classi cation result is obtained. The remaining two sections are devoted to the complete classi cation of one- and two-dimensional cocycles and to the relation of our results with the multiplicative ergodic theorem of Oseledets. Finally, we would like to mention an application of our work here: In a forthcoming paper [2] we use the normal form theory to prove that the cocycles with simple Lyapunov spectrum are L1 -dense in the space of all linear cocycles. We remark that the theory of Jordan form presented here is a re nement of the multiplicative ergodic theorem of Oseledets [23]. The ag (and splitting) we obtain is ner than the one of Oseledets; moreover, no integrability condition is required. Under the corresponding integrability conditions the theory developed in this paper facilitates another proof of the multiplicative ergodic theorem (more precisely, simpli es Oseledets' original proof [23]), by applying Oseledets' arguments to the Jordan normal form. Note that we need not extend cocycles by means of the orthogonal group as we already have (block-)triangular form with block-conformal subcocycles on the diagonal. In concluding the introduction we show that the general case of a linear cocycle can be easily reduced to the case of a unimodular cocycle. We rst give a general reduction theorem and then apply it to the case of the unimodular group. Let G be a Lie group with identity e and H; L G be Lie subgroups such that G = H L and H \ L = feg. Then any element g 2 G is uniquely represented in the form g = hl with h 2 H and l 2 L. Therefore, a G-cocycle A : ! G is uniquely represented in the form of the product of an H -cocycle AH and an L-cocycle AL such that A(!) = AH (!) AL (!) for all ! 2 .

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Lemma 1.2. Assume the above decomposition of G, and moreover that any h 2 H commutes with any l 2 L. Let A and B be two G-cocycles. Then A is G-

cohomologous to B if and only if AH is H -cohomologous to BH and AL is Lcohomologous to BL. Proof. If A is G-cohomologous to B then there is a measurable map C : ! G such that, for all ! 2 , A(!) = C (!)?1 B (!)C (!): Letting C (!) = CH (!)CL (!) be the decomposition of C according to G = H L, since any element of H commutes with any element of L this implies that AH (!) = CH (!)?1 BH (!)CH (!); AL (!) = CL (!)?1 BL(!)CL (!): The converse statement is clear. We denote by Sl(d) the Lie group of (d d)-matrices with determinant 1 (note that the value ?1 is allowed). Let R+ denote the (Abelian) multiplicative group of positive real numbers. Clearly, Gl(d) = Sl(d) R+ . Note that R+ is in the center of Gl(d), hence every element of it commutes with every element of Sl(d). Therefore, Lemma 1.2 is applicable. Corollary 1.3. Two Gl(d)-cocycles A and B are cohomologous if and only if the R+ -cocycles j det Aj and j det B j are cohomologous and the Sl(d)-cocycles j det Aj?1=d A and j det B j?1=d B are cohomologous.

2. Decomposition of linear cocycles In this section we show that the study of general linear cocycles can be reduced to the one of irreducible cocycles. The development is largely parallel to the theory of linear representations of groups, for which we refer to the textbooks by Kirillov [17] and Vinberg [29] for details. The analogy between the theory of cocycles and the theory of representations was observed by Mackey [19] and Zimmer [31, 32]. The aim of this section is to derive a version of the Jordan{Holder theorem for cocycles (Theorem 2.17). The long and rather tedious preparations serve to provide a mathematically rigorous base for Theorem 2.17. First we need the following notion of closed random sets and random subspaces. De nition 2.1. A map C : ! 2Rd taking values in the collection of all closed subsets of Rd is called a closed random set if for all x 2 Rd the function ! 7! d(x; C (!)) is measurable, where d(x; C (!)) := inf y2C (!) kx ? yk. C is called a random subspace if in addition C (!) is a linear subspace of Rd for any ! 2 . Proposition 2.2. C is a closed random set if and only if there exists a sequence cn , n 2 N , of measurable maps cn : ! Rd such that C (!) = fcn(!) j n 2 N g for all ! 2 ; where A denotes the closure of A Rd . In particular, if C is a closed random set then there exists a measurable selection, i.e., a measurable map c : ! Rd such that c(!) 2 C (!) for all ! 2 .


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For the proof see Castaing and Valadier [3, Theorem III.9, p. 67]. For random subspaces we can choose measurably an orthonormal basis as the following result of Walters [30] shows. Proposition 2.3. Let V be a random subspace. Then (i) r(!) := dim V (!) : ! N is measurable, and for each 1 k d, ! 7! V (!) is a measurable map from f! 2 j r(!) = kg into the Grassmannian manifold Grk (d) of all k-dimensional subspaces of Rd equipped with its Borel -algebra. (ii) For each 1 k d there are measurable maps v1 ; : : : ; vk : f! 2 j r(!) = kg ! Rd such that fv1 (!); : : : ; vk (!)g is an orthonormal basis of V (!). For more information on random sets we refer to Walters [30] and Crauel [5]. De nition 2.4. Let A 2 G (d). A random subspace U Rd is called invariant with respect to A if A(!)U (!) = U (!) for almost all ! 2 . Remark 2.5. Since A(!) preserves dimension and is ergodic, for any invariant random subspace U , the dimension dim U (!) does not depend on ! 2 . Let E Rd be a subspace and Rd =E the quotient subspace of Rd over E , which has dimension d ? dim E . By choosing a basis fe1; : : : ; edg of Rd such that fe1; : : : ; er g is a basis of E we can identify, by means of a linear isomorphism, Rd =E with the linear subspace of Rd spanned by fer+1 ; : : : ; ed g. If E F Rd are linear subspaces the quotient subspace F=E can be identi ed, by means of a linear isomorphism, with a linear subspace of F Rd of dimension dim F ? dim E . If U V Rd are random subspaces, then V=U de ned !-wise is a random linear space and can be identi ed, by means of a random linear isomorphism, with a random subspace of V Rd . Now let A 2 G (d) and U Rd be an invariant random subspace of A of dimension r. Then A induces a linear cocycle AU on U by AU (!)x := A(!)x, x 2 U (!), and a linear cocycle on Rd =U by ARd=U (!)y := A(!)y, y 2 Rd =U (!). Take a random basis f := ff1 (!); : : : ; fd(!)g of Rd such that ff1 (!); : : : ; fr (!)g is a basis of U (!) for all ! 2 . In this basis the cocycle A has the form A = A10(!) A (!) ; 2 where A1 is an r-dimensional random matrix and A2 a (d ? r)-dimensional random matrix. Obviously, A1 2 G (r) and A2 2 G (d ? r). Clearly, another choice of such a random basis f leads to new cocycles A01 2 G (r) and A02 2 G (d ? r) which are cohomologous to A1 and A2 , respectively. Moreover, there are random linear isomorphisms C1 (!) : U (!) ! Rr and C2 (!) : Rd =U (!) ! Rd?r mapping AU and ARd=U to A1 and A2 , i.e. for all ! 2 , AU (!) = C1 (!)?1 A1 (!) C1 (!); ARd=U (!) = C2 (!)?1 A2 (!) C2 (!): Thus we can identify AU and ARd=U with A1 and A2 , and arrive at the following de nition. De nition 2.6. The cocycle A1 2 G (r) is identi ed with AU and called the restriction of A to U ; its cohomology class is uniquely determined by A and U . The cocycle A2 2 G (d ? r) is identi ed with ARd=U and called the quotient cocycle of A over U ; its cohomology class is uniquely determined by A and U .

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Now let U and V be invariant random subspaces of A such that U (!) V (!) for all ! 2 , dim U (!) = r, dim V (!) = p, 0 r < p d. Then the cocycle B := AV on V is de ned and U is an invariant random subspace of B . Identifying V with Rp by means of a random linear isomorphism we have B 2 G (p) and U is transformed into an r-dimensional invariant random subspace of B in Rp , which we again denote by U , hence the quotient cocycle BRp=U 2 G (p ? r) is de ned and we call it the quotient cocycle of A on V=U and denote it by AV=U . It is easily seen that the cohomology class of the cocycle AV=U 2 G (p ? r) is uniquely determined by A, U and V . Obviously, AU = BU = (AV )U : All the above discussion becomes transparent as we choose a suitable random basis and reduce cocycles to the block-triangular form. If U and V are random spaces then U = V means that there is a random linear isomorphism between them. The following two lemmas expose the construction of quotient cocycles. Lemma 2.7. Let A; B 2 G (d) and U1; U2 Rd be invariant random subspaces of A such that U1 (!) U2 (!) for all ! 2 , dim Ui (!) = ri , i = 1; 2. Assume that there is a measurable map C : ! Gl(d) such that (1) A(!) = C (!)?1 B (!)C (!) for all ! 2 : Then: (i) Vi (!) := C (!)Ui (!) are random subspaces invariant with respect to B , dim Vi (!) = ri , i = 1; 2, and V1 (!) V2 (!) for all ! 2 ; (ii) AUi BVi as Gl(ri )-cocycles, i = 1; 2; (iii) AU =U BV =V as Gl(r2 ? r1 )-cocycles. Proof. Part (i) follows immediately from (1). Parts (ii) and (iii) follow from (1) by choosing appropriate random bases. Lemma 2.8. Let A 2 G (d). (i) If f = ff1 (!); : : : ; fd(!)g is a random basis of Rd in which the cocycle A has the matrix form A (!) 1 A= 0 A (!) 2

1

2

1

2

for some A1 2 G (r) and A2 2 G (d ? r), 1 r < d, then U (!) := spanff1(!); : : : ; fr (!)g is an r-dimensional invariant random subspace of A, and A1 = AU and A2 = ARd=U . Conversely, if U is an r-dimensional invariant random subspace of A, and f = ff1 (!); : : : ; fd(!)g is a random subspace of Rd such that spanff1(!); : : : ; fr (!)g = U (!) for all ! 2 , then in the basis f the cocycle A has the matrix form AU (!) 0 ARd=U (!) . (ii) Let U V W Rd be invariant random subspaces of A. Then AW=V = A(W=U )=(V=U ) : (iii) Let U; V; W be invariant random subspaces of A such that span(V (!) [ U (!)) = Rd and V (! ) W (! ) for all ! 2 . Then AW=V = A(W \U )=(V \U ) :


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Proof. Part (i) is obvious. Parts (ii) and (iii) can be easily seen by choosing appropriate random bases in Rd which are \adapted" to U; V; W in an obvious sense. Let A 2 G (d) have the form A = A10(!) A 0(!) 2 with A1 2 G (r) and A2 2 G (d ? r). Clearly, A1 and A2 are also the quotient cocycles of A over Rd?r = Rd =Rr and Rr = Rd =Rd?r , respectively. In this case we say that A is the direct sum of A1 and A2 and write A = A1 A2 . Now, if we are given two cocycles B1 2 G (r) and B2 2 G (d ? r), then the direct sum B = B1 B2 is de ned by B = B10(!) B 0(!) for all ! 2 . Clearly, B 0 := B2 B1 2 G (d) 2 is cohomologous to B . It is easily seen that if U and V are two invariant random subspaces of A such that U (!) V (!) = Rd for all ! 2 , then A A1 A2 , where A1 is the restriction of A to U and A2 is the restriction of A to V . Moreover, A1 is the quotient cocycle of A over V and A2 is the quotient cocycle of A over U . De nition 2.9. Let A 2 G (d). (i) The cocycle A is called irreducible if no random proper subspace of Rd is invariant under A. If A is not irreducible then we call A reducible. (ii) We say that A is strongly irreducible if no nite union of random proper subspaces of Rd is invariant under A. (iii) The cocycle A is called completely reducible if any invariant random subspace U of A admits invariant complement, i.e. there is an invariant random subspace U c of A such that U (!) U c (!) = Rd for all ! 2 . (iv) An invariant random subspace U of A is called minimal if there is no nontrivial invariant random subspace V of A such that V (!) is a proper subspace of U (!) for almost all ! 2 . The following lemma is immediate. Lemma 2.10. (i) The notions of irreducibility, strong irreducibility and complete reducibility are invariant with respect to cohomology. (ii) Strong irreducibility implies irreducibility; the converse assertion is false. Now we give here some elementary decomposition properties of cocycles. Lemma 2.11. Let A 2 G (d). Then (i) An r-dimensional invariant random subspace U of A is minimal if and only if AU is an irreducible Gl(r)-cocycle. In particular, A is irreducible if and only if Rd is minimal. (ii) A is completely reducible if and only if it is the direct sum of irreducible subcocycles. (iii) A is completely reducible if and only if Rd is the direct sum of some minimal invariant random subspaces of A. Proof. (i) is immediate from the de nition of minimal random subspace and of the restriction of a cocycle to an invariant random subspace. By virtue of (i), (ii) is equivalent to (iii), hence it remains to prove (iii). For the \if" part, let Rd = U 1 U n ;

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where Ui , i = 1; : : : ; n, are minimal subspaces of A. Let V be an arbitrary invariant random subspace of A. Put Vi := V \ Ui , i = 1; : : : ; n. Then Vi is an invariant subspaces of A contained in Ui . Since Ui is minimal Vi must either be null space or coincide with Ui . Therefore, V is the direct sum of some of Ui , i = 1; : : : ; n. Clearly the direct sum of those Ui , i = 1; : : : ; n, which do not enter the sum of V is an invariant subspace of A which is a complement of V in Rd . To prove the \only if" part, we proceed by induction on the dimension d of the completely reducible linear cocycles. The case d = 1 is trivial. Suppose we are done for all dimensions d ? 1. Let A 2 G (d) be completely reducible. If A is irreducible, then Rd is minimal and the sum reduces to one term. If A is reducible, then take a minimal subspace U of A of dimension 1 r < d. Since A is completely reducible, there is an invariant complement V of U for A. It is easily seen that AV is completely reducible, hence by the inductive hypothesis V is the direct sum of minimal subspaces of AV which are also minimal subspaces of A. Adding U to this sum we obtain that Rd is the direct sum of minimal subspaces of A. Lemma 2.12. Let A 2 G (d). Assume that Rd is a (not necessarily direct) sum of some minimal invariant random subspaces of A: Rd = U 1 + + U n : Let U be an arbitrary invariant random subspace of A. Then there are indices i1 ; : : : ip 2 f1; : : : ; ng such that (2) Rd = U U i U i p : In particular, Rd itself is the direct sum of some Ui , hence A is completely reducible. Proof. Let U Rd be an invariant random subspace of A. For ! 2 let 1 i1 (!) < i2 (!) < < ip(!) (!) n be the maximal set of indices such that the subspaces U (!), Ui (!) (!); : : : ; Uip ! (!) (!) are linearly independent. Since the linear operator A(!) is nonsingular and the subspaces U; U1 ; : : : ; Un are invariant we obtain that p() is -invariant, hence constant. Since there are only n possible indices for Ui it is easily seen that i(!) can be chosen independently of ! 2 . Thus we write i1 ; : : : ; ip instead of i1(!); : : : ; ip(!) (!). To prove (2) it suces to show that for any 1 i n, i 62 fi1; : : : ; ip g, (3) Ui (!) U (!) Ui (!) Uim (!) for all ! 2 : ? Since Ui (!) \ U (!) Ui (!) Uim (!) =: Wi (!) is an invariant subspace of A contained in Ui (!) it must either be null space or coincide with Ui (!). The maximality of p implies that Wi (!) is nontrivial, hence Wi = Ui , which proves (3). Proposition 2.13. Let A 2 G (d). Assume that A is cohomologous to the direct sums A1 Am and A01 A0l of irreducible subcocycles of dimensions d1 ; : : : ; dm and d01 ; : : : ; d0l , respectively. Then l = m and, in a suitable labeling, di = d0i and Ai is cohomologous to A0i as Gl(di )-cocycles, i = 1; : : : ; m. Proof. By assumption Rd admits two decompositions into direct sums of minimal subspaces Rd = V 1 V m = U 1 U l 1

1

( )

1

1


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such that AVi Ai and AUj A0j , i = 1; : : : ; m, j = 1; : : : ; l. We prove the proposition by induction on m. If m = 1 then A is irreducible, hence l = 1 and A1 A01 . Suppose the proposition has been proved for m ? 1. We apply Lemma 2.12 to U1 and obtain Rd = U 1 V i V i p for certain i1 ; : : : ; ip . Then 1

A01 AU ARd=Vi Vip Ak Akq ; where fk1 ; : : : ; kq g = f1; : : : ; mg n fi1; : : : ; ip g. Since A01 is irreducible, q = 1. Now let us relabel Ai so that k1 = 1. Then A01 A1 and Rd = U 1 V 2 V m : 1

