JORDAN NORMAL FORM FOR LINEAR COCYCLESLUDWIG ARNOLD, NGUYEN DINH CONG, AND VALERY IUSTINOVICH OSELEDETSAbstract. The paper is devoted to the problem of classi�cation of linear co-cycles up to cohomology. The main result is a theorem on the Jordan normalform saying that any linear cocycle is cohomologous to a block-triangular cocy-cle with irreducible block-conformal cocycles on the diagonal. Two invariantsof cocycle cohomology, the algebraic hull and the set of invariant measures, andtheir interrelations are studied. We show that all random invariant measuresof a cocycle are determined by the algebraic hull and, up to a cohomology, aredeterministic. For orthogonal cocycles the two invariants are equivalent andthey give a sub-relation of the equivalence relation of cocycle cohomology. Acomplete 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 invariantsubspaces corresponding to di�erent Lyapunov exponents. A by-product ofour theory is a classi�cation of amenable Lie subgroups of Gl(d;R).Contents1. Introduction 22. Decomposition of linear cocycles 43. The algebraic hull and invariant measures of a cocycle 123.1. De�nitions 123.2. Relation between algebraic hull and invariant measures 133.3. Structure of invariant measures 143.4. Criterion for orthogonal and conformal cocycles 164. Block-conformal cocycles 184.1. The linear cover of a subset of Sd�1 184.2. Structure of a cocycle on the span of the support of an invariantmeasure. Block-conformal cocycles 255. The Jordan normal form 325.1. The Jordan form 325.2. An algorithm for constructing the Jordan normal form 356. Orthogonal cocycles 376.1. The algebraic hull of an orthogonal cocycle 376.2. Invariant measures of orthogonal cocycles 386.3. Equivalence of orthogonal cocycles 396.4. About classi�cation of orthogonal cocycles 407. Classi�cation of low-dimensional cocycles 417.1. Classi�cation of one-dimensional linear cocycles 417.2. Classi�cation of two-dimensional linear cocycles 411991 Mathematics Subject Classi�cation. Primary 58F36, 58F35; Secondary 58F11, 28D05.Key words and phrases. Jordan normal form, cocycle, classi�cation, cohomology, algebraichull, invariant measure. 1

2 ARNOLD et al.8. Relation to the multiplicative ergodic theorem 44Acknowledgment 51References 511. IntroductionLet (;F ;P) be a probability space, and � an automorphism of (;F ;P) pre-serving the probability measure P. Throughout this paper, we assume that � isergodic. The non-ergodic case can be reduced to the ergodic one by using theergodic 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 measurablemapping 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. Itgenerates a linear cocycle over the dynamical system � via�A(n; !) :=8<: A(�n�1!) � : : : �A(!); n > 0;id ; n = 0;A�1(�n!) � : : : �A�1(��1!); n < 0:Conversely, if we are given a linear cocycle over �, then its time-one map is a linearrandom map. Therefore, the correspondence between A and �A is one-to-one andwe are free to choose one from them to work with. The above construction appliesto any topological group G in place of Gl(d) (in particular, G can be a Lie subgroupof 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 linearoperators 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 allGl(d)-cocycles will be denoted by G(d).Since we deal with discrete-time cocycles we can always neglect sets of nullmeasure, and we shall identify the randommappings 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 existsa 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 co-homology to block-triangular form with block-conformal irreducible subcocycles onthe diagonal; these subcocycles are uniquely, up to ordering, determined by A.Y. Guivarc'h and A. Raugi [13] considered cocycles which are products of in-dependent and identically distributed random matrices satisfying the integrabilityconditions of the multiplicative ergodic theorem and an additional condition called\total irreducibility". They proved that such cocycles are Lyapunov cohomologousto block-diagonal cocycles with conformal blocks.We would like to mention here a closely related paper of R. Zimmer [33], wherehe proved that every cocycle is cohomologous to a cocycle taking values in anamenable subgroup. We note that in the proof of Theorem 5.5 of [33] Zimmer has

JORDAN NORMAL FORM FOR LINEAR COCYCLES 3used a conformal structure on the ags which is somehow related to our block-conformal structure, but he did not derive a block-triangular form with block-conformal subcocycles on the diagonal, not to say a Jordan form (we note thatthe normal form problem was not his aim). Our Jordan form theory gives anotherproof of Zimmer's result about amenability. As a by-product we obtain the Jordannormal 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 measuresof a linear cocycle: in Section 3 we show that up to a cohomology all invariantmeasures are deterministic Lebesgue measures of algebraic manifolds.The key tools for our work are Furstenberg's lemma (Lemma 3.20), Zimmer'sconstruction of linear cover (see Section 4) and techniques from representationtheory (Section 2).This paper is organized as follows. In the remaining part of the introduction weprove a useful �rst reduction of cocycles. In Section 2 we present the techniques ofrepresentation theory applied to cocycles and derive a preliminary version of thenormal form, namely a block-triangular form with irreducible cocycles on the di-agonal. It remains to show that irreducible cocycles are block-conformal. To do sowe 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 mea-sure. In Section 5 we present the �nal Jordan normal form and an algorithm forconstructing it. Since the block-conformal form suggests to study conformal andorthogonal cocycles we devote Section 6 to the investigation of orthogonal cocyclesfor which a better classi�cation result is obtained. The remaining two sections aredevoted to the complete classi�cation of one- and two-dimensional cocycles and tothe 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 forth-coming paper [2] we use the normal form theory to prove that the cocycles withsimple 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 ofthe multiplicative ergodic theorem of Oseledets [23]. The ag (and splitting) weobtain is �ner than the one of Oseledets; moreover, no integrability condition isrequired. Under the corresponding integrability conditions the theory developedin this paper facilitates another proof of the multiplicative ergodic theorem (moreprecisely, simpli�es Oseledets' original proof [23]), by applying Oseledets' argumentsto the Jordan normal form. Note that we need not extend cocycles by means of theorthogonal group as we already have (block-)triangular form with block-conformalsubcocycles on the diagonal.In concluding the introduction we show that the general case of a linear cocyclecan be easily reduced to the case of a unimodular cocycle. We �rst give a generalreduction 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 thatG = H �L and H \L = feg. Then any element g 2 G is uniquely represented in theform g = hl with h 2 H and l 2 L. Therefore, a G-cocycle A : ! G is uniquelyrepresented in the form of the product of an H-cocycle AH and an L-cocycle ALsuch that A(!) = AH (!) � AL(!) for all ! 2 .

4 ARNOLD et al.Lemma 1.2. Assume the above decomposition of G, and moreover that any h 2H 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 L-cohomologous to BL.Proof. If A is G-cohomologous to B then there is a measurable map C : ! Gsuch that, for all ! 2 , A(!) = C(�!)�1B(!)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 thatAH(!) = CH(�!)�1BH(!)CH (!);AL(!) = CL(�!)�1BL(!)CL(!):The converse statement is clear.We denote by Sl(d) the Lie group of (d�d)-matrices with determinant �1 (notethat the value �1 is allowed). Let R+� denote the (Abelian) multiplicative group ofpositive real numbers. Clearly, Gl(d) �= Sl(d) �R+� . Note that R+� is in the center ofGl(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 theR+� -cocycles j detAj and j detBj are cohomologous and the Sl(d)-cocyclesj detAj�1=dA and j detBj�1=dB are cohomologous.2. Decomposition of linear cocyclesIn this section we show that the study of general linear cocycles can be reduced tothe one of irreducible cocycles. The development is largely parallel to the theory oflinear 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 thetheory of representations was observed by Mackey [19] and Zimmer [31, 32].The aim of this section is to derive a version of the Jordan{H�older theorem forcocycles (Theorem 2.17). The long and rather tedious preparations serve to providea 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 closedsubsets 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(!)) := infy2C(!) 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 sequencecn, n 2 N, of measurable maps cn : ! Rd such thatC(!) = fcn(!) j n 2 Ng for all ! 2 ;where �A denotes the closure of A � Rd . In particular, if C is a closed random setthen there exists a measurable selection, i.e., a measurable map c : ! Rd suchthat c(!) 2 C(!) for all ! 2 .