1

1

Therefore,

A2 Am ARd=U A02 A0l : The induction hypothesis implies that m = l and after a relabeling Ai A0i for i 2. De nition 2.14. Let A 2 G (d). Suppose that we have a strictly monotone collection of invariant random subspaces (invariant ag) of A: f0g = V0 V1 Vn?1 Vn = Rd such that the subcocycles Ai , i = 1; : : : ; n, appearing in Vi =Vi?1 |the quotient cocycles AVi =Vi? |are irreducible. Then the ag fVi g is called maximal (invariant)

ag (of A). The number n is called the length of the ag fVi g. The multi-index r = fr1; : : : ; rng, where ri := dim(Vi =Vi?1 ) = dim Vi ? dim Vi?1 , i = 1; : : : ; n, is called the index of the ag fVi g. We call any index of a maximal invariant ag of A an index of A. By choosing a random basis ff1(!); : : : ; fd(!)g of Rd such that the rst r1 + + ri basis vectors constitute a basis of the invariant random subspace Vi , i = 1; : : : ; n, we obtain that the cocycle A in this basis has block-triangular form with the subcocycles Ai on the diagonal. Lemma 2.15. Let A 2 G (d) be arbitrary. Then (i) A has at least one maximal invariant ag. (ii) Let U Rd be invariant random subspace of A and V0 Vk be a maximal invariant ag AU . Then there is a maximal invariant ag V00 Vk0+l of A such that Vi0 (!) = Vi (!) for all ! 2 and i = 0; : : : ; k. (iii) Let U Rd be invariant random subspace of A. Let V0 Vk and W0 Wl be maximal invariant ags of AU and ARd=U , respectively. Then A has a maximal ag V00 Vk0+l such that for i = 0; : : : ; k; Vi0 = VVik Wi?k for i = k + 1; : : : ; k + l: Proof. First we prove part (iii). Choose a random basis f = ff1(!); : : : ; fd(!)g of Rd such that spanff1 (!); : : : ; fri (!)g = Vi (!) forall ! 2 , i = 1; : : : ; k; here ri := dim Vi (!). By Lemma 2.8, A has the form AU0(!) A d (!) with 1

1

R =U

respect to the random basis f . It is easily seen that the basis f can be chosen to have, additionally, the property that the vectors ffk+1 (!); : : : ; fk+pj (!)g form

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a basis of Wj (!) (modulo a linear isomorphism) for all ! 2 and j = 1; : : : ; l; here pj := dim Wj (!). In this basis A has block-triangular form with irreducible subcocycles on the diagonal. Clearly the ag V00 Vk0+l de ned by Vi0 (!) = Vi (!) and Vk0+j (!) = spanff1 (!); : : : ; fk+pj (!)g for all ! 2 , i = 0; : : : ; k and j = 1; : : : ; l is a maximal invariant ag of A furnishing (iii), hence (iii) is proved. Part (ii) is a consequence of parts (i) and (iii), hence it remains to prove (i), which we do by induction on the dimension d of A. The case d = 1 is trivial. Suppose that we have proved (i) for all dimensions d ? 1, and A 2 G (d). Obviously A has at least one minimal invariant random subspace U Rd . If U = Rd then A is irreducible and the ag f0g Rd is maximal for A. If U 6= Rd , then AU and ARd=U are lower dimensional cocycles hence have maximal ags. Therefore, by (iii), A has a maximal ag. It is easily seen that a linear cocycle may have many maximal ags (e.g. diagonal cocycles) or only one maximal ag (e.g. irreducible cocycles), hence it may have several (but nitely many) indices. However, we have the following invariance of indices, which follows immediately from their de nition. Proposition 2.16. If A B and A has an index k then B has index k, too. Hence the indices of a cocycles are cohomology invariants. The following theorem is an analog of the Jordan{Holder theorem from the theory of representations (see Kirillov [17, Theorem 1, p. 116]) and is the main result of this section. Theorem 2.17. Let A 2 G (d), V0 Vn be a maximal invariant ag of A with index fr1 ; : : : ; rn g. Then (i) A is cohomologous to a block-triangular cocycle 1 0 A (!) V =V CA ; ... (4) AB @ 0 0 0 AVn =Vn? (!) where the cocycles AV =V ; : : : ; AVn =Vn? are irreducible. The form (4) depends only on the cohomology class of A. (ii) Suppose that W0 Wm is an arbitrary maximal invariant ag of A with index fq1 ; : : : ; qm g. Then m = n and, after a suitable relabeling, qi = ri and AVi =Vi? AWi =Wi? as Gl(ri )-cocycles for all i = 1; : : : ; n. In other words, all maximal ags of A have the same length which is a cohomology invariant, and the block-triangular form with irreducible subcocycles on the diagonal (4) of A is unique and determined by the cohomology class of A. Proof. (i) is obvious. We prove (ii) by induction on the length of the maximal invariant ag. Suppose that (ii) has been proved for all linear cocycles having a maximal invariant ag of length n ? 1. Let A 2 G (d) and V0 Vn be a maximal invariant ag of A. Put r := dim Vn?1 . Let W0 Wm be another maximal invariant ag of A. Put Wi0 = Wi \ Vn?1 ; i = 0; : : : ; m: Then Wi0 are invariant random subspaces of A and f0g = W00 W10 Wm0 = Vn?1 : 1

0

1

1

1

1

0

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We claim that there is an index j 2 f1; : : : ; mg such that Wj0?1 = Wj0 and that 0 00 k?1 is a proper subspace of Wk for k 6= j . To prove the claim, let Wi denote the image of Wi in the quotient space Rd =Vn?1 , i = 0; : : : ; m. It is easily seen that Wi00 (more precisely, it image in Rd?r by the random linear isomorphism mapping the quotient space Rd =Vn?1 into Rd?r ) is an invariant random subspace of the quotient cocycle ARd=Vn? . Since fVi g is a maximal invariant ag, the cocycle ARd=Vn? is irreducible, hence Wi00 either is the null space or coincides with Rd =Vn?1 . This implies that there is an index j 2 f1; : : : ; mg such that (5) W000 = : : : = Wj00?1 = f0g; Wj00 = : : : = Wm00 = Rd =Vn?1 : It is easily seen that, for any i = 1; : : : ; m, (6) Wi = (Wi \ Vn?1 ) Wi00 = Wi0 Wi00 : Therefore, (5) implies that for i 6= j we have Wi0 =Wi0?1 linearly isomorphic to Wi =Wi?1 . For the index j , since Wj?1 = Wj0?1 Wj0 Vn?1 and Wj = Wj0 Wj00 we have Wj =Wj?1 = (Wj0 =Wj0?1 ) Wj00 : Since the invariant ag fWi g is maximal and Wj00 6= f0g we must have Wj0 =Wj0?1 = f0g because the cocycle AWj =Wj? is irreducible and the lower dimensional random subspace Wj0 =Wj0?1 (modulo a random linear isomorphism) is invariant with respect to AWj =Wj? . Therefore, the sequence f0g = W00 W10 Wj0?1 Wj0+1 Wm0 = Vn?1 : is strictly monotone, and its image by the random linear isomorphism identifying Vn?1 with Rr is a maximal invariant ag of the cocycle AVn? . On the other hand, the image of the ag V0 Vn?1 by the random linear isomorphism identifying Vn?1 with Rr is another maximal invariant ag of the cocycle AVn? . Therefore, by the induction hypothesis, n = m and, up to their order, the cohomology classes of the cocycles (AVn? )Vi =Vi? , i = 1; : : : ; n ? 1, are the same as the cohomology classes of the cocycles (AVn? )Wi0 =Wi0? , i = 1; : : : ; j ? 1; j + 1; : : : ; m. By (5){(6) and Lemma 2.8, for i = 1; : : : ; j ? 1; j + 1; : : : ; m, we have (AVn? )Wi0 =Wi0? AWi0 =Wi0? AWi =Wi? : On the other hand, obviously (AVn? )Vi =Vi? AVi =Vi? for all i = 1; : : : ; n ? 1: Furthermore, clearly AWj =Wj? AVn =Vn? . Thus, to prove the theorem it remains to verify it for the case n = 1, which is trivial. Remark 2.18. (i) Theorem 2.17 allows us to reduce any linear cocycle A to the block-triangular form with irreducible subcocycles on the diagonal, where the cohomology classes of these subcocycles are uniquely determined by the cohomology class of A (up to their order). (ii) If A has an index fr1 ; : : : ; rn g then the subcocycles Ai of A, which come from an arbitrary maximal invariant ag of A, are (up to their order) of dimensions ri , i = 1; : : : ; n. Any two indices of A dier only by the order of their entries. We conclude this section by saying that the central task now is to classify the irreducible subcocycles on the diagonal of the form (4).

W0

1

1

1

1

1

1

1

1

1

1

1

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1

1

1

1

1

1

1

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3. The algebraic hull and invariant measures of a cocycle 3.1. De nitions. First we present the notion of the algebraic hull of a cocycle introduced by Zimmer, which is a useful cohomology invariant of linear cocycles. Here the notion of algebraic (i.e. closed in the Zariski topology) group is used. For the theory of algebraic groups we refer to Onishchik and Vinberg [22], and Humphreys [16]. Proposition 3.1. (Zimmer [35, Proposition 9.2.1]) Let G be a xed algebraic subgroup of Gl(d) and A be a G-cocycle. Then there exists an algebraic subgroup H of G such that A is G-cohomologous to a cocycle taking values in H but is not Gcohomologous to a cocycle taking values in a proper algebraic subgroup of H . The group H is unique up to conjugacy. De nition 3.2. (i) The conjugacy class in G of this algebraic group H is called algebraic hull of the G-cocycle A and is denoted by H(A). (ii) A G-cocycle A is called minimal if there is H 2 H(A) such that A(!) 2 H for all ! 2 . In this case we say that A is a minimal cocycle with range H . The following proposition is immediate. Proposition 3.3. Let G be an algebraic subgroup of Gl(d). (i) If A and B are two cohomologous G-cocycles, then H(A) = H(B ); (ii) Any G-cocycle is G-cohomologous to a minimal G-cocycle; (iii) Let A be a G-cocycle and H 2 H(A). Then A is G-cohomologous to a minimal cocycle with range H . The de nition of algebraic hull depends on the chosen algebraic group G. In this paper, unless otherwise speci ed explicitly (e.g., in Section 6) H(A) stands for the algebraic hull of a cocycle A in the space of Gl(d)-cocycles. Now, we turn to the notion of invariant measures of a cocycle. Let S d?1 denote the unit sphere of Rd . For a linear subspace V Rd we denote by [V ] its projection to S d?1, which is its intersection with S d?1 . In general, for a nonempty set f0g 6= Q Rd we denote by [Q] S d?1 its projection to S d?1 , hence [Q] is the subset of S d?1 consisting of those x 2 S d?1 for which rx 2 Q for some r > 0. Clearly, for 0 6= x 2 Rd , [x] = x=kxk. Let A 2 Gl(d). Denote by [A] the dieomorphism of S d?1 induced by A, i.e., [A](x) := [Ax]; for all x 2 S d?1 : We call [A] a linear transformation of S d?1. Clearly, [aA] = [A] for any a > 0, and [AB ] = [A] [B ] for any A; B 2 Gl(d). Denote by SGl(d) the group of all linear transformation of S d?1 . We have SGl(d) = Gl(d)=R+ I = Sl(d). d ? 1 Denote by P r(S ) the space of all probability measure on S d?1 equipped with the topology of weak convergence. Then P r(S d?1 ) is separable metrizable and compact since S d?1 has such properties (see Dellacherie and Meyer [6, p. 73]). It is well known that the topology of weak convergence on P r(S d?1 ) is generated by the following metric (see Dudley [7, x 11.3]): (; ) := inf f" > 0 j (U ) (U " ) + " for all U 2 B(S d?1)g; where U " := fy 2 G j ky ? xk < " for some x 2 U g and B(S d?1) denotes the Borel -algebra of S d?1. Fix this metric on P r(S d?1 ). Now lew A() : ! Gl(d) be a random linear map. Then [A()] is a random linear transformation of S d?1 and generates a (nonlinear) cocycle [A] on S d?1 .


13

Since S d?1 is compact there exists an invariant measure on ( S d?1; F B(S d?1)) for the skew-product of [A] which is de ned by : S d?1 ! S d?1, (!; x) 7! (!; [A(!)]x), i.e., = . Since S d?1 is a compact metric space can be disintegrated, and for P-almost all ! 2 we have a random probability measure ! on S d?1 such that (d!; dx) = ! (dx)P(d!): De nition 3.4. A measurable map : ! P r(S d?1 ); ! 7! ! ; is called invariant with respect to [A] if for P-almost all ! 2

! = [A(!)]! : This is equivalent to = . With a slight abuse of language we call an invariant measure of A, or of A as well as of [A] and of [A]. is called ergodic if it is ergodic with respect to . In order to obtain more information about a cocycle we study its action on dierent spaces. Therefore, for further use we generalize De nition 3.4 a little bit. Let X be a compact metrizable space and G Gl(d) a Lie subgroup. A continuous action of G on X is a group homeomorphism from G into the group Homeo(X ) of homeomorphisms of X such that the map G X ! X , (g; x) 7! (g)x, is continuous. In this case we call X a G-space. When a particular is not of importance or it is clear which action is meant we shall write simply gx instead of (g)x. If X is equipped with a metric %X which generates the given topology of X , then we call (X; %X ) a metric G-space. Denote by P r(X ) the space of all probability measure on X with the topology of weak convergence. Then P r(X ) is a compact metrizable space, since X is so. A G-cocycle A generates a skew-product dynamical system X : X ! X , (!; x) 7! (!; A(!)x). A matrix A 2 G acts on P r(X ) by the formula A(M ) := (A?1 M ) for any 2 P r(X ); M 2 B(X ). De nition 3.5. Let G Gl(d) be a Lie subgroup, X a G-space, and A a Gcocycle. A measurable map : ! P r(X ); ! 7! ! ; is called an invariant measure on X of A if for P-almost all ! 2

! = A(!)! : This is equivalent to the probability measure ! (dx)P(d!) being invariant with respect to X . We call ergodic if ! (dx)P(d!) is ergodic with respect to X . De nition 3.4 is a particular case of De nition 3.5 with G = Gl(d) and the action of Gl(d) on S d?1 de ned by x 7! [A]x for A 2 Gl(d); x 2 S d?1. Speaking of an invariant measure without mentioning X we mean that the situation of De nition 3.4 is considered. 3.2. Relation between algebraic hull and invariant measures. In this subsection we shall need the notion of smooth action. A group action of a locally compact group on a complete separable metric space is called smooth if all orbits are locally closed, i.e. open in their closures (for more information see Zimmer [35]). It is known that linear algebraic groups act smoothly on the space of measures on S d?1, a fact which is of crucial importance for studying invariant measures of linear cocycles.

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Lemma 3.6. Let G be a locally compact group, X a complete metric G-space, A a G-cocycle and ' : ! X a measurable invariant function of A, i.e., '(!) = A(!)'(!) for all ! 2 . If the action of G on X is smooth, then there exist an !0 2 and a measurable map D : ! G such that '(!) = D(!)'(!0 ) for almost all ! 2 .

Proof. See Furstenberg [10, Lemma 6.2, p. 284]. Corollary 3.7. Let A be a cocycle taking values in an algebraic subgroup G Gl(d) and ! be an invariant measure of A. Then there exist an !0 2 and a measurable map D : ! G such that ! = D(!)! for almost all ! 2 . Proof. Since G Gl(d) is algebraic, it acts smoothly on the space P r(S d?1 ) of probability measures of S d?1 (see Zimmer [35, Corollary 3.2.12, p. 45]). Hence the corollary follows from Lemma 3.6. The following theorem shows the invariance of invariant measures of a minimal cocycle with respect to the algebraic hull of the cocycle. It is the key relation between the algebraic hull and invariant measures. Theorem 3.8. Let A 2 G (d) be a minimal cocycle with range H and be an invariant measure of A. Then there is an !0 2 such that ! = ! for almost all ! 2 ; h! = ! for all h 2 H: In particular, all invariant measures of a minimal cocycle are deterministic. Proof. Since H is algebraic, by Corollary 3.7 there are !0 2 and a measurable map D : ! H such that ! = D(!)! almost surely. For all ! 2 , put B (!) := D(!)?1 A(!)D(!): Then B is cohomologous to A by D and for almost all ! 2 we have B (!)! = D(!)?1 A(!)! = D(!)?1 ! = ! : Since B (!) 2 H due to its de nition, for almost all ! 2 we have B (!) 2 fh 2 H j h! = ! g =: StabH (! ) =: H 0 : Changing B on a null set we get an H 0 -cocycle B 0 cohomologous to A. Since H 0 = StabH (! ) H is an algebraic subgroup (Zimmer [35, Corollary 3.2.4, p. 40]) and H 2 H(A) it follows that H 0 = H . Consequently, h! = ! for all h 2 H and ! = D(!)! = ! almost surely. 3.3. Structure of invariant measures. In this subsection we shall describe the structure of all invariant measures of a linear cocycle. First we describe the supports of ergodic invariant measures, for which we establish a one-to-one relation between these supports and minimal invariant sets, and then we describe all minimal sets. De nition 3.9. Let X be a compact metrizable Gl(d)-space. Let M1; M2 : ! X be two closed random sets (on X ). We say that M1 M2 if M1 (!) M2 (!) for almost all ! 2 . If M1 (!) = M2 (!) almost surely then we write M1 = M2 . If M1 M2 and M1 6= M2 then we write M1 < M2. De nition 3.10. An invariant (with respect to A) closed random set M is called minimal if M1 M and M1 invariant implies M1 = M . 0

0

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Lemma 3.11. For any invariant M there is a minimal M1 M . Proof. See Furstenberg [10, Lemma 3.1, p. 278].

Lemma 3.12. Let A be a Gl(d)-cocycle. (i) Let X be a compact metrizable Gl(d)-space and be an ergodic invariant measure on X of A. Then supp is minimal. (ii) Let M be a minimal set of A. Then M supports at least one ergodic invariant measure of A. If is a (not necessarily ergodic) invariant measure of A supported by M then supp = M . Proof. (i) If supp is not minimal then there is an invariant closed random set M < supp . Clearly M is a compact random set, hence there is an ergodic invariant measure on X of A supported by M (see Arnold [1]). Because both and are ergodic they either coincide or are singular to each other. The inequality supp M < supp then implies that and are singular to each other, hence ! (supp ! ) = 0 almost surely, which contradicts the inclusion supp M (!) supp . (ii) Since M is a compact random set it supports at least one ergodic invariant measure (see Arnold [1]). The last assertion follows immediately from the minimality of M . Lemma 3.13. Let A 2 G (d) be a minimal cocycle with range H . Then the minimal sets of A are exactly those orbits Hx, x 2 S d?1, of H on S d?1 which are closed. Proof. Clearly the orbits Hx, x 2 S d?1, are (not necessarily closed) deterministic invariant sets of A. Furthermore, they are nonsingular algebraic subvarieties of S d?1 (see Onishchik and Vinberg [22, Theorem 7, p. 104]). Let M (!) be an arbitrary invariant closed random set of A. Then there is an invariant measure of A such that supp ! M (!) for all ! 2 . By Theorem 3.8, there is !0 2 such that (7) ! = ! for almost all ! 2 ; (8) h! = ! for all h 2 H: Fix one y 2 supp ! S d?1 . Then (8) implies that supp ! Hy, which together with (7) implies that M Hy. Clearly, the boundary @Hy and closure Hy of Hy are closed deterministic invariant sets of A, and @Hy Hy M because M is closed. Now suppose that M is minimal, then there are two possibilities: either @Hy 6= ;, in this case M = @Hy = Hy, hence Hy = @Hy = Hy = M , or @Hy = ;, in which case Hy = Hy = M . In both cases M = Hy is a closed orbit. Conversely, if Hx is a closed orbit and M Hx is an arbitrary invariant closed random set of A, then y 2 Hx, which implies M Hy = Hx, hence M = Hx proving minimality of Hx. 0

0

0

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0

Now, having described the supports of all ergodic invariant measures we are in a position of describing all the invariant measures themselves. Theorem 3.14. Let A 2 G (d) be a minimal cocycle with range H . Then the ergodic invariant measures of A are exactly the deterministic normalized Lebesgue measures (i.e. the unique H -invariant measures) of the closed orbits of H on S d?1 .

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Proof. Let Hx be a closed orbit of H on S d?1 . By Lemma 3.13 the set Hx S d?1 is minimal for A, hence by Lemma 3.12, Hx supports at least one ergodic invariant measure of A, say . Since H 2 H(A) is an algebraic group the orbit Hx is a smooth algebraic submanifold of S d?1 which can be identi ed with the homogeneous space H=(StabH (x)), where StabH (x) is the closed subgroup of H which stabilizes x, hence its Lebesgue measure, which we denote by , is de ned and is quasi-invariant with respect to H , i.e. (E ) = 0 is equivalent to (hE ) = 0 for any h 2 H and Borel set E S d?1 (see Margulis [20, p. 33]). It is well known that the Lebesgue measure of Hx is the unique (regular) measure on Hx which is quasi-invariant with respect to H . Therefore, Theorem 3.8 implies that ! = almost surely, or = since we neglect null-sets. Thus the deterministic Lebesgue measure of Hx is the only invariant measure of A supported by Hx, hence is necessarily ergodic. Now, let 0 be an arbitrary ergodic invariant measure of A, then due to Lemmas 3.12 and 3.13, 0 is supported by a closed orbit Hy, hence the above argument shows that 0 coincides with the Lebesgue measure of Hy.

Corollary 3.15. Let A 2 G (d) and be an ergodic invariant measure of A. Then

almost surely supp ! is a smooth submanifold of S d?1 which is of constant dimension. Furthermore, ! is equivalent to Lebesgue measure of the submanifold supp ! S d?1. Proof. Let H 2 H(A), then A is cohomologous to a minimal H -cocycle A0 by a cohomology C . Since H(A0 ) = H(A) 3 H Theorem 3.14 is applicable to A0 . Noting that C is a linear, hence smooth, transformation of S d?1 mapping bijectively the ergodic invariant measures of A into those of A0 we have the corollary proved.

3.4. Criterion for orthogonal and conformal cocycles. 3.4.1. Criterion for orthogonal cocycle. Let O(d) Gl(d) denote the Lie subgroup of orthogonal matrices of Gl(d). We call a cocycle A 2 G (d) orthogonal if A(!) 2 O(d) for all ! 2 . Clearly, an orthogonal cocycle can be also viewed as an O(d)cocycle. It is known that if a Gl(d)-cocycle is bounded in a certain sense (see De nition 3.16) then it is cohomologous to a cocycle into a compact subgroup of Gl(d) (see Feldman and Moore [8], Schmidt [25], and Zimmer [32, 34]). Noting that any compact subgroup of Gl(d) is conjugate to a subgroup of O(d) (see, e.g., Hewitt and Ross [15, (22.23)]) we arrive at the following two criteria for orthogonal cocycles, one due to Schmidt [25], and the other one to Zimmer [32, 34]. We follow Schmidt [26] and introduce the following notion of a bounded cocycle. De nition 3.16. A Gl(d)-cocycle A is called bounded if for every " > 0 there exists a compact K" Gl(d) such that for every n 2 Z P(f! 2 j A (n; ! ) 62 K" g) < ": A reader familiar with probability theory can easily see that the boundedness of A is exactly the tightness of the sequence of distributions fL(A (n; ))gn2Z. Proposition 3.17. [25] A Gl(d)-cocycle is cohomologous to an orthogonal cocycle if and only if it is bounded.


17

Proof. Let A be a Gl(d)-cocycle. By Theorem 4.7 of Schmidt [25] A is bounded if and only if A is cohomologous to a cocycle taking values in a compact subgroup of Gl(d), hence if and only if A is cohomologous to an orthogonal cocycle.

Proposition 3.18. [32] Let A 2 G (d). If for almost all ! 2 the set fA (n; !) j n 2 Zg has compact closure in Gl(d), then A is Gl(d)-cohomologous to

an orthogonal cocycle. Since there is an increasing sequence of compacts covering Gl(d) it is easily seen that Proposition 3.17 implies Proposition 3.18. On the other hand, although Proposition 3.18 is weaker than Proposition 3.17 it gives a criterion which is, in a sense, easier to verify. Proposition 3.19. Let A 2 G (d) and H 2 H(A). Then A is cohomologous to an orthogonal cocycle if and only if H is compact. Proof. If H is compact then there is M 2 Gl(d) such that H 0 := M ?1HM O(d). Clearly H 0 2 H(A) and A is cohomologous to a H 0 -cocycle which is orthogonal. Conversely, if A is cohomologous to an orthogonal cocycle B , then there is a closed subgroup K O(d) which belongs to H(B ) = H(A). Clearly, any H 2 H(A) is compact because it is conjugate to the compact group K .

3.4.2. Criterion of conformal cocycle. Denote by CO(d) the group of conformal transformations, CO(d) := fA 2 Gl(d) j A A = c2 I; c > 0g ' O(d) R+ Gl(d): A cocycle A is called conformal if A(!) 2 CO(d) for all ! 2 . We shall derive a criterion for a cocycle to be cohomologous to a conformal cocycle. First we characterize nonorthogonal cocycles via their invariant measures. For this purpose we need the following fundamental lemma of Furstenberg [9] (see also Zimmer [35, pp. 39, 45]) which is the key tool for our investigation of invariant measures of linear cocycles. Lemma 3.20. (Furstenberg) Suppose [An ] 2 SGl(d), ; 2 P r(S d?1 ) and that [An ] ! . Then either (i) f[An ]g is bounded, i.e., has compact closure in SGl(d); or (ii) there exist linear subspaces V; W Rd with 1 dim V; dim W d ? 1 and dim V + dim W = d such that is supported on [V ] [ [W ]. Lemma 3.21. Let A be an Sl(d)-cocycle and be an arbitrary invariant measure of A. If A is not cohomologous to an orthogonal cocycle, then for P-almost all ! 2 the measure ! is supported on the union of two proper subspaces of Rd , the sum of whose dimensions is equal to d. Proof. Since Sl(d) is an algebraic group, A is cohomologous to a minimal cocycle B (!) 2 H 2 H(B ) = H(A) for all ! 2 such that H Sl(d). By Proposition 3.19, H is non-compact, hence contains an unbounded sequence hn , n 2 N . Now, noting that SGl(d) = Sl(d), Theorem 3.8 and Lemma 3.20 imply that any invariant measure of B is supported on the union of two proper subspaces of Rd , the sum of whose dimensions is equal to d. This readily proves the lemma. Now, we are able to prove our criterion in terms of invariant measures for a cocycle to be cohomologous to a conformal cocycle.

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Theorem 3.22. Let A 2 G (d). Then A is cohomologous to a conformal cocycle if and only if there exists an invariant measure ! of A which is equivalent to the Lebesgue measure of S d?1 for P-almost all ! 2 . Proof. Let A(!) = C (!)B (!)C (!)?1 , where B is conformal. Then ! := C (!)Leb is an invariant measure of A, where Leb denotes the Lebesgue measure of S d?1 . Clearly, ! is equivalent to Leb for P-almost all ! 2 . Suppose that A is not cohomologous to a conformal cocycle. Put for all ! 2

A^(!) := j det A(!)j?1=d A(!) 2 Sl(d): Clearly, for all ! 2 , [A^(!)] = [A(!)]. Hence, any invariant measure of A is an invariant measure of the cocycle A^ and vice verse. Furthermore, since A is not cohomologous to a conformal cocycle, A^ is not cohomologous to an orthogonal cocycle, because otherwise Corollary 1.3 would lead to a contradiction. This, by Lemma 3.21, implies that for any invariant measure ! of A^, for P-almost all ! 2 , supp (! ) 6= S d?1 , hence ! is not equivalent to Leb. 4. Block-conformal cocycles The aim of this section is to derive the block-conformal form for a cocycle on the span of the support of its invariant measure (Theorem 4.23). For this purpose, in Subsection 4.1 we study coverings of random sets by random linear spaces, where we show the measurability of the linear covers (Theorem 4.10) and the fact that these random covering linear subspaces have constant probability (Theorem 4.14). In Subsection 4.2 assuming an invariant splitting we introduce a lifting operation, i.e. we lift the given cocycle to a cocycle of smaller dimension over an extended dynamical system, and show that cocycle cohomology is invariant, in a sense, under the lifting operation (Theorem 4.20). This lifting operation allows us to apply the results of Section 3 to obtain the conformal form of the lifted cocycle and come back to prove the main Theorem 4.23. 4.1. The linear cover of a subset of S d?1 . In this subsection we use a construction of Zimmer [33, 35] and study the coverings of random sets by random linear subspaces. For a nonempty set f0g 6= X Rd we denote by span(X ) the linear subspace of Rd spanned by the vectors of X , and by [X ] its projection onto S d?1 (see Section 3). Linear subspaces U1 ; : : : ; Ur of Rd are called linearly independent if for any bi 2 Ui , i = 1; : : : ; r, the equality b1 + + br = 0 implies b1 = = br = 0. Denote by C the space of all closed nonempty subsets of S d?1. Then C is a compact metric space with the Hausdor metric. dH (E; F ) := sup fd(x; F ); d(y; E )g; x2E;y2F

where d(x; E ) := inf y2E d(x; y), and d(x; y) := kx ? yk is the Euclidean distance in

Rd .

Put

A :=

n [l i=1

[Vi ] j V1 ; : : : ; Vl are linear subspaces of Rd ;

Vi 6 Vj for i 6= j; and

X

o

dim Vi d :


19

P

ThenPA C is closed. P For E 2 A de ne n(E ) := l, d(E ) := dim Vi , D(E ) := dim( Vi ), where Vi := span([Vi ). Then 1 n(E ); d(E ); D(E ) d. We also introduce

Ab :=

[l [V ] 2 A j i i=1

the spaces V1 ; : : : ; Vl are linearly independent :

Let ; 6= M S d?1 be arbitrary. De ne d~(M ) := minfd(E ) j E 2 A and E M g; n~ (M ) := maxfn(E ) j E 2 A; E M and d(E ) = d~(M )g: Lemma 4.1. For any ; 6= M S d?1 there exists a unique E 2 A such that d(E ) = d~(M ) and n(E ) = n~ (M ). Proof. Let M S d?1 be arbitrary. Since d(); n(); d~(); n~ () take values in the nite set f1; : : : ; dg and the set S d?1 2 A covers M , there exists at least one E 2 A such that d(E ) =Sd~(M ) and n(E ) = n~ (M ). Let E = li=1 [Vi ] 2 A? S be such that E M , d(E ) = d~(M ) and n(E ) = n~ (M ). Observe that M [span li=1 Vi ], hence, by the de nition of d~(M ) we have

d(E ) D(E ) = d([span

? [l V ]) d(E ): i=1

i

Therefore, d(E ) = D(E ), which implies that the linear subspace Vi , i = 1; : : : ; l, are linearly independent. S [W ] is an element of A such that F M , d(F ) = d~(M ) Assume that F = m i=1 i and n(F ) = n~(M ). Then by the above argument the linear subspaces Wi , i = 1; : : : ; m, are linearly independent. Put Uj := V1 \ Wj ; j = 1; : : : ; m: Then the sets Ui , i = 1: : : : ; m, are independent linear subspaces of Rd , and m X i=1

dim Ui dim V1 ; M \ V1

m ?[ U : i=1

i

Therefore, among the Ui there is exactly one which is nontrivial with the same dimension as dim V1 , say Ui . Otherwise we replace [V1 ] by [m i=1 [Ui ] in E and get a contradiction to the maximality of n(E ) or the minimality of d(E ). This implies Ui = V1 , hence V1 Wi . Analogously, every of the Vi is contained in one of the Wi and vice versa. Consequently, E = F . d?1 the unique set E = Sl [Vi ] 2 A such De nition 4.2. For a set ; = 6 M S i=1 P that d(E ) = li=1 dim Vi = d~(M ) and l = n(E ) = n~(M ) is called the linear cover of M and denoted by Z (M ). For a probability measure 2 P r(S d?1 ) the linear cover of its support is called the linear cover of and denoted by Z (). In the proof of Lemma 4.1 we have already proved the following result. Lemma 4.3. Let ; 6= M S d?1 be arbitrary. S l (i) If Z (M ) = i=1 [Vi ] 2 A, then the linear subspaces Vi , i = 1; : : : ; l are linearly independent, hence Z (M ) 2 Ab and d(Z (M )) = dim span(Z (M )). (ii) If E 2 Ab and E M , then E Z (M ). 1

1

1

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>From Lemma 4.3 we obtain immediately the following assertion which can serve as an alternative de nition of the linear cover. Lemma 4.4. For any ; 6= M S d?1 the linear cover Z (M ) is the smallest (with respect to set inclusion) element of Ab covering M . Now we turn to the main part of this subsection, namely we shall prove that the linear cover of a closed random set is again a closed random set (Theorem 4.10). Moreover, we are able to pick the subspaces of the linear cover in a measurable way. First we need some auxiliary results. Lemma 4.5. Let E = Sli=1 [Vi ] 2 Ab. Assume that Uj , j = 1; : : : ; q, are linearly independent subspaces of Rd such P that [Uj ] E for all j = 1; : : : ; q. Assume further that there exists a vector a = qj=1 bj with bj 2 Uj , bj 6= 0, j = 1; : : : ; q, such that a 2 E . Then [q ? Uj ] E: [span j =1

Proof. Since Vi , i = 1; : : : ; l, are linearly independent, the sets [Vi ], i = 1; : : : ; l, are disjoint, and each of them is either connected or the union of two symmetric points. Hence each P [Uj ], j = 1; : : : ; q, must be contained in one of [Vi ], i = 1; : : : ; l. Now let a = li=1 ci be the unique representation of a with respect to span(E ) = li=1 Vi , i.e., ci 2 Vi is the projection of a into Vi along j6=i Vj . Since a 2 E there is only one non-vanishing ci , say c1 6= 0. It is easily seen that if Uk Vm then cm 6= 0, because X cm = br Ur Vm

and the Uj , j = 1; : : : ; q, are linearly independent. Therefore, Uj V1 for all j = 1; : : : ; q, which implies

[q ? U V ; span j =1

j

1

hence the lemma. Lemma 4.6. Let A 2 Gl(d). Sl d?1 (i) Sl If [;AV6= ]M. S is arbitrary and Z (M ) = i=1[Vi ], then Z ([A]M ) = [A]Z (M ) = i=1 i S (ii) 2 P r(S d?1 ) is arbitrary and Z () = li=1 [Vi ], then Z ([A]) = [A]Z () = Sl If[AV i=1 i ]. Proof. (i) [A] furnishes an automorphism of A preserving n(), d(), D() and maps M to [A]M . (ii) [A] maps supp to supp [A]. Lemma 4.7. For any ; 6= M S d?1, span(Z (M )) = span(M ), hence dim span(M ) = d(Z (M )). Proof. Since M Z (M ), span(M ) span(Z (M )). Since M span(M ) and [span(M )] 2 Ab, by Lemma 4.3, Z (M ) [span(M )], which implies span(Z (M )) span(M ). Therefore, span(Z (M )) = span(M ).