JORDAN NORMAL FORM FOR LINEAR COCYCLES 5For the proof see Castaing and Valadier [3, Theorem III.9, p. 67].For random subspaces we can choose measurably an orthonormal basis as thefollowing result of Walters [30] shows.Proposition 2.3. Let V be a random subspace. Then(i) r(!) := dimV (!) : ! N is measurable, and for each 1 � k � d, ! 7! V (!)is a measurable map from f! 2 j r(!) = kg into the Grassmannian manifoldGrk(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 invariantwith respect to A if A(!)U(!) = U(�!) for almost all ! 2 .Remark 2.5. Since A(!) preserves dimension and � is ergodic, for any invariantrandom subspace U , the dimension dimU(!) 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 � dimE. By choosing a basis fe1; : : : ; edg of Rd such thatfe1; : : : ; erg 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; : : : ; edg. If E � F � Rdare linear subspaces the quotient subspace F=E can be identi�ed, by means of alinear isomorphism, with a linear subspace of F � Rd of dimension dimF �dimE.If U � V � Rd are random subspaces, then V=U de�ned !-wise is a random linearspace and can be identi�ed, by means of a random linear isomorphism, with arandom subspace of V � Rd .Now let A 2 G(d) and U � Rd be an invariant random subspace of A of dimensionr. Then A induces a linear cocycle AU on U by AU (!)x := A(!)x, x 2 U(!), anda linear cocycle on Rd=U by ARd=U (!)y := A(!)y, y 2 Rd=U(!). Take a randombasis 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 formA = � A1(!) �0 A2(!) � ;where A1 is an r-dimensional random matrix and A2 a (d� r)-dimensional randommatrix. Obviously, A1 2 G(r) and A2 2 G(d � r). Clearly, another choice ofsuch a random basis f leads to new cocycles A01 2 G(r) and A02 2 G(d � r) whichare cohomologous to A1 and A2, respectively. Moreover, there are random linearisomorphisms C1(!) : U(!)! Rr and C2(!) : Rd=U(!)! Rd�r mapping AU andARd=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 followingde�nition.De�nition 2.6. The cocycle A1 2 G(r) is identi�ed with AU and called the re-striction 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 cocycleof A over U ; its cohomology class is uniquely determined by A and U .

6 ARNOLD et al.Now let U and V be invariant random subspaces of A such that U(!) � V (!)for all ! 2 , dimU(!) = r, dimV (!) = p, 0 � r < p � d. Then the cocycleB := AV on V is de�ned and U is an invariant random subspace of B. IdentifyingV with Rp by means of a random linear isomorphism we have B 2 G(p) and U istransformed into an r-dimensional invariant random subspace of B in Rp , whichwe again denote by U , hence the quotient cocycle BRp=U 2 G(p� r) is de�ned andwe call it the quotient cocycle of A on V=U and denote it by AV=U . It is easily seenthat the cohomology class of the cocycle AV=U 2 G(p � r) is uniquely determinedby A, U and V . Obviously, AU = BU = (AV )U :All the above discussion becomes transparent as we choose a suitable randombasis and reduce cocycles to the block-triangular form. If U and V are randomspaces then U �= V means that there is a random linear isomorphism betweenthem.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 ofA such that U1(!) � U2(!) for all ! 2 , dimUi(!) = ri, i = 1; 2. Assume thatthere is a measurable map C : ! Gl(d) such thatA(!) = C(�!)�1B(!)C(!) for all ! 2 :(1)Then:(i) Vi(!) := C(!)Ui(!) are random subspaces invariant with respect to B,dimVi(!) = ri, i = 1; 2, and V1(!) � V2(!) for all ! 2 ;(ii) AUi � BVi as Gl(ri)-cocycles, i = 1; 2;(iii) AU2=U1 � BV2=V1 as Gl(r2 � r1)-cocycles.Proof. Part (i) follows immediately from (1). Parts (ii) and (iii) follow from (1) bychoosing 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 hasthe matrix form A = � A1(!) �0 A2(!) �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 andA2 = 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. ThenAW=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 . ThenAW=V = A(W\U)=(V \U):

JORDAN NORMAL FORM FOR LINEAR COCYCLES 7Proof. Part (i) is obvious. Parts (ii) and (iii) can be easily seen by choosing appro-priate random bases in Rd which are \adapted" to U; V;W in an obvious sense.Let A 2 G(d) have the formA = � A1(!) 00 A2(!) �with A1 2 G(r) and A2 2 G(d�r). Clearly, A1 and A2 are also the quotient cocyclesof A over Rd�r = Rd=Rr and Rr = Rd=Rd�r , respectively. In this case we say thatA is the direct sum of A1 and A2 and write A = A1 � A2. Now, if we are giventwo cocycles B1 2 G(r) and B2 2 G(d � r), then the direct sum B = B1 � B2 isde�ned by B = � B1(!) 00 B2(!) � for all ! 2 . Clearly, B0 := B2 � B1 2 G(d)is cohomologous to B.It is easily seen that if U and V are two invariant random subspaces of A suchthat U(!)�V (!) = Rd for all ! 2 , then A � A1�A2, where A1 is the restrictionof A to U and A2 is the restriction of A to V . Moreover, A1 is the quotient cocycleof 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 invari-ant 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 sub-spaces of Rd is invariant under A.(iii) The cocycle A is called completely reducible if any invariant random subspaceU of A admits invariant complement, i.e. there is an invariant random subspace U cof 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 nontrivialinvariant random subspace V of A such that V (!) is a proper subspace of U(!) foralmost all ! 2 .The following lemma is immediate.Lemma 2.10. (i) The notions of irreducibility, strong irreducibility and completereducibility 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 ifAU is an irreducible Gl(r)-cocycle. In particular, A is irreducible if and only if Rdis minimal.(ii) A is completely reducible if and only if it is the direct sum of irreducible subco-cycles.(iii) A is completely reducible if and only if Rd is the direct sum of some minimalinvariant random subspaces of A.Proof. (i) is immediate from the de�nition of minimal random subspace and of therestriction 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 = U1 � � � � � Un;

8 ARNOLD et al.where Ui, i = 1; : : : ; n, are minimal subspaces of A. Let V be an arbitrary invariantrandom subspace of A. Put Vi := V \ Ui, i = 1; : : : ; n. Then Vi is an invariantsubspaces of A contained in Ui. Since Ui is minimal Vi must either be null spaceor 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 Vis 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 thecompletely reducible linear cocycles. The case d = 1 is trivial. Suppose we aredone for all dimensions � d � 1. Let A 2 G(d) be completely reducible. If A isirreducible, 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 completelyreducible, there is an invariant complement V of U for A. It is easily seen that AVis completely reducible, hence by the inductive hypothesis V is the direct sum ofminimal subspaces of AV which are also minimal subspaces of A. Adding U to thissum 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 ofsome minimal invariant random subspaces of A:Rd = U1 + � � �+ Un:Let U be an arbitrary invariant random subspace of A. Then there are indicesi1; : : : ip 2 f1; : : : ; ng such thatRd = U � Ui1 � � � � � Uip :(2)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 let1 � i1(!) < i2(!) < � � � < ip(!)(!) � nbe the maximal set of indices such that the subspaces U(!), Ui1(!)(!); : : : ;Uip(!)(!)(!) are linearly independent. Since the linear operator A(!) is nonsingularand 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 seenthat i(!) can be chosen independently of ! 2 . Thus we write i1; : : : ; ip insteadof i1(!); : : : ; ip(!)(!). To prove (2) it su�ces to show that for any 1 � i � n,i 62 fi1; : : : ; ipg,Ui(!) � U(!)� Ui1(!)� � � � � Uim(!) for all ! 2 :(3)Since Ui(!) \ �U(!) � Ui1(!) � � � � � Uim(!)� =: Wi(!) is an invariant subspaceof A contained in Ui(!) it must either be null space or coincide with Ui(!). Themaximality 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 directsums A1 � � � � � Am and A01 � � � � � A0l of irreducible subcocycles of dimensionsd1; : : : ; 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 minimalsubspaces Rd = V1 � � � � � Vm = U1 � � � � � Ul