21

Lemma 4.8. If M S d?1 is a closed random set, then span(M ) = span(Z (M ))

is a random subspace. Proof. Let cn , n 2 N , be a sequence of measurable maps cn : ! Rd such that M (!) = fcn (!) j n 2 N g for all ! 2 ; Clearly, the sequence pq?1 cn , p 2 Z, q; n 2 N , of measurable maps ordered by increasing jpj + q + n, is dense in span(M ). Therefore, by Proposition 2.2, span(M ) is a random subspace. Corollary 4.9. Let be an invariant measure of A. Then span(Z ()) is an invariant random subspace of A. Here is the main result of this subsection. Theorem 4.10. Let M S d?1 be a closed random set. Then there exist a measurable function

: ! f1; : : : ; dg and random subspaces U1 ; : : : ; U of Rd such that for all ! 2 the subspaces U1 (!); : : : ; U (!)(!) are linearly independent and

Z (M (!)) =

[ (!) i=1

[Ui (!)]:

In particular, Z (M ) is a closed random set. Proof. Let cn , n 2 N , be a sequence of measurable maps cn : ! Rd such that M (!) = fcn (!) j n 2 N g for all ! 2 : Fix an element ! 2 . For ease of notation we shall drop the argument !. We construct sequences of functions and subspaces as follows. Put

1 := 1; U11 := spanfa1 g; Ui1 := ; for i = 2; : : : ; d: Suppose we have constructed n 2 f1; : : : ; dg and subspaces Uin of Rd such that dim Uin 1 for i n ; Uin = ; for i > n ; and the Uin , i = 1; : : : ; n , are linearly independent. Now we construct n+1 2 f1; : : : ; dg and subspaces Uin+1 of Rd . Put

F n :=

n [

Uin :

i=1 n (i) In case cn+1 2 F , we put

n+1 = n ; Uin+1 = Uin for i = 1; : : : ; d: (ii) In case cn+1 62 span(F n ), noting that n < d we set = spanfcn+1g:

n+1 = n + 1; Uin+1 = Uin for i 6= n+1 ; U nn+1 +1 P

n b be the unique repre(iii) In case cn+1 2 span(F n ) n F n , let cn+1 = i=1 i n sentation of c with b 2 U . (Note that b is the projection of cn+1 into Uin n +1 i i i L n along j6=i Uj .) Let 1 i1 < < iq n be those indices for which bij 6= 0,

j = 1; : : : ; q. Put

n+1 = n ? q +1; U1n+1 = spanf[qj=1 Uinj g; Uin+1 = ; for i = n+1 +1; : : : ; d;

22

ARNOLD et al.

and the subspaces U2n+1 ; : : : ; U nn+1 are de ned to be equal to +1

U1n ; : : : ; Uin1 ?1 ; Uin1 +1 ; : : : ; Uin2 ?1 ; Uin2 +1 ; : : : ; Uinq ?1 ; Uinq +1 ; : : : ; U nn ;

respectively. Clearly, n+1 2 f1; : : : ; dg and the subspaces Uin+1 of Rd have properties similar to those of n ; Uin. Therefore, by induction we have constructed sequences of functions n , n 2 N , and subspaces Uin of Rd , n 2 N , i = 1; : : : ; d such that

n 2 f1; : : : ; dg and dim Uin 1 for i n ; Uin = ; for i > n ; and the Uin , i = 1; : : : ; n , are linearly independent. Since we have constructed these sequences for arbitrary xed ! 2 we obtain maps

n () : ! f1; : : : ; dg; n 2 N ; Uin() : ! 2Rd; n 2 N ; i 2 f1; : : : ; dg: As every step of the construction respects measurability, the functions n (), n 2 N , are measurable, the subspaces Uin of Rd , n 2 N , i = 1; : : : ; d, are random subspaces and

n [ F n () = Uin (); n 2 N ; i=1

are closed random sets. Obviously, for any ! 2 , n 2 N , F n (!) F n+1 (!); S n [U n(!)] 2 A for any n 2 N and ! 2 . This implies that either and [F n (!)] = i=1 i F n (!) = F n+1 (!) or there is at least one jump: either d([F n (!)]) = d([F n+1 (!)]) and n([F n (!)]) > n([F n+1 (!)]) or d([F n (!)]) < d([F n+1 (!)]). Since d() and n() take values in the nite set f1; : : : ; dg there are at most d2 jumps. Therefore, for any ! 2 there exists N (!) 2 N such that for all n N (!), F n (!) = F n+1 (!). Hence, Uin (!) = Uin+1 (!) for all n N (!), i = 1; : : : ; d. Clearly the function : ! f1; : : : ; dg de ned by

(!) := lim sup n (!) for all ! 2

n!1

is measurable. Obviously,

(!) = nlim !1 n (!) = N (!) (!) for all ! 2 : Put for any ! 2 , i 2 f1; : : : ; dg,

Ui (!) :=

1 [ \

m=1 km

Uik (!) = UiN (!)(!):

Then the Ui , i 2 f1; : : : ; dg, are closed random sets, because for any x 2 Rd d(x; Ui (!)) = d(x; UiN (!) (!)) = lim sup d(x; Uim (!)) m!1

is measurable. Clearly the Ui , i 2 f1; : : : ; dg, are random subspaces and Ui (!) = ; for i = (!) + 1; : : : ; d. Moreover, for any ! 2 , the subspaces Ui (!), i = 1; : : : ; (!), are linearly independent.


Put for any ! 2

E (!) :=

[ (!) i=1

23

[Ui (!)]:

Then E (!) is closed, hence E (!) M (!) since cn (!) 2 E (!) for all n 2 N , ! 2 . Therefore, since E (!) 2 Ab, E (!) Z (M (!)) due to Lemma 4.3. We show that E (!) Z (M (!)) for all ! 2 . Since E (!) = [F N (!)(!)] it suces to show that for each xed ! 2 , [F n (!)] Z (M (!)) for all n 2 N . We shall prove this by induction. Let n = 1. Then clearly [F 1 (!)] = fc1(!)g M (!) Z (M (!)). Now assume that [F n (!)] Z (M (!)). We shall show that [F n+1 (!)] Z (M (!)). There are three possibilities. (a) In case cn+1 (!) 62 span(F n (!)), we have [F n+1 (!)] = [F n (!)] [ fcn+1(!)g Z (M (!)): (b) In case cn+1 (!) 2 F n (!), we have [F n+1 (!)] = [F n (!)] Z (M (!)): (c) In case cn+1 (!) 2 span(F n (!)) n F n (!), let

cn+1 (!) =

X n (!) i=1

bi (!)

be the unique representation of cn+1 (!)Lwith bi (!) 2 Uin (!), hence bi (!) is the projection of cn+1 (!) into Uin (!) along j6=i Ujn (!). Let 1 i1 < < iq n be those indices such that bij 6= 0, j = 1; : : : ; q. Since U1n+1 (!) = spanf[qj=1 Uinj (!)g; Uinj (!) Z (M (!)), j = 1; : : : ; q, and cn+1 (!) 2 Z (M (!)), by Lemma 4.5, [U1n+1 (!)] Z (M (!)): Consequently, [F n+1 (!)] Z (M (!)): This nishes the proof. S (!)[U (!)] is uniquely determined by M (!), the subspaces While Z (M (!)) = i=1 i Ui (!) are, in general, not. They are unique up to some permutation of indices as the following lemma shows. Denote by (d) the group of permutations of f1; : : : ; dg equipped with the discrete -algebra. Lemma 4.11. Let M (!) be a closed random set and d(Z (M (!))) = (!) 2 f1; : : : ; dg. Assume that U1 (!); : : : ; U (!)(!) and V1 (!); : : : ; V (!) (!) are random subspaces such that for all ! 2

Z (M (!)) =

[ (!)

[ (!)

i=1

i=1

[Ui (!)] =

[Vi (!)]:

(i) There exists a measurable map : ! (d) such that (!)i = i for all i > (!), and Vi (!) = U (!)i (!) for all ! 2 . (ii) If : ! (d) is measurable and (!)i = i for all i > (!), then Wi (!) :=

24

ARNOLD et al.

U(!)i (!) for all ! 2 , i = 1; : : : ; (!), are random subspaces and Z (M (!)) = S (!)[W (!)]. i=1 i Proof. (i) By the uniqueness of the linear cover each Vi must coincide with some Uj and vice versa. Hence is well-de ned. The measurability of follows from the measurability of the Ui ; Vi , i = 1; : : : ; (!). (ii) Wi areSrandom subspaces, because Uj are and is measurable. The equality (!) [W (! )] is obvious. Z (M (!)) = i=1 i Corollary 4.12. Let A 2 G (d) and be an invariant measure of A. Then (i) Z () is an invariant closed random set of [A]. (ii) The maps ! 7! l = n(Z (! )) and ! 7! d(Z (! )) are measurable and invariant with respect to , hence constant. Proof. (i) Since is measurable, supp S d?1 is a closed random set. By Theorem 4.10, Z () is a closed random set. By Lemma 4.6 we have [A(!)]Z (! ) = Z (! ). Hence Z () is invariant with respect to A. (ii) By (i), the maps ! 7! n(Z (! )) and ! 7! d(Z (! )) are measurable. Their

invariance follows immediately from Lemma 4.6. 4.13. Let be an invariant measure of a cocycle A, and Z (! ) = SLemma l [V (! )]. Then for any k = 1; : : : ; d the (possibly empty) closed random set i=1 i [ [Vi (!)] Ek (!) := dim Vi (!)=k; i=1;::: ;l

is invariant P with respect to A. Furthermore, ! (Ek (!)) = pk are independent of ! and dk=1 pk = 1. If is ergodic, then pk = 1 for some k0 . Proof. Since A(!) 2 Gl(d) the subspaces Vi (!) and A(!)Vi (!) are of the same dimension. Therefore, we can group the subspaces of the same dimension to get an invariant set on S d?1 . Theorem Let A 2 G (d), let be an invariant measure of A with Z (! ) = Sl [V (!)]4.14. . Then there is a measurable permutation : ! (l) such that i i=1 Ui (!) := V(!S)i (!) are random subspaces having the following properties: (i) Z (! ) = li=1 [Ui (!)], i.e. the random subspaces Ui are rearrangements of the spaces Vi ; (ii) ! ([Ui (!)]) = constant =: mi ; (iii) A(!)Ui (!) = Uj (!) implies mi = mj , hence the union of those subspaces which have the same probability is invariant. Proof. Put f (!) := 1max f (!); fi (!) := ! ([Vi (!)]); i = 1; : : : ; l: il i 0

Obviously, fi (!) > 0 for all i = 1; : : : ; l and all ! 2 . It is easily seen that f is invariant with respect to , hence f (!) = constant =: c for all ! 2 . Let k(!) denote the total number of those fi (!) = c, then the function k is -invariant, hence constant, say k(!) = k 1 for all ! 2 . Let 1 i1 (!) < < ik (!) l be the indices such that fij (!) (!) = c. The functions ij (!), j = 1; : : : ; k, are measurable. Put (!)?1 j := ij (!); Uj (!) := Vij (!)(!); j = 1; : : : ; k:


Clearly, the random subspaces

E1 (!) := span

? [k V j =1

ij (!) (! )

?

and E2 (!) := span

[k m6=ij (!); m=1

25

Vij (!) (!)

are invariant random subspaces of A. Do the same procedure with the second greatest value of fi , i = 1; : : : ; l, i.e. consider g(!) := maxffi (!) j fi (!) < c; 1 i lg, and so on. After a nite number of steps we obtain a random permutation and random subspaces Ui . It is easily veri ed that for them the assertions of the theorem hold true. CorollaryS4.15. Let A 2 G (d), let be an ergodic invariant measure A with Z (! ) = li=1 [Vi (!)]. Then dim V1 (!) = = dim Vl (!) =: k independent of ! and ! ([Vi (!)]) = 1=l for all i; !. 4.2. Structure of a cocycle on the span of the support of an invariant measure. Block-conformal cocycles. First we give a criterion for a conformal cocycle in terms of the linear cover. Proposition 4.16. Let A 2 G (d). Then A is cohomologous to a conformal cocycle if and only if there exists an invariant measure ! of A such that Z (! ) = S d?1 for P-almost all ! 2 . Proof. Use the same argument as the one in the of the proof of Theorem 3.22. Now, let be an invariant measure of A such that S Z (! ) = li=1 [Ui (!)]; for all ! 2 ; (9) dim Ui (!) = k; for all ! 2 ; i = 1; : : : ; l: Note that given an invariant measure, by virtue of Lemma 4.13, we can consider its restriction to the union of those subspaces in the linear cover which are of the same dimension. This leads back to the case (9). By Lemmas 4.6 and 4.11, there exists a unique measurable map : ! (l) such that for all ! 2 and i 2 f1; : : : ; lg A(!)Ui (!) = U(!)i (!): In this situation we say that A permutes the Ui (or permutes the splitting li=1 Ui (!)) by the law of permutation . We note that the case where is equal to the identity element of (l) is equivalent to the splitting li=1 Ui (!) being invariant. Lemma 4.17. S Assume (9). Let Vi , i = 1; : : : ; l, be random subspaces such that Z (! ) = li=1 [Vi (!)], and let : ! (l) be the measurable map de ned by A(!)Vi (!) = V (!)i (!). Then is cohomologous to , i.e., there exists a measurable map : ! (l) such that (!) = (!)?1 (!) (!) for all ! 2 : Proof. By Lemma 4.11 there exists a unique measurable map : ! (l) such that for all ! 2 and i 2 f1; : : : ; lg Vi (!) = U(!)i (!): By de nition, A(!)Ui (!) = U(!)i (!); A(!)Vi (!) = V (!)i (!):

26

ARNOLD et al.

Therefore, for all ! 2 and i 2 f1; : : : ; lg (!) (!)i = (!) (!)i: Consequently, for all ! 2 , (!) = (!)?1 (!) (!). Proposition 4.18. Assume (9). Let A and be as above, and A B by a cohomology . Then B has an invariant measure = C such that Z (! ) = Sl [W (!)]Cwith dim Wi (!) = k, i = 1; : : : ; l, and B permutes the Wi by . In i=1 i short: l; k; () are cohomology invariants. Proof. Let C : ! Gl(d) be a random map such that for all ! 2

A(!) = C ?1 (!)B (!)C (!): Put ! := C (!)! . Clearly, is an invariant measure of B . Choose Wi (!) = C (!)Ui (!); for all ! 2 ; i = 1; : : : ; l: Then for all ! 2

[l Z (! ) = [Wi (!)]: i=1

Clearly, dim Wi (!) = dim Ui (!) and B (!)Wi (!) = W(!)i (!) for all ! 2 , i = 1; : : : ; l. Next we shall investigate the restriction of A to span(Z ()) but we keep the situation (9), hence, for simplicity we assume that span(Z (! )) = Rd for all ! 2 . Clearly the splitting Rd = li=1 Ui (!) and the random permutation () are determined by A, hence so are also the restrictions of A to the Ui . It is easily seen that given a splitting, a law of permutation and l cocycles of dimension k there is a unique Gl(d)-cocycle which permutes the splitting by the given law of permutation and has the restrictions to the subspaces of the splitting coinciding with the given Gl(k)-cocycles. We formalize this observation in the following construction of lifting a cocycle and show that cohomology respects the lifting construction. In short, the operation of lifting is an extension of the cocycle by making the law of permutation a part of the underlying dynamical system . This construction is a crucial tool of this paper. Roughly speaking, it makes the linear cover bigger relative to the new phase space, hence enables us to use Proposition 4.16 to describe the structure of the linear cocycle. We now describe the lifting construction in detail. We forget for a while about the invariant measure , assume that we are given a splitting Rd = li=1 Ui (!) of Rd into l random k-dimensional subspaces, a law of random permutation : ! (l), and that A permutes the given random subspaces by the law , i.e., A(!)Ui (!) = U(!)i (!) for all ! 2 : We de ne the following skew-product:

:= f1; : : : ; lg ; P(fig E ) := l?1 P(E ) for all i = 1; : : : ; l; E 2 F ; : ! ; where (i; !) := ((!)i; !): It is easily seen that is an automorphism of preserving P . Choose andL x a random basis f := ff1(!); : : : ; fd(!)g of Rd adapted to the splitting Rd = Ui (!), i.e. such that ff(i?1)k+1 (!); : : : ; fik (!)g is a basis of Ui (!)


27

for any i = 1; : : : ; l, ! 2 . Let fe1; : : : ; ek g denote the standard Euclidean basis of Rk . De ne linear isomorphisms Li (!) : Ui (!) ! Rk , ! 2 , i 2 f1; : : : ; lg, by Li (!)f(i?1)k+m (!) = em ; ! 2 ; m 2 f1; : : : ; kg; i 2 f1; : : : ; lg: We de ne a linear random map A : ! Gl(k) by setting for all ! 2 , i 2 f1; : : : ; lg, A (i; !) = L(!)i (!) A(!) Li (!)?1 : We note that Li (!) is an identi cation of the space Ui (!) with the Euclidean space Rk , and A (i; !) is the matrix form of the restriction of A(!) to Ui (!). Thus we have lifted the cocycle A over to the cocycle A over , and the above procedure is called a lifting operation. Clearly, for all n 2 Z and (i; !) 2 , A (n; (i; !)) = L (n;!)i (n !) A (n; !) Li (!)?1 : Lemma 4.19. Let gL := fg1 (!); : : : ; gd(!)g be another random basis of Rd adapted to the splitting Rd = Ui (!). Then A A as Gl(k)-cocycles over . Proof. We have Li (!)g(i?1)k+m (!) = em ; ! 2 ; m 2 f1; : : : ; kg; i 2 f1; : : : ; lg; A (i; !) = L(!)i (!) A(!) Li (!)?1 : De ne a linear random map P : ! Gl(k) by putting for all ! 2 , i 2 f1; : : : ; lg, P (i; !) := Li (!) Li (!)?1 : Then P ((i; !)) := L(!)i (!) L(!)i (!)?1 : It is easily seen that for all ! 2 , i 2 f1; : : : ; lg, A (i; !) = P ((i; !)) A (i; !) P (i; !)?1: So A is cohomologous to A . Theorem 4.20. Let A and f be as above. (i) If A B , then there is a splitting for B which B permutes by the same law and a random basis g such that B = A . (ii) If A A~ as Gl(k)-cocycles over , then there exists a Gl(d)-cocycle B over such that B A, B permutes the splitting Rd = li=1 Ui (!) by the law and B = A~. Proof. (i) Let C : ! Gl(d) be a random map such that for all ! 2

A(!) = C ?1 (!)B (!)C (!): Put Wi (!) = C (!)Ui (!); for all ! 2 ; i = 1; : : : ; l: Obviously, for all ! 2 , f

f

f

f

f

f

f

f

f

f

f

f

g

f

g

g

g

g

g

f

g

f

f

g

g

f

g

f

f

f

Rd =

l M i=1

Wi (!):

28

ARNOLD et al.

It is easily seen that B permutes Wi (!) by , i.e., B (!)Wi (!) = W(!)i (!) for all ! 2 , i = 1; : : : ; l. Put gi (!) := C (!)fi (!) for all ! 2 , i = 1; : : : ; d. Clearly, g = fg1(!); : : : ; gd(!)g is a random basis of Rd adapted to the splitting Rd = li=1 Wi (!). We have Li (!)g(i?1)k+m (!) = em ; ! 2 ; m 2 f1; : : : ; kg; i 2 f1; : : : ; lg; B (i; !) = L(!)i (!) B (!) Li (!)?1 : Clearly Li (!) = Li (!) C (!)?1 : Therefore, B (i; !) = L(!)i (!) C (!)?1 B (!) C (!)Li (!)?1 = A (i; !): (ii) Let A~ and A be cohomologous by P : ! Gl(k), i.e., for all (i; !) 2 A~(i; !) = P ((i; !)) A (i; !) P (i; !)?1 : Put Ci (!) := P (i; !) 2 Gl(k); ! 2 ; i = 1; : : : ; l: Clearly, P ((i; !)) = P ((!)i; !) = C(!)i (!). So A~(i; !) = C(!)i (!) A (i; !) Ci (!)?1 : Now, by de nition, A (i; !) = L(!)i (!) A(!) Li (!)?1 : Hence, A~(i; !) = C(!)i (!) L(!)i (!) A(!) Li (!)?1 Ci (!)?1 : De ne Cî (!) : Ui (!) ! Ui (!), i = 1; : : : ; l, ! 2 , by Cî (!) := Li (!)?1 Ci (!) Li (!): Put f^j (!) := Cî (!)fj (!) for i = 1; : : : ; l; m = (i ? 1)k + 1; : : : ; ik: De ne a linear random map D : ! Gl(d) by setting for all ! 2 , j = 1; : : : ; d D(!)fj (!) = f^j (!): D(!) leaves Ui (!), i = 1; : : : ; l, invariant and the restriction of D(!) to Ui (!) coincides with Cî (!). Put B (!) := D(!)A(!)D(!)?1 : We have B (!)Ui (!) = U(!)i (!), i.e. B has the same splitting as A and B permutes the splitting Rd = li=1 Ui (!) by the same law as A does. Now, for any (i; !) 2 B (i; !) = L(!)i (!)B (!)Li (!)?1 = L(!)i (!)D(!)A(!)D(!)?1 Li (!)?1 = C(!) (!)L(!)i (!)A(!)Li (!)?1 Ci (!)?1 = A~(i; !): Hence the theorem is proved. g

g

g

g

g

f

f

f

g

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f


29

4.21. Suppose that A has an invariant measure such that Z (! ) = SProposition l [U (! )] with dim U (! ) = k , kl = d, i = 1; : : : ; l, and f is a random basis of Rd i=1 i

i

adapted to the splitting Rd = li=1 Ui (!). Then Af is cohomologous to a conformal cocycle.

Proof. By Theorem 4.20 we can assume that ff(i?1)k+1 (!); : : : ; fik (!)g is an orthonormal basis of Ui (!) for any i = 1; : : : ; l, ! 2 . Hence Lfi (!) are isometries. The case k = 1 is trivial, because any one-dimensional matrix is conformal. Let k > 1. Using Theorem 4.14 we can assume without loss of generality that

! ([Ui (!)]) = 1=l; for all ! 2 ; i = 1; : : : ; l: De ne a random probability measure : ! P r(S k?1 ) by setting for all (i; !) 2 , M 2 B(S k?1 ),

(i;!) (M ) := l! ([Li (!)?1 (M )]): f

Clearly, (i;!) (S k?1 ) = 1 for all (i; !) 2 , hence (i;!) 2 P r(S k?1 ). The measurability of the map is obvious. We claim that is an invariant measure of A . For, since A (i; !) = L(!)i (!) A(!) Li (!)?1 we have, for any M 2 B(S k?1 ), f

f

f

f

A(!) Li (!)?1 (M ) = L(!)i (!)?1 A (i; !)(M ): f

f

f

This, due to the invariance of , implies

(i;!) (M ) = l! ([Li (!)?1 (M )]) = l! ([A(!)][Li (!)?1 (M )]) = l! (L(!)i (!)?1 [A (i; !)](M )) = (!)i;! ([A (i; !)](M )): f

f

f

f

f

Thus the claim is proved. Clearly, supp (i;!) = [Li (!)(supp ! \ Ui (!))] for all i = 1; : : : ; l, ! 2 . Hence, by Lemma 4.6, Z ((i;!) ) = S k?1 for all (i; !) 2 . By Proposition 4.16, A is cohomologous to a conformal cocycle. f

f

Proposition 4.21 shows that if we lift a linear cocycle to the linear cover of its invariant measure then we obtain a conformal cocycle, but over an extended dynamical system. We shall prove that when looking back to the original dynamical system we will derive a so-called block-conformal cocycle. First we introduce the notion of a block-conformal cocycle.

De nition 4.22. Let d = kl. A random (d d)-matrix B is called block-conformal (with k-dimensional blocks) if there is a measurable mapping : ! (l) such

30

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that B has the following form 0 . BB ... BB .. BB .. B .

1 .. .. . . CC .. .. CC B(!)2 (!) . . CC .. .. 0 . . CC BB .. .. .. CC . . . BB 0 CC .. .. .. B (!) = B (10) . . . BB B(!)1 (!) CC .. .. .. BB 0 CC . . . BB . CC .. .. BB .. CC . . 0 .. .. BB ... C . . B(!)l (!) C @ . A .. .. .. .. . . . with B(!)i (!) 2 CO(k) for all ! 2 and i = 1; : : : ; l. Here B is of the form of an (l l)-matrix each entry of which is a (k k)-matrix, and in the m-th column of B the only non-trivial entry is B(!)m (!) in the (!)m position, similarly for the rows. A linear cocycle A is called block-conformal if the matrix representation of A in the standard Euclidean basis of Rd is a random block-conformal matrix. Now we are in a position to prove the main result of this section, namely the following theorem on the reduction of a linear cocycle to block-conformal form. Theorem 4.23. Let A 2 G (d). S (i) If A has an invariant measure such that Z (! ) = li=1 [Ui (!)] with dim Ui (!) = k, kl = d, i = 1; : : : ; l, then A is cohomologous to a block-conformal cocycle. (ii) If is an arbitrary (not necessarily ergodic) invariant measure of A, then the restriction of A to span(supp ! ) is cohomologous to a direct sum of block-conformal subcocycles. (iii) If A is cohomologous to a direct sum of block-conformal subcocycles then A has an invariant measure such that span(supp ! ) = Rd for almost all ! 2 . L Proof. First note that by Lemma 4.7, span(supp ! ) = span(Z (! )) = li=1 Ui (!). (i) Let A permute the Ui by the law . By Theorem 4.20 and Proposition 4.21 there are a Gl(d)-cocycle C which permutes the Ui by the law and L a random basis f := ff1(!); : : : ; fd(!)g of Rd adapted to the splitting Rd = Ui (!) such that C is cohomologous to A and C (i; !) is a conformal matrix (with respect to the standard Euclidean basis of Rk ) for any (i; !) 2 . Let e = fe1 ; : : : ; edg denote the standard Euclidean basis of Rd . De ne a random linear map R of Rd by setting R(!)fi (!) = ei for all i = 1; : : : ; l; ! 2 : Put for all ! 2

B (!) := R(!)C (!)R(!)?1 : Then B is cohomologous to C , hence to A. We show that B has the desired block-conformal form. By the de nition of C , it has a matrix representation with respect to the random basis f = ff1 (!); : : : ; fd (!)g .. .

f


31

of the form (10) with B(!)i (!) replaced with C(!)i (!), the matrix representation of C (i; !) in the standard Euclidean basis of Rk (hence is conformal). Further, by the de nition of B , its matrix representation with respect to the basis e coincides with the matrix representation of C with respect to the basis f . Thus (i) is proved. Part (ii) follows from (i) by grouping those random subspaces Vi 's which have the same dimension. Part (iii) follows from the de nition of a block-conformal cocycle by taking the Lebesgue measures on the intersections of the unit sphere with the subspaces corresponding to the blocks. f

Since, as we proved, a linear cocycle has a \good" structure on spans of supports of its invariant measures it is natural to look for the biggest-possible support. Note that the supports of invariant measures are what can be \seen" by long-term observations of the system. In the remaining part of this section we settle this problem. Lemma 4.24. (i) For any nonempty sets M1; M2 S d?1 we have Z (M1 ) [ Z (M2 ) Z (M1 [ M2) and span(Z (M1 ) [ Z (M2 )) = span(Z (M1 [ M2 )): (ii) Let A be a linear cocycle and 1 and 2 be two invariant measures of A. Then there exists an invariant measure of A such that for P-almost all ! 2

span(Z (! )) = span(Z (1! ) [ Z (2! )): Proof. (i) We have M1 Z (M1 [ M2 ) and Z (M1 [ M2 ) 2 Ab, hence, by Lemma 4.3, Z (M1 ) Z (M1 [ M2 ). Similarly, Z (M2 ) Z (M1 [ M2), hence Z (M1 ) [ Z (M2 ) Z (M1 [ M2 ). By Lemma 4.7 span(Z (M1 [ M2 )) = span(M1 [ M2 ) span(Z (M1) [ Z (M2 )) span(Z (M1 [ M2 )): (ii) For all ! 2 , put ! () := 21 (1! () + 2! ()): Then is an invariant measure of A with supp = supp 1 [ supp 2 . Therefore, due to (i), is the required invariant measure.

Theorem 4.25. Every cocycle A 2 G (d) has a maximal invariant measure max with the following properties: max (i) supp max ! = Z (! ) for P-almost all ! 2 , (ii) Any invariant measure of A is absolutely continuous with respect to max, i.e. ! max ! for P-almost all ! 2 . Proof. Since the functions d() and n() take values in the nite set f1; : : : ; dg there is an invariant measure of A such that for any invariant measure 0 of A either d(Z ( 0 ! )) < d(Z (! )) =: r or d(Z (!0 )) = r and n(Z (!0 )) n(Z (! )) =: l. By Theorem 4.23, the restriction of A to span(Z (! )) is cohomologous to a cocycle B 2 G (d) which is a direct sum of block-conformal subcocycles. Taking the Lebesgue

measures on the intersections of the unit sphere with the subspaces corresponding to the blocks of the subcocycles of B we see that B has an invariant measure 1 with supp 1 = Z ( 1 ), d(Z ( 1 )) = r, and n(Z ( 1 )) = l. Transforming 1 back to

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an invariant measure on A we nd an invariant measure max of A such that, for almost all ! 2 , max max max supp max ! = Z (! ); d(Z (! )) = r; n(Z (! )) = l: We show that this max is the maximal measure we are looking for. Part (i) is immediate. Let be an arbitrary invariant measure of A. Then ! := (! + max ! )=2 is an invariant measure of A. If Z (! ) 6 span(Z (max implies ! )) then Lemma 4.24 max that d(Z (! )) > r which contradicts the choice of r. Therefore, span(Z (! )) = span(Z (! )). Consequently, by the choice of r, l and we have for almost all ! 2

max d(Z (! )) = d(Z (max ! )); n(Z (! )) n(Z (! )); max Z (! ) supp ! supp ! : Since Z (! ) 2 A and Z (! ) supp max ! )) = ! , Lemma 4.1 implies that n(Z (max max n(Z (max )) and Z ( ) = Z ( ). This, by Lemma 4.24, implies that Z ( ! ! ! ! ) Z (! ) supp ! . Due to (i), ! (supp max ) = 1 for P -almost all ! 2

, hence ! part (ii) is proved.

Remark 4.26. A linear cocycle may have many maximal measures, but their supports are the same. Thus any linear cocycle has a unique good maximal invariant set supp max ! on the span of which it has the form of a block conformal cocycle. 5. The Jordan normal form In this section we prove our nal theorem on the Jordan normal form of a linear cocycle (Theorem 5.6) and give an algorithm of constructing the Jordan normal form via invariant measures. 5.1. The Jordan form. We shall show that the structure of the algebraic hull of a linear cocycle determines the structure of the cocycle itself. De nition 5.1. A closed subgroup G of Gl(d) is called irreducible if no proper subspace of Rd is invariant under G. We say that G is strongly irreducible if no nite union of proper subspaces of Rd is invariant under G. It is easily seen that for a connected subgroup G Gl(d) irreducibility is equivalent to strong irreducibility. Recall De nition 2.9 concerning irreducibility etc. of a cocycle. We note that the de nition of these particular classes of linear groups is borrowed from the theory of representations of groups. The following lemma is immediate. Lemma 5.2. (i) The notions of irreducibility and strong irreducibility are invariant with respect to group conjugacy. (ii) Strong irreducibility implies irreducibility, the converse assertion is false. The following theorem establishes the equivalence of irreducibility of the cocycle and of its algebraic hull. Theorem 5.3. Let A 2 G (d). Then A is (strongly) irreducible if and only if some, hence any H 2 H(A) is (strongly) irreducible.