JORDAN NORMAL FORM FOR LINEAR COCYCLES 9such 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. Weapply Lemma 2.12 to U1 and obtainRd = U1 � Vi1 � � � � � Vipfor certain i1; : : : ; ip. ThenA01 � AU1 � ARd=Vi1�����Vip � Ak1 � � � � � Akq ;where fk1; : : : ; kqg = f1; : : : ;mg n fi1; : : : ; ipg. Since A01 is irreducible, q = 1.Now let us relabel Ai so that k1 = 1. Then A01 � A1 andRd = U1 � V2 � � � � � Vm:Therefore, A2 � � � � �Am � ARd=U1 � A02 � � � � �A0l:The induction hypothesis implies that m = l and after a relabeling Ai � A0i fori � 2.De�nition 2.14. Let A 2 G(d). Suppose that we have a strictly monotone collec-tion of invariant random subspaces (invariant ag) of A:f0g = V0 � V1 � � � � Vn�1 � Vn = Rdsuch that the subcocycles Ai, i = 1; : : : ; n, appearing in Vi=Vi�1|the quotientcocycles AVi=Vi�1|are irreducible. Then the ag fVig is called maximal (invariant) ag (of A). The number n is called the length of the ag fVig. The multi-indexr = fr1; : : : ; rng, where ri := dim(Vi=Vi�1) = dimVi � dimVi�1, i = 1; : : : ; n, iscalled the index of the ag fVig.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 withthe 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 maximalinvariant ag AU . Then there is a maximal invariant ag V 00 � � � � � V 0k+l of Asuch that V 0i (!) = Vi(!) for all ! 2 and i = 0; : : : ; k.(iii) Let U � Rd be invariant random subspace of A. Let V0 � � � � � Vk andW0 � � � � �Wl be maximal invariant ags of AU and ARd=U , respectively. Then Ahas a maximal ag V 00 � � � � � V 0k+l such thatV 0i �= � Vi for i = 0; : : : ; k;Vk �Wi�k for i = k + 1; : : : ; k + l:Proof. First we prove part (iii). Choose a random basis f = ff1(!); : : : ; fd(!)gof Rd such that spanff1(!); : : : ; fri(!)g = Vi(!) for all ! 2 , i = 1; : : : ; k; hereri := dimVi(!). By Lemma 2.8, A has the form � AU (!) �0 ARd=U (!) � withrespect to the random basis f . It is easily seen that the basis f can be chosento have, additionally, the property that the vectors ffk+1(!); : : : ; fk+pj (!)g form

10 ARNOLD et al.a basis of Wj(!) (modulo a linear isomorphism) for all ! 2 and j = 1; : : : ; l;here pj := dimWj(!). In this basis A has block-triangular form with irreduciblesubcocycles on the diagonal. Clearly the ag V 00 � � � � � V 0k+l de�ned by V 0i (!) =Vi(!) and V 0k+j(!) = spanff1(!); : : : ; fk+pj (!)g for all ! 2 , i = 0; : : : ; k andj = 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), whichwe do by induction on the dimension d of A. The case d = 1 is trivial. Supposethat we have proved (i) for all dimensions � d � 1, and A 2 G(d). Obviously Ahas at least one minimal invariant random subspace U � Rd . If U = Rd then A isirreducible and the ag f0g � Rd is maximal for A. If U 6= Rd , then AU and ARd=Uare lower dimensional cocycles hence have maximal ags. Therefore, by (iii), A hasa maximal ag.It is easily seen that a linear cocycle may have many maximal ags (e.g. diagonalcocycles) or only one maximal ag (e.g. irreducible cocycles), hence it may haveseveral (but �nitely many) indices. However, we have the following invariance ofindices, 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{H�older theorem from thetheory of representations (see Kirillov [17, Theorem 1, p. 116]) and is the mainresult of this section.Theorem 2.17. Let A 2 G(d), V0 � � � � � Vn be a maximal invariant ag of Awith index fr1; : : : ; rng. Then(i) A is cohomologous to a block-triangular cocycleA � 0B@ AV1=V0(!) � �0 . . . �0 0 AVn=Vn�1(!) 1CA ;(4)where the cocycles AV1=V0 ; : : : ; AVn=Vn�1 are irreducible. The form (4) depends onlyon the cohomology class of A.(ii) Suppose that W0 � � � � � Wm is an arbitrary maximal invariant ag of Awith index fq1; : : : ; qmg. Then m = n and, after a suitable relabeling, qi = ri andAVi=Vi�1 � AWi=Wi�1 as Gl(ri)-cocycles for all i = 1; : : : ; n.In other words, all maximal ags of A have the same length which is a cohomologyinvariant, 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 maximalinvariant ag. Suppose that (ii) has been proved for all linear cocycles having amaximal invariant ag of length n � 1. Let A 2 G(d) and V0 � � � � � Vn be amaximal invariant ag of A. Put r := dimVn�1.Let W0 � � � � �Wm be another maximal invariant ag of A. PutW 0i =Wi \ Vn�1; i = 0; : : : ;m:Then W 0i are invariant random subspaces of A andf0g =W 00 �W 01 � � � �W 0m = Vn�1:

JORDAN NORMAL FORM FOR LINEAR COCYCLES 11We claim that there is an index j 2 f1; : : : ;mg such that W 0j�1 = W 0j and thatW 0k�1 is a proper subspace of W 0k for k 6= j. To prove the claim, let W 00i denote theimage of Wi in the quotient space Rd=Vn�1, i = 0; : : : ;m. It is easily seen that W 00i(more precisely, it image in Rd�r by the random linear isomorphism mapping thequotient space Rd=Vn�1 into Rd�r ) is an invariant random subspace of the quotientcocycle ARd=Vn�1 . Since fVig is a maximal invariant ag, the cocycle ARd=Vn�1 isirreducible, hence W 00i either is the null space or coincides with Rd=Vn�1. Thisimplies that there is an index j 2 f1; : : : ;mg such thatW 000 = : : : =W 00j�1 = f0g; W 00j = : : : =W 00m = Rd=Vn�1:(5)It is easily seen that, for any i = 1; : : : ;m,Wi �= (Wi \ Vn�1)�W 00i =W 0i �W 00i :(6)Therefore, (5) implies that for i 6= j we have W 0i=W 0i�1 linearly isomorphic toWi=Wi�1.For the index j, since Wj�1 =W 0j�1 �W 0j � Vn�1 and Wj �=W 0j �W 00j we haveWj=Wj�1 �= (W 0j=W 0j�1)�W 00j :Since the invariant ag fWig is maximal and W 00j 6= f0g we must have W 0j=W 0j�1 =f0g because the cocycle AWj=Wj�1 is irreducible and the lower dimensional randomsubspaceW 0j=W 0j�1 (modulo a random linear isomorphism) is invariant with respectto AWj=Wj�1 . Therefore, the sequencef0g =W 00 �W 01 � � � �W 0j�1 �W 0j+1 � � � �W 0m = Vn�1:is strictly monotone, and its image by the random linear isomorphism identifyingVn�1 with Rr is a maximal invariant ag of the cocycle AVn�1 .On the other hand, the image of the ag V0 � � � � � Vn�1 by the random linearisomorphism identifying Vn�1 with Rr is another maximal invariant ag of thecocycle AVn�1 . Therefore, by the induction hypothesis, n = m and, up to theirorder, the cohomology classes of the cocycles (AVn�1)Vi=Vi�1 , i = 1; : : : ; n� 1, arethe same as the cohomology classes of the cocycles (AVn�1)W 0i=W 0i�1 , i = 1; : : : ; j �1; j + 1; : : : ;m. By (5){(6) and Lemma 2.8, for i = 1; : : : ; j � 1; j + 1; : : : ;m, wehave (AVn�1)W 0i=W 0i�1 � AW 0i=W 0i�1 � AWi=Wi�1 :On the other hand, obviously(AVn�1)Vi=Vi�1 � AVi=Vi�1 for all i = 1; : : : ; n� 1:Furthermore, clearly AWj=Wj�1 � AVn=Vn�1 . Thus, to prove the theorem it remainsto 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 theblock-triangular form with irreducible subcocycles on the diagonal, where the co-homology classes of these subcocycles are uniquely determined by the cohomologyclass of A (up to their order).(ii) If A has an index fr1; : : : ; rng then the subcocycles Ai of A, which come froman arbitrary maximal invariant ag of A, are (up to their order) of dimensions ri,i = 1; : : : ; n. Any two indices of A di�er only by the order of their entries.We conclude this section by saying that the central task now is to classify theirreducible subcocycles on the diagonal of the form (4).