33

Proof. Let H 2 H(A). There is B A such that B (!) 2 H for all ! 2 . If H is reducible then there is a proper subspace U Rd which is invariant with respect to H . Clearly U is invariant with respect to B , hence B is reducible, whence A is reducible. Suppose that A is reducible, then B is reducible, hence B has an invariant random proper subspace U (!) Rd . Note that dim U (!) is constant. Take and x a nonrandom proper subspace V Rd of the same dimension. Choose a random basis of U (!) and a nonrandom basis of V . Extend them to a random basis and a nonrandom basis of Rd , respectively. Denote by C (!) the random linear operator of Rd which map the constructed random basis to the nonrandom basis. Clearly C furnishes a cohomology from B to a cocycle D 2 G (d) which preserves V for all ! 2 . It is easily seen that V is invariant with respect to some H 0 2 H(D) = H(A), hence H 0 is reducible, which implies reducibility of H . The case of strong irreducibility is analogous. Theorem 5.4. Let A 2 G (d). If A is strongly irreducible, then for any invariant measure of A we have Z (! ) = S d?1, hence A is cohomologous to a conformal cocycle. Proof. Suppose that A is stronglySirreducible. Let be an ergodic invariant measure of A with linear cover Z (! ) = li=1 [Vi (!)]. If dim span(Z (! )) < d then A is not irreducible because the proper random subspace span(Z (! )) is invariant. If l > 1 then A is not strongly irreducible. Thus, dim span(Z (! )) = d and l = 1, hence Z (! ) = S d?1 . Therefore, by Proposition 4.16, A is cohomologous to a conformal cocycle. Theorem 5.5. Let A 2 G (d). If A is irreducible, then for any ergodic invariS ant measure of A we have Z (! ) = li=1 [Vi (!)] with dim(V1 (!)) = = dim(Vl (!)) = k independent of ! and kl = d, hence A is cohomologous to a blockconformal cocycle. Proof. be an ergodic invariant measure of A with linear cover Z (! ) = Sl [V Let ( ! )]. If dim span(Z (! )) < d then A is not irreducible because the proper i i=1 random subspace span(Z (! )) is invariant. By the ergodicity of the subspaces Vi (!) have the same dimension, hence, by Theorem 4.23, A is cohomologous to a block-conformal cocycle. Theorem 5.6. (Jordan normal form for a linear cocycle) Let A 2 G (d). Then A is cohomologous to a block-triangular cocycle 0 A (!) 1 BB 10 A2(!) CC A^ = B C; ... @ 0 A 0 0 0 Ap (!) where the subcocycles Ai 2 G (di ) are irreducible block-conformal cocycles, i = 1; : : : ; p. The number p is the length of any maximal invariant ag of A. The cohomology classes of the subcocycles Ai 2 G (di ), up to their order, are uniquely determined by the cohomology class of A. Further, if V0 Vp is an arbitrary maximal invariant ag of A, then, after a suitable reordering of indices, AVi =Vi? Ai for all i = 1; : : : ; p. 1

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Proof. Take and x a maximal invariant ag U0 Up of A, the existence of which is ensured by Lemma 2.15. Put di := dim Ui ? dim Ui?1 , i = 1; : : : ; p. Choose and x a random basis f := ff1(!); : : : ; fd(!)g of Rd such that ff1(!); : : : ; fd ++di (!)g is a basis of Ui (!) for all ! 2 , i = 1; : : : ; p. Let e := fe1 ; : : : ; ed g denote the standard Euclidean basis of Rd . De ne a random linear map C : ! Gl(d) by setting C (!)ei = fi (!) for all ! 2 and i = 1; : : : ; d. Put B (!) := C (!)?1 A(!)C (!) for all ! 2 : Obviously, B 2 G (d) and B A. It is easily seen that the (nonrandom) spaces Wi := spanfe1; : : : ; ed ++di g, i = 1; : : : ; p, are invariant with respect to B and they constitute a maximal invariant ag of B . Therefore, B has the following matrix form: 0 B (!) 1 BB 10 B2(!) CC B=B C; ... @ 0 A 0 0 0 Bp (!) where Bi = BWi =Wi? 2 G (di ), i = 1; : : : ; p, are irreducible. By Theorem 5.5, the cocycles Bi 2 G (di ), i = 1; : : : ; p, are block-conformal. By Theorem 2.17, the cohomology classes of the subcocycles Bi 2 G (di ), up to their order, are uniquely determined by the cohomology class of B . Set A^ := B . Clearly B A. The last statement follows from Theorem 2.17. De nition 5.7. Any block-triangular matrix cocycle with the properties of A^ in Theorem 5.6 cohomologous to A is called a Jordan normal form of the cocycle A. Remark 5.8. (i) We expect that the normal form for linear cocycles is not simpler than that for linear groups|as it will be clear from the considerations of Section 6 dealing with orthogonal cocycles, hence Theorem 5.6 probably gives us the simplest possible normal form for a general linear cocycle. (ii) Unlike the Jordan normal form of a matrix, there is no normal Jordan block for the cocycle case|only the diagonal is determined by the cocycle, the elements above the diagonal are far from unique. As seen from the two-dimensional cocycles treated in Section 7, the rule of change of the entry above the diagonal is a cohomological equation and there is no unique way of nding the simplest form of this entry. The higher dimensional case is of course more complicated. Theorem 5.9. Let A 2 G (d) and max be a maximal invariant measure of A. If d span(supp max ! ) = R for almost all ! 2 , then the Jordan normal form of A is block-diagonal with block-conformal entries. Proof. Use the lifting operation and the fact that orthogonal cocycles are completely reducible. As a by-product we derive the following Jordan form for amenable subgroups of Gl(d). Recall the classical de nition of amenability: A locally compact group G is called amenable if for every continuous G-action on a compact metrizable space X , there is a G-invariant probability measure on X (see, e.g., Zimmer [35]). The notion of the Jordan normal form for a linear group is the same as that for a linear representation; see Kirillov [17, p. 116]. 1

1

1


35

Theorem 5.10. Any amenable Lie subgroup H of Gl(d) has (i.e. reduced to by means of group conjugacy) the following Jordan (matrix) form

0H 1 BB 01 H2 CC H^ = B @ 0 0 . . . CA ;

0 0 Hp where Hi , i = 1; : : : ; p, are groups of block-conformal matrices. Proof. By a theorem of Golodets and Sinelshchikov [12], H is the range of an H cocycle A (over some dynamical system ( ; )) in the sense that the skew-product action of A on H is ergodic. Obviously, H 2 H(A), where H is the smallest algebraic group containing H . Apply Theorem 5.6 to A and reduce it to the Jordan form A^. Clearly a group ^ H 2 H(A^) = H(A) satis es the conclusion of the theorem. Since H is conjugate to H^ we have proved the theorem. In a coordinate-free language, H^ has an invariant ag, where on each factor of the ag H^ preserves a nite union of linearly independent subspaces of the same dimension together with a conformal structure (i.e. H^ acts on these subspaces by conformal maps which by de nition are scalar multiples of isometries). Moreover, the action of H^ on the factors is irreducible. 5.2. An algorithm for constructing the Jordan normal form. We give here an algorithm of reducing an arbitrary linear cocycle to the block-triangular form with block-conformal subcocycles on the diagonal, hence in particular to the Jordan normal form. 5.2.1. The algorithm. Let A 2 G (d). It is well known that A has at least one invariant ergodic measure. Let be one such measure. Denote by e = fe1 ; : : : ; edg the standard Euclidean basis of Rd . By Lemma 4.13, since is ergodic its linear cover consists of random linear subspaces of the same dimension, say

Z (! ) =

[l

i=1

[Ui (!)]

with dim Ui (!) = k for all i = 1; : : : ; l, ! 2 . Put m := kl. Choose a random basis f := ff1 (!); : : : ; fd (!)g of Rd such that ff1(!); : : : ; fm (!)g is a basis of span(Z (! )) =: E (!) for any ! 2 . De ne a random linear map R : ! Gl(d) by setting R(!)ei = fi (!) for all i = 1; : : : ; d; ! 2 : Put for all ! 2

B (!) := R(!)?1 A(!)R(!): Clearly B is cohomologous to A and has the following matrix representation in the basis e B (!) B (!) 1 12 B (!) = 0 B (!) ; 2

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where B1 (!) is a (m m)-matrix and has R(!)! =: ! as its invariant measure with [l Z (! ) = [R(!)?1 Ui (!)]; span(Z (! )) = span(e1 ; : : : ; em): i=1

By Theorem 4.23, B1 is cohomologous to a block-conformal Gl(m)-cocycle. Let B1 (!) = P (!)?1 C1 (!)P (!) for a random m-dimensional matrix P and a random block-conformal (m m)-matrix C1 . Put, for all ! 2 , 2 Gl(d); P~ (!) := P (0!) I 0 d?m where Ir denotes the unit (r r)- matrix. Set for all ! 2

C (!) := P~ (!)B (!)P~ (!)?1 ; C12 (!) := P (!)B12 (!): Straightforward computations shows that C (!) C (!) 1 12 C (!) = 0 B2 (!) : The cocycle C is cohomologous to B by construction. If m = d we are done. Let m < d. Consider B2 as Gl(d ? m)-cocycle on Rd?m := span(em+1 ; : : : ; ed ). By the above argument we can nd a random (d?m)dimensional matrix Q : ! Gl(d ? m), a random block-conformal (n n)-matrix C2 , 1 n d ? m, a random ((d ? m ? n) (d ? m ? n))-matrix C3 and a random (n (d ? m ? n))-matrix C23 such that for all ! 2

Q(!)?1 B2 (!)Q(!) = C20(!) CC23((!!)) : 3 Put for all ! 2

D(!) := Q~ (!)C (!)Q~ (!); where Q~ (!) := I0m Q(0!) 2 Gl(d): Clearly, D is cohomologous to A and has the form 1 0 C1 (!) C12 (!)Q(!) D(!) = B @ 0 C2 (!) C23(!) CA : 0 0 C3 (!) Continuing the process if m + n < d. After d steps we obtain the desired blocktriangular cocycle which is cohomologous to A. 5.2.2. Construction of the Jordan normal form. First we note that the linear span of an ergodic invariant measure is not always a minimal invariant subspace of the cocycle, hence the above algorithm does not always give the Jordan normal form of A though the form it gives is block-triangular with block-conformal subcocycles on the diagonal. However, if in that algorithm we choose in each step an ergodic invariant measure with the linear span of its support having the smallest possible dimension, then the quotient subcocycles on the diagonal will be irreducible and the algorithm produces the Jordan normal form of A. Another way of constructing the Jordan normal form is to rst apply the above algorithm with arbitrary (not necessarily ergodic) invariant measures, and then apply Theorem 5.9 to the subcocycles on the diagonal.


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In case A satis es the integrability condition of the multiplicative ergodic theorem (see Oseledets [23]), we can decompose A into a direct sum of subcocycles according to the Oseledets splitting of A, and then combine it with the Jordan normal forms of the subcocycles. 6. Orthogonal cocycles The results of Section 4 reduce the investigation of linear cocycles on the span of the supports of their invariant measures to the investigation of block-conformal cocycles a nite extension of which are conformal cocycles. Furthermore, since we can factorize the determinant the problem reduces to the study of orthogonal cocycles, which are the subject of this section. Criteria for a cocycle to be cohomologous to an orthogonal cocycle are given in Subsection 3.4. We recall an important fact that any closed (hence compact) subgroup of O(d) is algebraic (see Onishchik and Vinberg [22, Theorem 5, p. 133]). This fact will be of crucial importance in this section. 6.1. The algebraic hull of an orthogonal cocycle. Recall De nition 3.2 of the algebraic hull. First we note that the de nition of the algebraic hull of a G-cocycle depends on the group G: If we are given two algebraic subgroups G1 G2 Gl(d), then for a G1 -cocycle A its algebraic hull depends on whether we consider A as G1 valued or G2 -valued cocycle. The latter case gives, in general, a smaller algebraic hull because we have a bigger choice of cohomologies. However, we are able to prove that they coincide (inside G1 ) in the case G1 = O(d) G2 = Gl(d). Proposition 6.1. If A and B are orthogonal and they are Gl(d)-cohomologous then they are O(d)-cohomologous. Proof. Let C : ! Gl(d) be measurable and let for all ! 2

(11) A(!) = C (!)?1 B (!)C (!): We need to show that there is a measurable map C 0 : ! O(d) such that for all !2

A(!) = C 0 (!)?1 B (!)C 0 (!): By the theorem on polar decomposition (see Gantmacher [11, p. 286]) each C (!) 2 Gl(d) can be represented in the form C (!) = U (!)D(!)V (!); where U (!); V (!) 2 O(d) and D(!) = diagfx1 (!); x2 (!); : : : ; xd (!)g with x1 (!) x2 (!) xd (!) > 0 being the singular values of C (!). Moreover, U; V; D can be chosen to be measurable. It is easily seen that

x1 (!) = kC (!)k; xk (!) = k

^k

C (!)k=k

^k?1

C (!)k for k = 2; : : : ; d:

Taking the exterior product and then the norm on both sides of equation (11) and remembering that A and B are orthogonal we obtain xk (!) = xk (!) for all k = 1; : : : ; d: This, due to the ergodicity of , implies that xk (!), k = 1; : : : ; d, are constant, hence we write xk (!) =: xk and D(!) = diagfx1 ; x2 ; : : : ; xd g =: D.

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Now (11) takes the form U (!)DV (!)A(!) = B (!)U (!)DV (!); which is equivalent to (12) DV (!)A(!)V (!)?1 D?1 = U (!)?1 B (!)U (!): Next we we need the following elementary assertion. Claim. Let D be as above and X 2 O(d). If DXD?1 2 O(d) then DXD?1 = X . To prove the claim, group the equal entries of D: the rst l1 entries are equal y1 and correspond to the subspace E1 of the rst l1 basis vectors e1 ; : : : ; el , the next l2 entries are equal y2 and correspond to the subspace E2 of the next l2 basis vectors, : : : , the last lm entries are equal ym and correspond to the subspace Em of the last lm basis vectors, y1 > y2 > > ym > 0. Now Pwe show that the spaces E1; : : : ; Em are invariant with respect to X . Let Xe1 = dj=1 j ej , then 1

v u d uX ? 1 1 = ke1 k = kDXD e1 k = t 2j x2j x?1 2 ; j =1

from which it follows that j = 0 for j > l1 , hence Xe1 2 E1 . Analogously, Xej 2 E1 for j = 2; : : : ; l1, whence XE1 = E1 . Since X is orthogonal it preserves also the orthogonal complement of E1 which is the direct sum of E2 ; : : : ; Em . Restrict X to this sum and proceed further we obtain that all the spaces Ek are invariant. Since the restrictions of D to Ek are scalars we see that X commutes with D, hence DXD?1 = X . Thus the claim is proved. Applying this result to (12) and putting C 0 (!) := U (!)V (!) nishes the proof. We would like to mention here that Knill [18, Lemma 7.1, p. 80] has proved the special case d = 2 of Proposition 6.1. It immediately follows from Proposition 6.1 that the algebraic hull of an orthogonal cocycle A in the space of O(d)-cocycles is the intersection with O(d) of its algebraic hull in the space of Gl(d)-cocycles. Furthermore, A is minimal as a Gl(d)-cocycle if and only if A is a minimal O(d)-cocycle. In this section, we shall restrict ourselves to orthogonal cocycles and H(A) will stand for the algebraic hull of A in the space of O(d)-cocycles. 6.2. Invariant measures of orthogonal cocycles. Utilizing the compactness of O(d) we can obtain more information about invariant measures of orthogonal cocycles. Proposition 6.2. Assume that A is a minimal orthogonal cocycle with range H 2 H(A). Then the ergodic invariant measures of A are exactly the normalized Lebesgue measures of the orbits of H in S d?1. In particular, all ergodic invariant measures of A are deterministic and the union of their supports equals S d?1. Proof. Immediate from Theorem 3.14 and the fact that any orbit of H on S d?1 is closed due to the compactness of H . Theorem 6.3. Assume that A is a minimal orthogonal cocycle with range H 2 H(A). Let L; G O(d) be closed subgroups, such that G H; L. Put X := G=L and assume that G acts naturally on X . Then the ergodic invariant measures on


39

X of A are exactly (deterministic) normalized Lebesgue measures (i.e. the unique H -invariant measures) of the orbits Hx, x 2 X , of H on X . Proof. The same argument as in the proof of Lemma 3.13 shows that for any x 2 X the set Hx is minimal. Now, the same argument as the one for the proof of Theorem 3.14 also proves this theorem. 6.3. Equivalence of orthogonal cocycles. We show that two invariants of orthogonal cocycles (algebraic hull and invariant measures) are equivalent in some sense. De nition 6.4. Let H 2 Gl(d) be a Lie subgroup, X an H -space, A and B two H cocycles. We say that A is equivalent to B with respect to invariant measures on X if there exists a measurable map D : ! H such that the mapping ! 7! D(!)! furnishes a one-to-one correspondence between invariant measures on X of A and those of B . In the case of classical measures (on X = S d?1) we shall not mention X at all. The following lemma is immediate. Lemma 6.5. (i) \Equivalent with respect to invariant measures on X " is an equivalence relation. (ii) If A is cohomologous to B then they are equivalent with respect to invariant measures on X for any X . (iii) A is equivalent to B with respect to invariant measures on X if and only if there is a H -cocycle C which is cohomologous to A as H -cocycles and has the same collection of invariant measures on X as B does. Proposition 6.6. Let A and B be orthogonal cocycles. If H(A) = H(B) then there is an orthogonal cocycle C cohomologous to A such that C has the same collection of ergodic invariant measures as B , hence A is equivalent to B with respect to invariant measures. Proof. Take and x K 2 H(A) = H(B ), K O(d). Then there are K -cocycles A0 and B 0 cohomologous to A and B . Clearly A0 and B 0 are orthogonal minimal cocycles, hence the ergodic invariant measures of A0 and B 0 are exactly the deterministic normalized Lebesgue measures of the orbits of K on S d?1 , hence they have the same collection of ergodic invariant measures. The proposition follows now from Lemma 6.5. Remark 6.7. The condition H(A) = H(B) is strictly stronger then the equivalence with respect to invariant measures of A and B . For, if H(A) 6= H(B ) but both of them act transitively on S d?1 (e.g., O(d) 2 H(A) and SO(d) 2 H(B )) then both A and B have a unique invariant measure, namely the Lebesgue measure of S d?1 , hence they are equivalent with respect to invariant measures. Theorem 6.8. Let A and B be two orthogonal cocycles. Then H(A) = H(B) if and only if A and B are equivalent with respect to invariant measures on X for any homogeneous space X = O(d)=L with L being a closed subgroup of O(d) and the action of O(d) on X being natural. Proof. Let H(A) = H(B ). Take and x H 2 H(A) = H(B ). Then there exist minimal H -cocycles A0 and B 0 which are cohomologous to A and B , respectively.

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Let L O(d) be an arbitrary closed subgroup, X = O(d)=L, and O(d) act naturally on X . By Theorem 6.3, A0 and B 0 have the same collection of ergodic invariant measures on X , which by Lemma 6.5 implies that A and B are equivalent with respect to invariant measures on X . Now, assume that A and B are equivalent with respect to invariant measures on X for any homogeneous space X = O(d)=L with L being a closed subgroup of O(d) and the action of O(d) on X being natural. Take and x H1 2 H(A) and H2 2 H(B ). Clearly, there are a minimal H1 -cocycle C and a minimal H2 -cocycle D which are cohomologous to A and B respectively. Since H1 2 H(C ) = H(A) and H2 2 H(D) = H(B ) we can apply Theorem 6.3 to C and D with X = O(d)=H2 and obtain that the ergodic invariant measures on X of C and D are exactly the normalized Lebesgue measures of the orbits of H1 and H2 on X , respectively. Since the orbit H2 (eH2 ) through the point eH2 2 X , where e is the unity element of O(d), consists of the single point eH2 , the cocycle D has the deterministic invariant Dirac measure eH on X . Now, C and D are equivalent with respect to invariant measures on X by the assumption on A, B and Lemma 6.5, hence C has a (possibly random) invariant Dirac measure. Since all the ergodic invariant measures of C are deterministic, C has a deterministic invariant Dirac measure. By Theorem 6.3, this deterministic invariant Dirac measure is supported by an orbit of H1 on X . Therefore, there is x = gH2 2 X , g 2 O(d), such that H1 x = x. This implies that for any h 2 H1 we have hgH2 = gH2 , hence g?1 hg 2 H2 , whence g?1 H1 g H2 . Changing the role of H1 and H2 we nd g1 2 O(d) such that g1?1 H2 g1 H1 . Therefore, 2

which implies that

g?1 g1?1 H2 g1 g g?1 H1 g H2 ;

g?1 g1?1 H2 g1 g = g?1 H1 g = H2 : Thus H1 is conjugate to H2 implying H(A) = H(B ).

6.4. About classi cation of orthogonal cocycles. The remarks at the beginning of this section should convince the reader that it is of crucial importance to classify orthogonal cocycles as completely as possible. We have found two invariants of cohomology, namely the equivalence of invariant measure and the algebraic hull, which turned out to be equivalent in the orthogonal case. We hence can rst classify their algebraic hulls which are (conjugacy classes of) closed subgroups of O(d). This is a very well studied classical problem. Now we single out the so-called elementary orthogonal cocycles by the following theorem. Theorem 6.9. Let A be an orthogonal cocycle. The following statements are equivalent: (i) A is elementary in the sense that the Lebesgue measure Leb of S d?1 is the only invariant measure of A; (ii) The Lebesgue measure Leb of S d?1 is an ergodic invariant measure of A; (iii) Any group H 2 H(A) acts transitively on S d?1 . Proof. Looking at cohomologous cocycles we see that it suces to prove the theorem for the case where A is minimal with range H 2 H(A). Due to Proposition 6.2, it follows from (iii) that Leb is the unique ergodic invariant measure of A, hence the unique invariant measure of A. Also from Proposition 6.2, if Leb is an ergodic invariant measure of A then S d?1 is an orbit of H ,


41

hence H acts transitively on S d?1. Thus (ii) is equivalent to (iii), which implies (i). If Leb is the unique invariant measure of A then obviously it is ergodic, hence (i) implies (ii). The name \elementary" is justi ed by the fact that such a cocycle has only one invariant measure. However, for a given algebraic hull there are, in general, in nitely many cohomology classes of orthogonal cocycles corresponding to this hull, as the following example shows. Example 6.10. Let d = 2 and let be the trivial one-point space (hence the deterministic case). For each irrational 2 R we set ) sin(2) : A := ?cos(2 sin(2) cos(2) Then SO(2) 2 H(A ) for any , hence A is elementary. However, there are in nitely (continuously) many cohomology classes of such cocycles. 7. Classification of low-dimensional cocycles To classify linear cocycle we shall need the following notion of a coboundary. De nition 7.1. Let G Gl(d) be a Lie subgroup. A G-cocycle is called a Gcoboundary (or simply a coboundary if it is clear which group G is meant) if it is G-cohomologous to the trivial G-cocycle, i.e. the G-cocycle taking the value I |the identity of G|for all ! 2 . 7.1. Classi cation of one-dimensional linear cocycles. Note that Gl(1) is the Abelian multiplicative group R of non-vanishing real numbers. Clearly, Gl(1) = Z2 R+ ; and for any a 2 Gl(1) we have the decomposition (13) a = sign(a) jaj; sign(a) 2 Z2; jaj 2 R+ : Theorem 7.2. Let A; B 2 G (1). Then they are Gl(1)-cohomologous if and only if (i) The Z2-cocycle sign(A(!)B (!)) is a Z2-coboundary; (ii) The R+ -cocycle jA(!)B ?1 (!)j is an R+ -coboundary. Proof. Immediate from Lemma 1.2 and the fact that Z2 and R+ are abelian. Remark 7.3. (i) In the geometrical sense, the condition (i) is an orientation condition (note that it is exactly condition (4.8.33) of Theorem 4.8.1 of [21]), whereas the condition (ii) is a radial (growth rate) one. (ii) Since the only compact subgroup of R+ is the trivial subgroup, the results of Subsection 3.4 readily give criteria for an R+ -cocycle to be an R+ -coboundary (see also Schmidt [25] and Zimmer [32]). 7.2. Classi cation of two-dimensional linear cocycles. We now give a complete classi cation of the two-dimensional linear cocycles. In particular, we will improve the results of Thieullen [28] and Oseledets [24]. We mention that Thieullen and Oseledets have used the method of barycenters to classify two-dimensional cocycles in terms of invariant measures, as we also do below in de ning the classes IIi , i = 0; 1; 2. They dealt with cocycles satisfying the integrability conditions of the multiplicative ergodic theorem, but their results hold verbatim for general

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two-dimensional linear cocycles. We also note that the results of Thieullen and Oseledets follow immediately from Zimmer [33]. Using our ndings from Section 7.1, we can give a complete classi cation of G (2). De nition 7.4. A two-dimensional cocycle A 2 G (2) is called of class IIi , i = 0; 1; 2, if A has, respectively, no, exactly one or at least two invariant one-dimensional random subspaces. Proposition 7.5. (i) The classes IIi, i = 0; 1; 2, are disjoint and invariant with respect to cohomology. Their union is the whole of G (2). (ii) A cocycle A 2 G (2) is of class IIi , i = 0; 1; 2, if and only if the action of A on the projective space RP 1 has, respectively, no, exactly one or at least two invariant random Dirac measures. Proof. Obvious. Theorem 7.6. (i) A cocycle A 2 G (2) is of class II2 if and only if A is cohomologous to a diagonal cocycle diag(a1 ; a2 ). (ii) Two cocycles A = diag(a1 ; a2 ) and B = diag(b1 ; b2 ) of class II2 are Gl(2)cohomologous if and only if either a1 b1 and a2 b2 in G (1) or a1 b2 and a2 b1 in G (1). Proof. Part (i) and the \if" part of (ii) are easy. The \only if" part of (ii) is an immediate consequence of Proposition 2.13. Theorem 7.7. (i) A cocycle A 2 G(2) is of class II1 if and only if A is Gl(2)cohomologous to a triangular cocycle a10(!) aa12((!!)) and A is not Gl(2)-coho2 mologous to a diagonal cocycle, which is equivalent to the condition that the functional equation (14) f (!) = aa1 ((!!)) f (!) + aa12((!!)) 2 2 has no measurable solution. (ii) Two cocycles a (!) a (!) b (!) b (!) 1 12 12 A= and B = 1 0 0 a2 (!) b2 (!) of class II1 are Gl(2)-cohomologous if and only if ai is Gl(1)-cohomologous to bi by some cohomology ci , i = 1; 2, and the functional equation (15) f (!) = ab1 ((!!)) f (!) + b12 (a!()!c2)(!) ? b1(!a )(c!1 ()!a)a(!12)(!) 2 2 1 2 has a measurable solution. Proof. Let A 2 G (2) be of class II1 . Then it has a one-dimensional invariant random subspace U (!) = spanff1(!)g. Extend f1 to a random basis f = ff1(!); f2 (!)g of aR2(.!)It ais easily seen that in the basis f the cocycle A has the matrix form 1 12 (! ) . Since A is not of the class II , it is not Gl(2)-cohomologous 2 0 a2 (!) to a diagonal cocycle. Now, if A is Gl(2)-cohomologous to a diagonal cocycle B by a random linear map C , then B has two one-dimensional invariant random subspaces. Obviously C (!)f1 (!) := g1 (!) is also a one-dimensional invariant random subspace of B .


43

Clearly, we can nd another one-dimensional random subspace spanfg2(!)g of B such that fg1 (!); g2 (!)g is a basis of R2 for all ! 2 . Therefore, appropri a by(!an ) a ( ! ) 1 12 ate choice of a random basis, the triangular cocycle A(!) = 0 a2 (!) is Gl(2)-cohomologous to a diagonal cocycle if and onlyif there is a measurable map f : ! R such that the random matrix C (!) = 10 f (1!) furnishes a a (!) 0 Gl(2)-cohomology between A and the diagonal cocycle 10 a (!) . This is 2 equivalent to the condition that f is a measurable solution of the equation f (!)a2 (!) = a1 (!)f (!) + a12 (!); which is equivalent to (14). Thus (i) is proved. To prove (ii), note that a Gl(2)-cohomology C between two cocycles A and B of the class II1 must transform their unique one-dimensional invariant random subspaces one c (!) into another. Therefore, the cohomology C has a triangular f ( ! ) 1 form 0 c2 (!) in an appropriate random basis. Furthermore, clearly ci furnishes a Gl(1)-cohomology between ai and bi , i = 1; 2. >From the cohomology equation it follows that the measurable function f must satisfy the equation (15). De nition 7.8. A cocycle A 2 G (2) of class II0 is said to be of class II00 if A admits an invariant random measure which is equivalent to the Lebesgue measure of S 1 ; otherwise A is said to be of class II10 Theorem 7.9. (i) The classes IIi0, i = 0; 1, are disjoint and invariant with respect to cohomology. Their union is the whole of the class II0 . (ii) Let A be of class II0 . Then A is of class II00 if and only if A is cohomologous to a conformal cocycle. (iii) Let A be of class II0 . Then A is of class II10 if and only if the action of A on the projective space RP 1 admits exactly one invariant random measure supported on an inseparable pair of random points of RP 1 . Proof. Easy. Theorem 7.10. (i) A cocycle A is of class II10 if and only if A is Gl(2)-cohomologous to a block-conformal cocycle with one-dimensional blocks, nontrivial law of permutation and A is not cohomologous to a conformal cocycle. (ii) Two cocycles A and B of class II10 are Gl(2)-cohomologous if and only if their laws of permutation are cohomologous (as cocycles taking values in the group of permutations (2) ' Z2) and the Gl(1)-cocycles generated by them over the extended probability space := f1; 2g are Gl(1)-cohomologous. Proof. Easy. De nition 7.11. Let A be a cocycle of class II00. We say that A is of class II00;n if some, hence any H 2 H([A]) has exactly n elements, n 2 N . If H 2 H([A]) has in nitely many elements then we say that A is of class II00;1 . Lemma 7.12. The classes II00;n , n = 1; 2; : : : ; 1, are disjoint and invariant with respect to cohomology. Their union is the whole of the class II00 .

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Proof. Obvious.

Theorem 7.13. (i) Let A be a cocycle of class II00;n , n = 1; 2; : : : ; 1. Then A~(!) := j det A(!)j?1=2 A(!) 2 G (2) is Gl(2)-cohomologous to an orthogonal cocycle of class II00;n . If 1 < n < 1 then the cocycle A~ is Gl(2)-cohomologous to a cocycle taking values in the Abelian group Gn := fexp(k 2ni ) j k = 0; : : : ; n ? 1g of rotations on angles k 2n of S 1 , and Gn 2 H(A~). The class II00;1 is empty. (ii) Two cocycles A and B of class II00;n , n < 1, are Gl(2)-cohomologous if and only if the R+ -cocycles j det A(!)j and j det B (!)j are R+ -cohomologous and the co-

cycles A~ and B~ (which, by (i), can be considered as Gn -cocycles) are cohomologous. (iii) A cocycle A 2 G (2) is of class II00;1 if and only if A~ is cohomologous to an elementary cocycle. Proof. (i) By Theorem 7.9, A is cohomologous to a conformal cocycle, hence A~ is cohomologous to an orthogonal cocycle. For simplicity we assume that A is conformal, hence [A(!)] = j det A(!)j?1=2 A(!) = A~(!) for all ! 2 and [A] is orthogonal. Clearly, [A] is of class II00;n . If n < 1, since H([A]) 3 H O(2) has n elements, it coincides with the group Gn . This implies that the cocycle [A] is cohomologous to a cocycle taking values in Gn . The class II00;1 is empty because any A 2 II00;1 would leave any one-dimensional subspace invariant, and would hence belong to II2 . Part (ii) follows immediately from Corollary 1.3 and (i). To prove (iii), note that the condition that the closed subgroup H([A]) 3 H O(2) has in nitely many elements is equivalent to the condition that either H = O(2) or H = SO(2) which in turn is equivalent to the condition that the cocycle A is elementary.

8. Relation to the multiplicative ergodic theorem Let A 2 G (d). Assume now that A satis es the following integrability conditions: (16) log+ kA()1 k 2 L1 (P): Then the multiplicative ergodic theorem (see Oseledets [23], and also Arnold [1]), which we shall abbreviate as MET, applies to the cocycle A. According to the MET, A has Lyapunov exponents 1 > : : : > p with multiplicities d1 ; : : : ; dp , which are independent of ! due to the ergodicity of . Furthermore, the phase space Rd is decomposed into the direct sum of Oseledets subspaces Ei (!) of dimensions di corresponding to the Lyapunov exponents i , i = 1; : : : ; p, i.e. lim n?1 log kA (n; !)xk = i (!) () x 2 Ei (!)nf0g; n!1

where k k denotes the standard Euclidean norm of Rd . The subspaces Ei (!) are measurable and invariant with respect to A, i.e., A(!)Ei (!) = Ei (!). We note that the statements of the MET hold on an invariant set of full P-measure. The Lyapunov spectrum f(i ; di ); i = 1; : : : ; pg of A consists of the Lyapunov exponents 1 ; : : : ; p and their multiplicities d1 ; : : : ; dp . Denote by GIC (d) G (d) the space of those cocycles which satisfy the integrability conditions (16) of the MET. Obviously, if A is orthogonal, then A 2 GIC (d). Let B 2 G (d); we say that B has a spectral theory if the assertions of the MET hold true for B . Denote by GSP (d) the space of all Gl(d)-cocycles which have a spectral theory. Clearly, GIC (d) GSP (d) G (d).