12 ARNOLD et al.3. The algebraic hull and invariant measures of a cocycle3.1. De�nitions. First we present the notion of the algebraic hull of a cocycleintroduced 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], andHumphreys [16].Proposition 3.1. (Zimmer [35, Proposition 9.2.1]) Let G be a �xed algebraic sub-group of Gl(d) and A be a G-cocycle. Then there exists an algebraic subgroup Hof G such that A is G-cohomologous to a cocycle taking values in H but is not G-cohomologous to a cocycle taking values in a proper algebraic subgroup of H. Thegroup H is unique up to conjugacy.De�nition 3.2. (i) The conjugacy class in G of this algebraic group H is calledalgebraic 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 forall ! 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 minimalcocycle with range H.The de�nition of algebraic hull depends on the chosen algebraic group G. In thispaper, unless otherwise speci�ed explicitly (e.g., in Section 6) H(A) stands for thealgebraic 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 Sd�1 denotethe unit sphere of Rd . For a linear subspace V � Rd we denote by [V ] its projectionto Sd�1, which is its intersection with Sd�1. In general, for a nonempty set f0g 6=Q � Rd we denote by [Q] � Sd�1 its projection to Sd�1, hence [Q] is the subset ofSd�1 consisting of those x 2 Sd�1 for which rx 2 Q for some r > 0. Clearly, for0 6= x 2 Rd , [x] = x=kxk.Let A 2 Gl(d). Denote by [A] the di�eomorphism of Sd�1 induced by A, i.e.,[A](x) := [Ax]; for all x 2 Sd�1:We call [A] a linear transformation of Sd�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 lineartransformation of Sd�1. We have SGl(d) �= Gl(d)=R+� � I �= Sl(d).Denote by Pr(Sd�1) the space of all probability measure on Sd�1 equipped withthe topology of weak convergence. Then Pr(Sd�1) is separable metrizable andcompact since Sd�1 has such properties (see Dellacherie and Meyer [6, p. 73]). Itis well known that the topology of weak convergence on Pr(Sd�1) is generated bythe following metric (see Dudley [7, x 11.3]):�(�; �) := inff" > 0 j �(U) � �(U") + " for all U 2 B(Sd�1)g;where U" := fy 2 G j ky � xk < " for some x 2 Ug and B(Sd�1) denotes theBorel �-algebra of Sd�1. Fix this metric � on Pr(Sd�1).Now lew A(�) : ! Gl(d) be a random linear map. Then [A(�)] is a randomlinear transformation of Sd�1 and generates a (nonlinear) cocycle �[A] on Sd�1.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 13Since Sd�1 is compact there exists an invariant measure � on ( � Sd�1;F �B(Sd�1)) for the skew-product � of [A] which is de�ned by � : �Sd�1 ! �Sd�1,(!; x) 7! (�!; [A(!)]x), i.e., � = ��. Since Sd�1 is a compact metric space � canbe disintegrated, and for P-almost all ! 2 we have a random probability measure�! on Sd�1 such that �(d!; dx) = �!(dx)P(d!):De�nition 3.4. A measurable map� : ! Pr(Sd�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 invariantmeasure 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 ondi�erent 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 continuousaction 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, iscontinuous. In this case we call X a G-space. When a particular � is not ofimportance or it is clear which action is meant we shall write simply gx insteadof �(g)x. If X is equipped with a metric %X which generates the given topologyof X , then we call (X; %X) a metric G-space. Denote by Pr(X) the space of allprobability measure on X with the topology of weak convergence. Then Pr(X) is acompact metrizable space, since X is so. A G-cocycle A generates a skew-productdynamical system �X : �X ! �X , (!; x) 7! (�!;A(!)x). A matrix A 2 Gacts on Pr(X) by the formula A�(M) := �(A�1M) for any � 2 Pr(X);M 2 B(X).De�nition 3.5. Let G � Gl(d) be a Lie subgroup, X a G-space, and A a G-cocycle. A measurable map� : ! Pr(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 withrespect 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 actionof Gl(d) on Sd�1 de�ned by x 7! [A]x for A 2 Gl(d); x 2 Sd�1. Speaking of an in-variant measure without mentioning X we mean that the situation of De�nition 3.4is considered.3.2. Relation between algebraic hull and invariant measures. In this sub-section we shall need the notion of smooth action. A group action of a locallycompact group on a complete separable metric space is called smooth if all orbitsare 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 onSd�1, a fact which is of crucial importance for studying invariant measures of linearcocycles.

14 ARNOLD et al.Lemma 3.6. Let G be a locally compact group, X a complete metric G-space, Aa 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 almostall ! 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 measurablemap D : ! G such that �! = D(!)�!0 for almost all ! 2 .Proof. Since G � Gl(d) is algebraic, it acts smoothly on the space Pr(Sd�1) ofprobability measures of Sd�1 (see Zimmer [35, Corollary 3.2.12, p. 45]). Hence thecorollary follows from Lemma 3.6.The following theorem shows the invariance of invariant measures of a minimalcocycle with respect to the algebraic hull of the cocycle. It is the key relationbetween the algebraic hull and invariant measures.Theorem 3.8. Let A 2 G(d) be a minimal cocycle with range H and � be aninvariant measure of A. Then there is an !0 2 such that�! = �!0 for almost all ! 2 ;h�!0 = �!0 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 measurablemap D : ! H such that �! = D(!)�!0 almost surely. For all ! 2 , putB(!) := D(�!)�1A(!)D(!):Then B is cohomologous to A by D and for almost all ! 2 we haveB(!)�!0 = D(�!)�1A(!)�! = D(�!)�1��! = �!0 :Since B(!) 2 H due to its de�nition, for almost all ! 2 we haveB(!) 2 fh 2 H j h�!0 = �!0g =: StabH(�!0) =: H 0:Changing B on a null set we get an H 0-cocycle B0 cohomologous to A. SinceH 0 = StabH(�!0) � 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�!0 = �!0 for allh 2 H and �! = D(!)�!0 = �!0 almost surely.3.3. Structure of invariant measures. In this subsection we shall describe thestructure of all invariant measures of a linear cocycle. First we describe the supportsof ergodic invariant measures, for which we establish a one-to-one relation betweenthese supports and minimal invariant sets, and then we describe all minimal sets.De�nition 3.9. LetX be a compact metrizable Gl(d)-space. LetM1;M2 : ! Xbe two closed random sets (on X). We say that M1 � M2 if M1(!) � M2(!) foralmost all ! 2 . If M1(!) = M2(!) almost surely then we write M1 = M2. IfM1 �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 calledminimal if M1 �M and M1 invariant implies M1 =M .

JORDAN NORMAL FORM FOR LINEAR COCYCLES 15Lemma 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 measureon X of A. Then supp� is minimal.(ii) Let M be a minimal set of A. Then M supports at least one ergodic invariantmeasure of A. If � is a (not necessarily ergodic) invariant measure of A supportedby M then supp� =M .Proof. (i) If supp� is not minimal then there is an invariant closed random setM < supp�. Clearly M is a compact random set, hence there is an ergodicinvariant 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 inequalitysupp � � 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 invariantmeasure (see Arnold [1]). The last assertion follows immediately from the minimal-ity of M .Lemma 3.13. Let A 2 G(d) be a minimal cocycle with range H. Then the minimalsets of A are exactly those orbits Hx, x 2 Sd�1, of H on Sd�1 which are closed.Proof. Clearly the orbits Hx, x 2 Sd�1, are (not necessarily closed) deterministicinvariant sets of A. Furthermore, they are nonsingular algebraic subvarieties ofSd�1 (see Onishchik and Vinberg [22, Theorem 7, p. 104]).Let M(!) be an arbitrary invariant closed random set of A. Then there is aninvariant measure � of A such that supp�! �M(!) for all ! 2 . By Theorem 3.8,there is !0 2 such that�! = �!0 for almost all ! 2 ;(7) h�!0 = �!0 for all h 2 H:(8)Fix one y 2 supp�!0 � Sd�1. Then (8) implies that supp�!0 � Hy, which togetherwith (7) implies that M � Hy. Clearly, the boundary @Hy and closure Hy of Hyare closed deterministic invariant sets of A, and @Hy � Hy � M because M isclosed.Now suppose thatM is minimal, then there are two possibilities: either @Hy 6= ;,in this case M = @Hy = Hy, hence Hy = @Hy = Hy =M , or @Hy = ;, in whichcase Hy = Hy = M . In both cases M = Hy is a closed orbit. Conversely, if Hxis 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 ofHx.Now, having described the supports of all ergodic invariant measures we are ina 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 ergodicinvariant measures of A are exactly the deterministic normalized Lebesgue measures(i.e. the unique H-invariant measures) of the closed orbits of H on Sd�1.

16 ARNOLD et al.Proof. Let Hx be a closed orbit of H on Sd�1. By Lemma 3.13 the set Hx � Sd�1is minimal for A, hence by Lemma 3.12, Hx supports at least one ergodic invariantmeasure of A, say �.SinceH 2 H(A) is an algebraic group the orbitHx is a smooth algebraic subman-ifold of Sd�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 Lebesguemeasure, which we denote by �, is de�ned and is quasi-invariant with respect toH , i.e. �(E) = 0 is equivalent to �(hE) = 0 for any h 2 H and Borel set E � Sd�1(see Margulis [20, p. 33]). It is well known that the Lebesgue measure of Hx is theunique (regular) measure on Hx which is quasi-invariant with respect to H . There-fore, Theorem 3.8 implies that �! = � almost surely, or � = � since we neglectnull-sets. Thus the deterministic Lebesgue measure of Hx is the only invariantmeasure of A supported by Hx, hence is necessarily ergodic.Now, let �0 be an arbitrary ergodic invariant measure of A, then due to Lem-mas 3.12 and 3.13, �0 is supported by a closed orbit Hy, hence the above argumentshows 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. Thenalmost surely supp�! is a smooth submanifold of Sd�1 which is of constant di-mension. Furthermore, �! is equivalent to Lebesgue measure of the submanifoldsupp�! � Sd�1.Proof. Let H 2 H(A), then A is cohomologous to a minimal H-cocycle A0 by acohomology C. Since H(A0) = H(A) 3 H Theorem 3.14 is applicable to A0. Notingthat C is a linear, hence smooth, transformation of Sd�1 mapping bijectively theergodic 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 subgroupof orthogonal matrices of Gl(d). We call a cocycle A 2 G(d) orthogonal if A(!) 2O(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�ni-tion 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 anycompact subgroup of Gl(d) is conjugate to a subgroup of O(d) (see, e.g., Hewitt andRoss [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 existsa compact K" � Gl(d) such that for every n 2 ZP(f! 2 j �A(n; !) 62 K"g) < ":A reader familiar with probability theory can easily see that the boundedness ofA 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 cocycleif and only if it is bounded.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 17Proof. Let A be a Gl(d)-cocycle. By Theorem 4.7 of Schmidt [25] A is bounded ifand only if A is cohomologous to a cocycle taking values in a compact subgroup ofGl(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 setf�A(n; !) j n 2 Zg has compact closure in Gl(d), then A is Gl(d)-cohomologous toan orthogonal cocycle.Since there is an increasing sequence of compacts covering Gl(d) it is easilyseen that Proposition 3.17 implies Proposition 3.18. On the other hand, althoughProposition 3.18 is weaker than Proposition 3.17 it gives a criterion which is, in asense, easier to verify.Proposition 3.19. Let A 2 G(d) and H 2 H(A). Then A is cohomologous to anorthogonal 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 aclosed subgroupK � O(d) which belongs toH(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 conformaltransformations,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 conformalcocycle. First we characterize nonorthogonal cocycles via their invariant measures.For this purpose we need the following fundamental lemma of Furstenberg [9] (seealso Zimmer [35, pp. 39, 45]) which is the key tool for our investigation of invariantmeasures of linear cocycles.Lemma 3.20. (Furstenberg) Suppose [An] 2 SGl(d), �; � 2 Pr(Sd�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 � dimV; dimW � d � 1 anddimV + dimW = d such that � is supported on [V ] [ [W ].Lemma 3.21. Let A be an Sl(d)-cocycle and � be an arbitrary invariant measureof 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 , thesum of whose dimensions is equal to d.Proof. Since Sl(d) is an algebraic group, A is cohomologous to a minimal cocycleB(!) 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 anyinvariant 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 acocycle to be cohomologous to a conformal cocycle.

18 ARNOLD et al.Theorem 3.22. Let A 2 G(d). Then A is cohomologous to a conformal cocycleif and only if there exists an invariant measure �! of A which is equivalent to theLebesgue measure of Sd�1 for P-almost all ! 2 .Proof. Let A(!) = C(�!)B(!)C(!)�1, where B is conformal. Then �! := C(!)Lebis an invariant measure of A, where Leb denotes the Lebesgue measure of Sd�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 detA(!)j�1=dA(!) 2 Sl(d):Clearly, for all ! 2 , [A(!)] = [A(!)]. Hence, any invariant measure of A isan invariant measure of the cocycle A and vice verse. Furthermore, since A isnot cohomologous to a conformal cocycle, A is not cohomologous to an orthogonalcocycle, because otherwise Corollary 1.3 would lead to a contradiction. This, byLemma 3.21, implies that for any invariant measure �! of A, for P-almost all ! 2 ,supp (�!) 6= Sd�1, hence �! is not equivalent to Leb.4. Block-conformal cocyclesThe aim of this section is to derive the block-conformal form for a cocycle on thespan of the support of its invariant measure (Theorem 4.23). For this purpose, inSubsection 4.1 we study coverings of random sets by random linear spaces, wherewe show the measurability of the linear covers (Theorem 4.10) and the fact thatthese 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 extendeddynamical system, and show that cocycle cohomology is invariant, in a sense, underthe lifting operation (Theorem 4.20). This lifting operation allows us to apply theresults of Section 3 to obtain the conformal form of the lifted cocycle and comeback to prove the main Theorem 4.23.4.1. The linear cover of a subset of Sd�1. In this subsection we use a construc-tion of Zimmer [33, 35] and study the coverings of random sets by random linearsubspaces. For a nonempty set f0g 6= X � Rd we denote by span(X) the linearsubspace of Rd spanned by the vectors of X , and by [X ] its projection onto Sd�1(see Section 3). Linear subspaces U1; : : : ; Ur of Rd are called linearly independent iffor 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 Sd�1. Then C is acompact metric space with the Hausdor� metric.dH(E;F ) := supx2E;y2Ffd(x; F ); d(y; E)g;where d(x;E) := infy2E d(x; y), and d(x; y) := kx� yk is the Euclidean distance inRd .Put A := n l[i=1[Vi] j V1; : : : ; Vl are linear subspaces of Rd ;Vi 6� Vj for i 6= j; and X dimVi � do:

JORDAN NORMAL FORM FOR LINEAR COCYCLES 19Then A � C is closed. For E 2 A de�ne n(E) := l, d(E) := P dimVi, D(E) :=dim(PVi), where PVi := span([Vi). Then 1 � n(E); d(E); D(E) � d. We alsointroducebA := � l[i=1[Vi] 2 A j the spaces V1; : : : ; Vl are linearly independent:Let ; 6=M � Sd�1 be arbitrary. De�ne~d(M) := minfd(E) j E 2 A and E �Mg;~n(M) := maxfn(E) j E 2 A; E �M and d(E) = ~d(M)g:Lemma 4.1. For any ; 6= M � Sd�1 there exists a unique E 2 A such thatd(E) = ~d(M) and n(E) = ~n(M).Proof. LetM � Sd�1 be arbitrary. Since d(�); n(�); ~d(�); ~n(�) take values in the �niteset f1; : : : ; dg and the set Sd�1 2 A coversM , there exists at least one E 2 A suchthat d(E) = ~d(M) and n(E) = ~n(M).Let E = Sli=1[Vi] 2 A be such that E � M , d(E) = ~d(M) and n(E) = ~n(M).Observe that M � [span�Sli=1 Vi�], hence, by the de�nition of ~d(M) we haved(E) � D(E) = d([span� l[i=1Vi�]) � d(E):Therefore, d(E) = D(E), which implies that the linear subspace Vi, i = 1; : : : ; l,are linearly independent.Assume that F = Smi=1[Wi] is an element of A such that F �M , d(F ) = ~d(M)and n(F ) = ~n(M). Then by the above argument the linear subspaces Wi, i =1; : : : ;m, are linearly independent. PutUj := V1 \Wj ; j = 1; : : : ;m:Then the sets Ui, i = 1: : : : ;m, are independent linear subspaces of Rd , andmXi=1 dimUi � dimV1; M \ V1 � � m[i=1Ui�:Therefore, among the Ui there is exactly one which is nontrivial with the samedimension as dimV1, say Ui1 . Otherwise we replace [V1] by [mi=1[Ui] in E and geta contradiction to the maximality of n(E) or the minimality of d(E). This impliesUi1 = V1, hence V1 � Wi1 . Analogously, every of the Vi is contained in one of theWi and vice versa. Consequently, E = F .De�nition 4.2. For a set ; 6= M � Sd�1 the unique set E = Sli=1[Vi] 2 A suchthat d(E) =Pli=1 dimVi = ~d(M) and l = n(E) = ~n(M) is called the linear coverof M and denoted by Z(M).For a probability measure � 2 Pr(Sd�1) the linear cover of its support is called thelinear 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 � Sd�1 be arbitrary.(i) If Z(M) = Sli=1[Vi] 2 A, then the linear subspaces Vi, i = 1; : : : ; l are linearlyindependent, hence Z(M) 2 bA and d(Z(M)) = dim span(Z(M)).(ii) If E 2 bA and E �M , then E � Z(M).