45

Obviously, if A 2 GSP (d) has more than one Lyapunov exponent, then A is decomposable. Therefore, if A 2 GSP (d) is irreducible then A has only one Lyapunov exponent. De nition 8.1. Two cocycles A; B 2 G (d) are called Lyapunov cohomologous if they are cohomologous by a random linear operator C : ! Gl(d) which satis es the following conditions for almost all ! 2 : lim 1 log kC 1 (n !)k = 0: n!1 n In this case C is called a Lyapunov cohomology. Clearly, if A 2 GSP (d), B 2 G (d) and B is Lyapunov cohomologous to A, then B 2 GSP (d). Therefore, the space GSP (d) is invariant with respect to Lyapunov cohomology. Proposition 8.2. If two linear cocycles A and B from GSP (d) are cohomologous by a random linear map C : ! Gl(d), then C is a Lyapunov cohomology, and hence A and B are Lyapunov cohomologous, whence they have the same Lyapunov spectrum. Proof. See Arnold [1, Proposition 4.1.9]. De nition 8.3. (i) Let x and y be two non-vanishing vectors of Rd , then the angle \(x; y) is the conventional angle between vectors x and y in the plane spanned by x and y, i.e., := \(x; y) is the unique number from the closed interval [0; =2] ij such that cos = kjhxx;y k kyk , where kk and h; i denote the standard Euclidean norm and Euclidean scalar product of Rd . (ii) Let U and V be nontrivial independent linear subspaces of Rd , then the angle \(U; V ) between them is \(U; V ) := inf f\(x; y) j 0 6= x 2 U; 0 6= y 2 V g: Proposition 8.4. Assume that A 2 GSP (d) and let U1(!); : : : ; Un(!) be independent r-dimensional random linear subspaces of Rd such that, for any ! 2 , ? A(!) U1 (!) [ [ Un (!) = U1 (!) [ [ Un(!): Then there is a set ~ of full P-measure such that, for any i 2 f1; : : : ; ng and all ! 2 ~ , ?1 log \ Ui (n !); [ Uj (n !) = 0: lim n n!1 j 6=i

Proof. First we prove the proposition for the case n = 2. Let U; V Rd be linearly independent invariant random subspaces of A. We will show that, for almost all ! 2 , (17) lim n?1 log \(U (n !); V (n !)) = 0: n!1 A little elementary geometry and trigonometry show that, for any two nonvanishing vectors x and y of Rd and for any M 2 Gl(d), \(x; y) ?1 (18) 2kM k kM ?1k \(Mx; My) 2kM k kM k\(x; y): For all ! 2 , put (!) := \(U (!); V (!)):

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>From (18) it follows that, for almost all ! 2 , (!) ?1 2kA(!)k kA(!)?1k (!) 2kA(!)k kA(!) k (!): Therefore, for almost all ! 2 and all n 2 N , (!) n ?1 2kA(n; !)k kA(n; !)?1 k ( !) 2kA(n; !)k kA(n; !) k (!): Since A 2 GSP (d), this implies that, for almost all ! 2 , n ? lim sup log ( !) lim sup 1 log k (n; !)k + log k (n; !)?1 k < 1: n!1

n

n!1

n

A

A

Therefore, a theorem of Tanny [27] implies that, for almost all ! 2 , n lim sup log ( !) = 0:

n n!1 ? 1 n Similarly, lim inf n!1 n log ( !) = 0 for almost all !

2 . The estimates for the case n ! ?1 is analogous. Thus, the proposition is proved for the case n = 2. Now, we address the general case of arbitrary n 2. For all ! 2 , put [ f (!) := 1min \ U ( ! ) ; U ( ! ) : i j in j 6=i

>From the assumption it follows that, for any ! 2 there is a permutation fk1 ; : : : ; kn g of f1; : : : ; ng such that A(!)Ui = Uki (!) for all i = 1; : : : ; n: This, together with (18), implies that, for almost all ! 2 , f (!) ?1 2kA(!)k kA(!)?1k f (!) 2kA(!)k kA(!) k f (!): The same argument as above for the case n = 2 implies that almost surely limn!1 n?1 log f (n !) = 0. Proposition 8.5. Let A 2 GSP (d) be irreducible. Then there exists a block-conformal cocycle B 2 GSP (d) which is Lyapunov cohomologous to A. Proof. Since A is irreducible, it is cohomologous to a block-conformal cocycle, hence there are r-dimensional linearly independent random subspaces U1 (!); : : : ; Un (!) of Rd such that, for any ! 2 , ? A(!) U1 (!) [ [ Un (!) = U1 (!) [ [ Un(!): Take and x random orthonormal bases ff1 (!); : : : ; fr (!)g; : : : ; ff(n?1)r+1(!); : : : ; fd(!)g of U1 (!); : : : ; Un (!), respectively. Then f := ff1(!); : : : ; fd(!)g is a random basis of Rd . Denote by e = fe1 ; : : : ; edg the standard Euclidean basis of Rd . By Proposition 8.4, the linear random map C : ! Gl(d) which maps the basis f to the basis e for all ! 2 is a Lyapunov cohomology. Clearly, A is cohomologous to a conformal Gl(r)-cocycle. It is easily seen that we can nd an r-dimensional Lyapunov cohomology which transforms A into a conformal matrix Gl(r)-cocycle. Combining this Lyapunov cohomology with C we obtain a Lyapunov cohomology which transforms A into a block-conformal matrix cocycle, which is of course in GSP (d). We next address the question whether GSP (d) is closed under cohomology. f

f


47

Lemma 8.6. Let ( ; F ; P) be a standard probability space and be ergodic and aperiodic. Then there exists a measurable map f : ! R+ such that, for almost all ! 2 , n lim sup log fn( !) = 1: n!1

Proof. Since ( ; F ; P) is a standard probability space and is ergodic and aperiodic, the Rohlin-Halmos lemma (see Cornfeld at al. [4]) implies that for any n 2 N there is a set En 2 F such that the sets En ; En ; : : : ; n?1 En are disjoint and ? S n ? P i=01 i En ) > 3=4, which implies that, for all k = 0; 1; : : : ; n ? 1, 3 1 k (19) 4n P( En ) n : Now, by induction, we shall construct two sequences of measurable sets fFi g and fGi g satisfying the following conditions: (a) For any forSany i 2 N there is ki 2 f0; 1; : : : ; 2i+2 ? 1g such that Fi = ki E2i ; (b) Gi = Fi n ji?=11 Fi ; (c) P(Gi ) 2?i?3 . Put F1 = G1 = E2 = E8 . By (19), F1 and G1 satisfy the conditions (a) to (c). Suppose that we have constructed F1 ; : : : ; Fi?1 and G1 ; : : : ; Gi?i 1 satisfying the conditions (a) to (c). Clearly, from the disjoint sets E2i? ; : : : ; 2 ??S1 E2i there is a set?ki E2i , ki 2 f0; 1; : : : ; 2i+2 ? 1g, such that P ki E2i \ ji?=11 Fj S S 2?i?2 P ji?=11 Fj . Put Fi := ki E2i and Gi := Fi n ji?=11 Fj . By (19) and the choice of Fi ; Gi we have +2

3

+2

+2

+2

+2

+2

+2

P(Gi ) =

i[ ?1 i?1 X ? ? 3 ? i ? 2 Fj 2i+4 ? 2 P(Fi ) P(Fi ) ? P Fi \ j =1 i ? 1 3 ? 2?i?2 X 2?j?2 3 2i+4 2i+4 j =1

j =1

? 2?i?2 2?2 = 2?i?3 :

Thus, the sequences fFi g and fGi g satisfy the conditions (a) to (c). Obviously, the sets Gi , i 2 N , are disjoint. Construct a function g : ! R+ by setting ( 2i+3 for ! 2 G ; i 2 N; i g(!) := S G: 1 for ! 2 n 1 i=1 i Obviously, g(!) 1 for all ! 2 , and g is measurable and is not integrable. We claim that, for almost all ! 2 , (20) lim sup n?1 g(n !) = 1: n!1

For, suppose the opposite is true. Then, by a theorem of Tanny [27], there is ~ 2 F such that P( ~ ) = 1 and for all ! 2 ~ lim n?1 g(n !) = 0: n!1 For all K 2 N , put U (K ) := f! 2 ~ j n?1 g(n !) < 1=100 for all n K g:

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S

Clearly, the sets U (K ), K 2 N , are measurable, U (K ) U (K +1) and 1 i=1 U (K ) =

~ . Therefore, there is 100 < L 2 N such that P(U (L)) > 3=4. Now, for any ! 2 U (L) and n 2 fL; L + 1; : : : ; 81Lg we have (21) g(n !) < n=100 < L: Clearly, there is 3 < i 2 N such that 2i L < 2i+1 . By (21) and the de nition of g, for any ! 2 U (L) and n 2 fL; L + 1; : : : ; 81Lg, n ! 62 Gi , equivalently n U (L) \ Gi = ;. Put V := 40L U (L). Then m V \ Gi = ; for all ?40L < m < 40, equivalently V \ l Gi = ; for all ?40L < l < 40L. Thus, (22) P(V ) > 3=4; and V \ l Gi = ; for all ? 40L < l < 40L: By the de nition of Gi we have Gi Fi = ki E2i , hence the sets

?ki Gi ; ?ki +1 Gi ; : : : ;

are disjoint, so that for the set

G := we have

2i+2[ ?ki ?1

j =?ki

+2

2i+2 ?ki ?1

Gi

j Gi

P(G) = 2i+2 P(Gi ) 2i+2 2?i?3 = 1=2:

Furthermore, from (22), from the de nition of G and the inequality 40L > 2i+2 it follows that V \ G = ;, which implies P(V [ G) = P(V )+ P(G) > 1. Thus we arrive at a contradiction, and the claim is proved. Put f (!) := exp(g(!)), and we have proved the lemma. Proposition 8.7. Let be a nonatomic space. (i) The space GSP (d) is not invariant with respect to cohomology. (ii) For any A 2 GSP (d) there exist M > 0 and B 2 GIC (d) such that B A and kB (!)k < M for all ! 2 . (iii) There is an A 2 G (d) such that the entire cohomology class of A lies outside GSP (d). Proof. (i) Take the function f from Lemma 8.6. For all ! 2 , set B (!) = f (!)A(!)f (!)?1 : Clearly B 2 G (d) and B is cohomologous to A. Suppose that B 2 GSP (d). Then, by Proposition 8.2, the random linear map C (!) := f (!)I is a Lyapunov cohomology, hence, for almost all ! 2 , n lim log kC ( !)k = 0: n!1

This implies, for almost all ! 2 ,

n

n

log f ( !) = 0; nlim !1 n which contradicts the choice of f . Therefore, B 62 GSP (d). (ii) See Nguyen Dinh Cong [21, Proposition 2.4.14, p. 38]. (iii) Clearly there is a measurable function g : ! (1; 1) such that almost surely nX ?1 lim 1 log g(i !) = +1: n!1 n i=0


49

Put A(!) := g(!) id for all ! 2 . Suppose that there is a measurable map C : ! Gl(d) such that B (!) := C (!)?1 A(!)C (!) 2 GSP (d). Then nX ?1 nX ?1 1 1 i log g( !) = lim inf log kA(n; !)k lim inf n!1 n i=0 n!1 n i=0

?1 1 nX log kC (n?1 !)B (n; !)C (!)k = lim inf n!1 n i=0 n ? X 1 1

?1 1 nX log k ( n; ! ) k + lim inf log kC (n?1 !)k < +1: = nlim B n!1 n i=0 !1 n i=0 Thus we arrive at a contradiction.

Proposition 8.8. There is a dynamical system ( ; ) such that for any A 2 GIC (d) there is B 2 GSP (d) n GIC (d) such that B A. Thus GIC (d) is not invariant with respect to Lyapunov cohomology. Proof. We use a construction of Gerstenhaber (see Halmos [14, p. 32]). Let a2n?1 = a2n = (2a)?1 n?3=2 ; n 2 N ; where a =

1 X i=1

n?3=2 :

De ne to be the union of intervals Xn = [0; an ), n 2 N , presented in the form

= f! = (n; x) j n 2 N ; x 2 Xn g, with the Borel -algebra and Lebesgue measure. Clearly is a probability space. Let T be an arbitrary ergodic transformation of X1 preserving the Lebesgue measure. De ne to be the (induced) transformation of acting by the formula x < an+1 ; (n; x) := ((1n; +Tx1); x) if if x an+1 : Then is an ergodic measure-preserving transformation of . De ne a measurable map f : ! R+ by putting f (!) = n1=2 for ! 2 X2n and f (!) = ?n1=2 for ! 2 X2n?1 . Then it is easily seen that f is not integrable and, for all ! 2 , lim 1 f (n !) = 0: (23) n!1 n

Put g(!) := exp(f (!)). Let A 2 GIC (d) be arbitrary, put B (!) := g(!)g(!)?1 A(!): Then B is Lyapunov cohomologous to A due to (23), hence B 2 GSP (d). Suppose that B 2 GIC (d), then the function f (!) ? f (!) is integrable because A 2 GIC (d), which is easily seen not to be the case. Therefore, B 62 GIC (d).

Remark 8.9. Using the construction of Lemma 8.6 one can show that the dynamical system ( ; ) in Proposition 8.8 can be an arbitrary aperiodic ergodic dynamical system on a Lebesgue space. We shall use the following fact about quotient cocycles.

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Lemma 8.10. Let A 2 GSP (d) and f = ff1(!); : : : ; fd(!)g be a random orthonormal basis of Rd . Assume that A has the following matrix representation with respect to f A (!) A (!) 1 12 A(!) = 0 A (!) 2

with A1 (!) 2 Gl(m) and A2 (!) 2 Gl(d ? m) for all ! 2 . Then A1 2 GSP (m), A2 2 GSP (d ? m) and the Lyapunov spectrum of A is the union of the Lyapunov spectra of A1 and A2 . We refer for the rather simple proof to Arnold [1]. Lemma 8.11. Let A 2 GSP (d) and let U Rd be an invariant random subspace of A. Then there is a Lyapunov cohomology C : ! Gl(d) such that, for all ! 2 , A (!) U ? 1 C (!) A(!)C (!) = 0 ARd=U (!) :

Proof. Take a random orthonormal basis f := ff1 (!); : : : ; fd(!)g of Rd such that ff1(!); : : : ; fr (!)g is a basis of U (!) for all ! 2 and de ne C to be the basis change from f to the standard Euclidean basis of Rd . Theorem 8.12. For any A 2 GSP (d), there is a Lyapunov cohomology C : !

Gl(d) transforming A into its Jordan normal form.

Proof. Use the argument similar to that of the proof of Lemma 8.11 for a maximal invariant ag of A. Then apply Proposition 8.5 to the irreducible subcocycles on the diagonal. Corollary 8.13. For any A 2 GSP (d), there are A^ 2 GSP (d) and A~ 2 G (d)nGSP (d) which are both Jordan normal forms of A. Proof. By Theorem 8.12, there is A^ 2 GSP (d) which is a Jordan normal form of A. Take a function f from Lemma 8.6. For all ! 2 , set A~(!) = f (!)A^(!)f (!)?1 : Clearly A~ 2 G (d) n GSP (d) and is a Jordan normal form of A.

Proposition 8.14. Suppose that A 2 GSP (d), and 0 A (!) 1 BB 10 A2 (!) CC A^ = B C ... @ 0 A 0 0 0 Ap (!) is a Jordan normal form of A which belongs to GSP (d). Then Ai 2 GSP (di ),

di := dim Ai , i = 1; : : : ; p. Moreover, Ai has one-point Lyapunov spectrum, and Lyapunov exponent i is a Lyapunov exponent of A, i = 1; : : : ; p; the Lyapunov spectrum of A is the collection of the Lyapunov exponents i with multiplicities di , i = 1; : : : ; p (here possibly i = j for i 6= j ). Proof. Use Lemma 8.10.


51

Acknowledgment

The work of the second-named author is supported by the Deutsche Forschungsgemeinschaft, Germany and the work of the third-named author was partially supported by the Volkwagen-Stiftung, Germany, the University of Bremen, Germany, and the Cariplo Foundation for Scienti c Research, Italy. The authors would like to thank H. Crauel for providing the proof of Lemma 4.8 and helpful conversations on closed random sets, A. L. Onishchik, E. B. Vinberg and Le Hong Van for helpful discussions on the theory of Lie groups and algebraic groups, K. Schmidt and W. Krieger for helpful discussions on algebraic ergodic theory, and G. A. Margulis and A. Furman for helpful conversations on algebraic ergodic theory and group actions.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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Institute for Dynamical Systems, University of Bremen, P. O. Box 330 440, 28334 Bremen, Germany and Institute for Applied Mathematics, University of Heidelberg, Germany

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Chair of Probability, Department of Mechanics and Mathematics, Moscow State University, Moscow 119 899, Russia

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