20 ARNOLD et al.>From Lemma 4.3 we obtain immediately the following assertion which can serveas an alternative de�nition of the linear cover.Lemma 4.4. For any ; 6=M � Sd�1 the linear cover Z(M) is the smallest (withrespect to set inclusion) element of bA covering M .Now we turn to the main part of this subsection, namely we shall prove that thelinear 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 bA. Assume that Uj , j = 1; : : : ; q, are linearlyindependent subspaces of Rd such that [Uj ] � E for all j = 1; : : : ; q. Assume furtherthat there exists a vector a =Pqj=1 bj with bj 2 Uj, bj 6= 0, j = 1; : : : ; q, such thata 2 E. Then [span� q[j=1Uj�] � E: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 symmetricpoints. Hence each [Uj ], j = 1; : : : ; q, must be contained in one of [Vi], i = 1; : : : ; l.Now let a =Pli=1 ci be the unique representation of a with respect to span(E) =�li=1Vi, i.e., ci 2 Vi is the projection of a into Vi along �j 6=iVj . Since a 2 E thereis only one non-vanishing ci, say c1 6= 0. It is easily seen that if Uk � Vm thencm 6= 0, because cm = XUr�Vm brand the Uj , j = 1; : : : ; q, are linearly independent. Therefore, Uj � V1 for allj = 1; : : : ; q, which implies span� q[j=1Uj� � V1;hence the lemma.Lemma 4.6. Let A 2 Gl(d).(i) If ; 6=M � Sd�1 is arbitrary and Z(M) = Sli=1[Vi], then Z([A]M) = [A]Z(M) =Sli=1[AVi].(ii) If � 2 Pr(Sd�1) is arbitrary and Z(�) = Sli=1[Vi], then Z([A]�) = [A]Z(�) =Sli=1[AVi].Proof. (i) [A] furnishes an automorphism of A preserving n(�), d(�), D(�) and mapsM to [A]M .(ii) [A] maps supp� to supp [A]�.Lemma 4.7. For any ; 6= M � Sd�1, span(Z(M)) = span(M), hencedim span(M) = d(Z(M)).Proof. Since M � Z(M), span(M) � span(Z(M)). Since M � span(M) and[span(M)] 2 bA, by Lemma 4.3, Z(M) � [span(M)], which implies span(Z(M)) �span(M). Therefore, span(Z(M)) = span(M).

JORDAN NORMAL FORM FOR LINEAR COCYCLES 21Lemma 4.8. If M � Sd�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 thatM(!) = fcn(!) j n 2 Ng for all ! 2 ;Clearly, the sequence pq�1cn, p 2 Z, q; n 2 N, of measurable maps ordered byincreasing 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 in-variant random subspace of A.Here is the main result of this subsection.Theorem 4.10. Let M � Sd�1 be a closed random set. Then there exist a mea-surable function : ! f1; : : : ; dgand random subspaces U1; : : : ; U of Rd such that for all ! 2 the subspacesU1(!); : : : ; U (!)(!) are linearly independent andZ(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 thatM(!) = fcn(!) j n 2 Ng for all ! 2 :Fix an element ! 2 . For ease of notation we shall drop the argument !. Weconstruct sequences of functions and subspaces as follows. Put 1 := 1; U11 := spanfa1g; U1i := ; for i = 2; : : : ; d:Suppose we have constructed n 2 f1; : : : ; dg and subspaces Uni of Rd such thatdimUni � 1 for i � n; Uni = ; for i > n;and the Uni , i = 1; : : : ; n, are linearly independent. Now we construct n+1 2f1; : : : ; dg and subspaces Un+1i of Rd . PutFn := n[i=1Uni :(i) In case cn+1 2 Fn, we put n+1 = n; Un+1i = Uni for i = 1; : : : ; d:(ii) In case cn+1 62 span(Fn), noting that n < d we set n+1 = n + 1; Un+1i = Uni for i 6= n+1; Un+1 n+1 = spanfcn+1g:(iii) In case cn+1 2 span(Fn) n Fn, let cn+1 = P ni=1 bi be the unique repre-sentation of cn+1 with bi 2 Uni . (Note that bi is the projection of cn+1 into Unialong Lj 6=i Unj .) Let 1 � i1 < � � � < iq � n be those indices for which bij 6= 0,j = 1; : : : ; q. Put n+1 = n�q+1; Un+11 = spanf[qj=1Unijg; Un+1i = ; for i = n+1+1; : : : ; d;

22 ARNOLD et al.and the subspaces Un+12 ; : : : ; Un+1 n+1 are de�ned to be equal toUn1 ; : : : ; Uni1�1; Uni1+1; : : : ; Uni2�1; Uni2+1; : : : ; Uniq�1; Uniq+1; : : : ; Un n ;respectively.Clearly, n+1 2 f1; : : : ; dg and the subspaces Un+1i of Rd have properties sim-ilar to those of n; Uni . Therefore, by induction we have constructed sequencesof functions n, n 2 N, and subspaces Uni of Rd , n 2 N, i = 1; : : : ; d such that n 2 f1; : : : ; dg anddimUni � 1 for i � n; Uni = ; for i > n;and the Uni , i = 1; : : : ; n, are linearly independent.Since we have constructed these sequences for arbitrary �xed ! 2 we obtainmaps n(�) : ! f1; : : : ; dg; n 2 N;Uni (�) : ! 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 Uni of Rd , n 2 N, i = 1; : : : ; d, are random subspacesand Fn(�) = n[i=1Uni (�); n 2 N;are closed random sets.Obviously, for any ! 2 , n 2 N,Fn(!) � Fn+1(!);and [Fn(!)] = S ni=1[Uni (!)] 2 A for any n 2 N and ! 2 . This implies that eitherFn(!) = Fn+1(!) or there is at least one jump: either d([Fn(!)]) = d([Fn+1(!)])and n([Fn(!)]) > n([Fn+1(!)]) or d([Fn(!)]) < d([Fn+1(!)]). Since d(�) and n(�)take values in the �nite set f1; : : : ; dg there are at most d2 jumps. Therefore, forany ! 2 there exists N(!) 2 N such that for all n � N(!), Fn(!) = Fn+1(!).Hence, Uni (!) = Un+1i (!) for all n � N(!), i = 1; : : : ; d.Clearly the function : ! f1; : : : ; dg de�ned by (!) := lim supn!1 n(!) for all ! 2 is measurable. Obviously, (!) = limn!1 n(!) = N(!)(!) for all ! 2 :Put for any ! 2 , i 2 f1; : : : ; dg,Ui(!) := 1\m=1 [k�mUki (!) = UN(!)i (!):Then the Ui, i 2 f1; : : : ; dg, are closed random sets, because for any x 2 Rdd(x; Ui(!)) = d(x; UN(!)i (!)) = lim supm!1 d(x; Umi (!))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.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 23Put for any ! 2 E(!) := (!)[i=1 [Ui(!)]:Then E(!) is closed, hence E(!) �M(!) since cn(!) 2 E(!) for all n 2 N, ! 2 .Therefore, since E(!) 2 bA, E(!) � Z(M(!)) due to Lemma 4.3. We show thatE(!) � Z(M(!)) for all ! 2 . Since E(!) = [FN(!)(!)] it su�ces to show thatfor each �xed ! 2 , [Fn(!)] � Z(M(!)) for all n 2 N. We shall prove this byinduction.Let n = 1. Then clearly [F 1(!)] = fc1(!)g �M(!) � Z(M(!)).Now assume that [Fn(!)] � Z(M(!)). We shall show that [Fn+1(!)] � Z(M(!)).There are three possibilities.(a) In case cn+1(!) 62 span(Fn(!)), we have[Fn+1(!)] = [Fn(!)] [ fcn+1(!)g � Z(M(!)):(b) In case cn+1(!) 2 Fn(!), we have[Fn+1(!)] = [Fn(!)] � Z(M(!)):(c) In case cn+1(!) 2 span(Fn(!)) n Fn(!), letcn+1(!) = n(!)Xi=1 bi(!)be the unique representation of cn+1(!) with bi(!) 2 Uni (!), hence bi(!) is theprojection of cn+1(!) into Uni (!) along Lj 6=i Unj (!). Let 1 � i1 < � � � < iq � nbe those indices such that bij 6= 0, j = 1; : : : ; q. SinceUn+11 (!) = spanf[qj=1Unij (!)g;Unij (!) � Z(M(!)), j = 1; : : : ; q, and cn+1(!) 2 Z(M(!)), by Lemma 4.5,[Un+11 (!)] � Z(M(!)):Consequently, [Fn+1(!)] � Z(M(!)):This �nishes the proof.While Z(M(!)) = S (!)i=1 [Ui(!)] is uniquely determined by M(!), the subspacesUi(!) are, in general, not. They are unique up to some permutation of indices asthe following lemma shows.Denote by �(d) the group of permutations of f1; : : : ; dg equipped with the dis-crete �-algebra.Lemma 4.11. Let M(!) be a closed random set and d(Z(M(!))) = (!) 2f1; : : : ; dg. Assume that U1(!); : : : ; U (!)(!) and V1(!); : : : ; V (!)(!) are randomsubspaces such that for all ! 2 Z(M(!)) = (!)[i=1 [Ui(!)] = (!)[i=1 [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 (!)i=1 [Wi(!)].Proof. (i) By the uniqueness of the linear cover each Vi must coincide with someUj and vice versa. Hence � is well-de�ned. The measurability of � follows from themeasurability of the Ui; Vi, i = 1; : : : ; (!).(ii) Wi are random subspaces, because Uj are and � is measurable. The equalityZ(M(!)) = S (!)i=1 [Wi(!)] is obvious.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 invariantwith respect to �, hence constant.Proof. (i) Since � is measurable, supp� � Sd�1 is a closed random set. By The-orem 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. Theirinvariance follows immediately from Lemma 4.6.Lemma 4.13. Let � be an invariant measure of a cocycle A, and Z(�!) =Sli=1[Vi(!)]. Then for any k = 1; : : : ; d the (possibly empty) closed random setEk(!) := [dimVi(!)=k; i=1;::: ;l[Vi(!)]is invariant with respect to A. Furthermore, �!(Ek(!)) = pk are independent of !and Pdk=1 pk = 1. If � is ergodic, then pk0 = 1 for some k0.Proof. Since A(!) 2 Gl(d) the subspaces Vi(!) and A(!)Vi(!) are of the samedimension. Therefore, we can group the subspaces of the same dimension to get aninvariant set on Sd�1.Theorem 4.14. Let A 2 G(d), let � be an invariant measure of A with Z(�!) =Sli=1[Vi(!)]. Then there is a measurable permutation � : ! �(l) such thatUi(!) := V�(!)i(!) are random subspaces having the following properties:(i) Z(�!) = Sli=1[Ui(!)], i.e. the random subspaces Ui are rearrangements of thespaces Vi;(ii) �!([Ui(!)]) = constant =: mi;(iii) A(!)Ui(!) = Uj(�!) implies mi = mj , hence the union of those subspaceswhich have the same probability is invariant.Proof. Put f(!) := max1�i�l fi(!); fi(!) := �!([Vi(!)]); i = 1; : : : ; l:Obviously, fi(!) > 0 for all i = 1; : : : ; l and all ! 2 . It is easily seen that f isinvariant 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, henceconstant, say k(!) = k � 1 for all ! 2 . Let 1 � i1(!) < � � � < ik(!) � l be theindices such that fij(!)(!) = c. The functions ij(!), j = 1; : : : ; k, are measurable.Put �(!)�1j := ij(!); Uj(!) := Vij (!)(!); j = 1; : : : ; k:

JORDAN NORMAL FORM FOR LINEAR COCYCLES 25Clearly, the random subspacesE1(!) := span� k[j=1Vij (!)(!)� and E2(!) := span� k[m6=ij (!); m=1Vij (!)(!)�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 �nitenumber of steps we obtain a random permutation � and random subspaces Ui. Itis easily veri�ed that for them the assertions of the theorem hold true.Corollary 4.15. Let A 2 G(d), let � be an ergodic invariant measure A withZ(�!) = Sli=1[Vi(!)]. Then dimV1(!) = � � � = dimVl(!) =: k independent of! and �!([Vi(!)]) = 1=l for all i; !.4.2. Structure of a cocycle on the span of the support of an invariantmeasure. Block-conformal cocycles. First we give a criterion for a conformalcocycle in terms of the linear cover.Proposition 4.16. Let A 2 G(d). Then A is cohomologous to a conformal cocycleif and only if there exists an invariant measure �! of A such that Z(�!) = Sd�1for 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 thatZ(�!) = Sli=1[Ui(!)]; for all ! 2 ;dimUi(!) = k; for all ! 2 ; i = 1; : : : ; l:(9)Note that given an invariant measure, by virtue of Lemma 4.13, we can considerits restriction to the union of those subspaces in the linear cover which are of thesame 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; : : : ; lgA(!)Ui(!) = U�(!)i(�!):In this situation we say thatA permutes the Ui (or permutes the splitting�li=1Ui(!))by the law of permutation �. We note that the case where � is equal to the identityelement of �(l) is equivalent to the splitting �li=1Ui(!) being invariant.Lemma 4.17. Assume (9). Let Vi, i = 1; : : : ; l, be random subspaces such thatZ(�!) = Sli=1[Vi(!)], and let � : ! �(l) be the measurable map de�ned byA(!)Vi(!) = V�(!)i(�!). Then � is cohomologous to �, i.e., there exists a measur-able map � : ! �(l) such that�(!) = �(�!)�1 � �(!) � �(!) for all ! 2 :Proof. By Lemma 4.11 there exists a unique measurable map � : ! �(l) suchthat for all ! 2 and i 2 f1; : : : ; lgVi(!) = 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 acohomology C. Then B has an invariant measure � = C� such that Z(�!) =Sli=1[Wi(!)] with dimWi(!) = k, i = 1; : : : ; l, and B permutes the Wi by �. Inshort: 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. ChooseWi(!) = C(!)Ui(!); for all ! 2 ; i = 1; : : : ; l:Then for all ! 2 Z(�!) = l[i=1[Wi(!)]:Clearly, dimWi(!) = dimUi(!) 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 thesituation (9), hence, for simplicity we assume that span(Z(�!)) = Rd for all ! 2 .Clearly the splitting Rd = �li=1Ui(!) and the random permutation �(�) aredetermined by A, hence so are also the restrictions of A to the Ui. It is easily seenthat given a splitting, a law of permutation � and l cocycles of dimension k there is aunique Gl(d)-cocycle which permutes the splitting by the given law of permutationand has the restrictions to the subspaces of the splitting coinciding with the givenGl(k)-cocycles. We formalize this observation in the following construction of liftinga cocycle and show that cohomology respects the lifting construction. In short, theoperation 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 toolof this paper. Roughly speaking, it makes the linear cover bigger relative to thenew phase space, hence enables us to use Proposition 4.16 to describe the structureof the linear cocycle.We now describe the lifting construction in detail. We forget for a while aboutthe invariant measure �, assume that we are given a splitting Rd = �li=1Ui(!) of Rdinto 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�1P(E) for all i = 1; : : : ; l; E 2 F ;�� : �! �; where ��(i; !) := (�(!)i; �!):It is easily seen that �� is an automorphism of � preserving �P.Choose and �x a random basis f := ff1(!); : : : ; fd(!)g of Rd adapted to thesplitting Rd =LUi(!), i.e. such that ff(i�1)k+1(!); : : : ; fik(!)g is a basis of Ui(!)

JORDAN NORMAL FORM FOR LINEAR COCYCLES 27for any i = 1; : : : ; l, ! 2 . Let fe1; : : : ; ekg denote the standard Euclidean basisof Rk . De�ne linear isomorphisms Lfi (!) : Ui(!)! Rk , ! 2 , i 2 f1; : : : ; lg, byLfi (!)f(i�1)k+m(!) = em; ! 2 ; m 2 f1; : : : ; kg; i 2 f1; : : : ; lg:We de�ne a linear random map Af : �! Gl(k)by setting for all ! 2 , i 2 f1; : : : ; lg,Af (i; !) = Lf�(!)i(�!) �A(!) � Lfi (!)�1:We note that Lfi (!) is an identi�cation of the space Ui(!) with the Euclideanspace Rk , and Af (i; !) is the matrix form of the restriction of A(!) to Ui(!). Thuswe have lifted the cocycle A over � to the cocycle Af over ��, and the above procedureis called a lifting operation.Clearly, for all n 2 Z and (i; !) 2 �,�Af (n; (i; !)) = Lf��(n;!)i(�n!) ��A(n; !) � Lfi (!)�1:Lemma 4.19. Let g := fg1(!); : : : ; gd(!)g be another random basis of Rd adaptedto the splitting Rd =LUi(!). Then Ag � Af as Gl(k)-cocycles over ��.Proof. We haveLgi (!)g(i�1)k+m(!) = em; ! 2 ; m 2 f1; : : : ; kg; i 2 f1; : : : ; lg;Ag(i; !) = Lg�(!)i(�!) �A(!) � Lgi (!)�1:De�ne a linear random map P : �! Gl(k) by putting for all ! 2 , i 2 f1; : : : ; lg,P (i; !) := Lfi (!) � Lgi (!)�1:Then P (��(i; !)) := Lf�(!)i(�!) � Lg�(!)i(�!)�1:It is easily seen that for all ! 2 , i 2 f1; : : : ; lg,Af (i; !) = P (��(i; !)) �Ag(i; !) � P (i; !)�1:So Ag is cohomologous to Af .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 Bg = Af .(ii) If Af � ~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=1Ui(!) by the law � andBf = ~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 , Rd = lMi=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=1Wi(!). We haveLgi (!)g(i�1)k+m(!) = em; ! 2 ; m 2 f1; : : : ; kg; i 2 f1; : : : ; lg;Bg(i; !) = Lg�(!)i(�!) �B(!) � Lgi (!)�1:Clearly Lgi (!) = Lfi (!) � C(!)�1:Therefore,Bg(i; !) = Lf�(!)i(�!) � C(�!)�1 �B(!) � C(!)Lfi (!)�1 = Af (i; !):(ii) Let ~A and Af be cohomologous by P : �! Gl(k), i.e., for all (i; !) 2 �~A(i; !) = P (��(i; !)) �Af (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(�!) �Af (i; !) � Ci(!)�1:Now, by de�nition, Af (i; !) = Lf�(!)i(�!) �A(!) � Lfi (!)�1:Hence, ~A(i; !) = C�(!)i(�!) � Lf�(!)i(�!) �A(!) � Lfi (!)�1 � Ci(!)�1:De�ne Ci(!) : Ui(!)! Ui(!), i = 1; : : : ; l, ! 2 , byCi(!) := Lfi (!)�1 � Ci(!) � Lfi (!):Put fj(!) := Ci(!)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; : : : ; dD(!)fj(!) = fj(!):D(!) leaves Ui(!), i = 1; : : : ; l, invariant and the restriction of D(!) to Ui(!)coincides with Ci(!). Put B(!) := D(�!)A(!)D(!)�1:We haveB(!)Ui(!) = U�(!)i(�!), i.e. B has the same splitting asA andB permutesthe splitting Rd = �li=1Ui(!) by the same law � as A does. Now, for any (i; !) 2 �Bf (i; !) = Lf�(!)i(�!)B(!)Lfi (!)�1= Lf�(!)i(�!)D(�!)A(!)D(!)�1Lfi (!)�1= C�(!)(�!)Lf�(!)i(�!)A(!)Lfi (!)�1Ci(!)�1= ~A(i; !):Hence the theorem is proved.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 29Proposition 4.21. Suppose that A has an invariant measure � such that Z(�!) =Sli=1[Ui(!)] with dimUi(!) = k, kl = d, i = 1; : : : ; l, and f is a random basis of Rdadapted to the splitting Rd = �li=1Ui(!). Then Af is cohomologous to a conformalcocycle.Proof. By Theorem 4.20 we can assume that ff(i�1)k+1(!); : : : ; fik(!)g is an or-thonormal 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 � : �! Pr(Sk�1) by setting for all (i; !) 2 �,M 2 B(Sk�1), �(i;!)(M) := l�!([Lfi (!)�1(M)]):Clearly, �(i;!)(Sk�1) = 1 for all (i; !) 2 �, hence �(i;!) 2 Pr(Sk�1). The measura-bility of the map � is obvious. We claim that � is an invariant measure of Af . For,since Af (i; !) = Lf�(!)i(�!) �A(!) � Lfi (!)�1 we have, for any M 2 B(Sk�1),A(!) � Lfi (!)�1(M) = Lf�(!)i(�!)�1Af (i; !)(M):This, due to the invariance of �, implies�(i;!)(M) = l�!([Lfi (!)�1(M)])= l��!([A(!)][Lfi (!)�1(M)])= l��!(Lf�(!)i(�!)�1[Af (i; !)](M))= ��(!)i;�!([Af (i; !)](M)):Thus the claim is proved.Clearly, supp �(i;!) = [Lfi (!)(supp�!\Ui(!))] for all i = 1; : : : ; l, ! 2 . Hence,by Lemma 4.6, Z(�(i;!)) = Sk�1 for all (i; !) 2 �. By Proposition 4.16, Af iscohomologous to a conformal cocycle.Proposition 4.21 shows that if we lift a linear cocycle to the linear cover ofits invariant measure then we obtain a conformal cocycle, but over an extendeddynamical system. We shall prove that when looking back to the original dynamicalsystem � we will derive a so-called block-conformal cocycle. First we introduce thenotion 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 ARNOLD et al.that B has the following formB(!) =

0BBBBBBBBBBBBBBBBBBBBBB@... ... ... � � � ...... B�(!)2(!) ... � � � ...... 0 ... � � � ...0 ... ... � � � ...B�(!)1(!) ... ... � � � ...0 ... ... � � � ...... ... ... � � � 0... ... ... � � � B�(!)l(!)... ... ... � � � ...

1CCCCCCCCCCCCCCCCCCCCCCA(10)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 ofB the only non-trivial entry is B�(!)m(!) in the �(!)m position, similarly for therows.A linear cocycle A is called block-conformal if the matrix representation of A in thestandard 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 thefollowing theorem on the reduction of a linear cocycle to block-conformal form.Theorem 4.23. Let A 2 G(d).(i) If A has an invariant measure � such that Z(�!) = Sli=1[Ui(!)] with dimUi(!) =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 therestriction of A to span(supp�!) is cohomologous to a direct sum of block-conformalsubcocycles.(iii) If A is cohomologous to a direct sum of block-conformal subcocycles then A hasan invariant measure � such that span(supp �!) = Rd for almost all ! 2 .Proof. First note that by Lemma 4.7, span(supp�!) = span(Z(�!)) =Lli=1 Ui(!).(i) Let A permute the Ui by the law �. By Theorem 4.20 and Proposition 4.21there are a Gl(d)-cocycle C which permutes the Ui by the law � and a randombasis f := ff1(!); : : : ; fd(!)g of Rd adapted to the splitting Rd = LUi(!) suchthat C is cohomologous to A and Cf (i; !) is a conformal matrix (with respect tothe standard Euclidean basis of Rk ) for any (i; !) 2 �.Let e = fe1; : : : ; edg denote the standard Euclidean basis of Rd . De�ne a randomlinear map R of Rd by settingR(!)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, ithas a matrix representation with respect to the random basis f = ff1(!); : : : ; fd(!)g

JORDAN NORMAL FORM FOR LINEAR COCYCLES 31of the form (10) with B�(!)i(!) replaced with C�(!)i(!), the matrix representationof Cf (i; !) in the standard Euclidean basis of Rk (hence is conformal). Further, bythe de�nition of B, its matrix representation with respect to the basis e coincideswith 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 havethe same dimension.Part (iii) follows from the de�nition of a block-conformal cocycle by taking theLebesgue measures on the intersections of the unit sphere with the subspaces cor-responding to the blocks.Since, as we proved, a linear cocycle has a \good" structure on spans of sup-ports 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-termobservations of the system. In the remaining part of this section we settle thisproblem.Lemma 4.24. (i) For any nonempty sets M1;M2 � Sd�1 we haveZ(M1) [ Z(M2) � Z(M1 [M2) andspan(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. Thenthere exists an invariant measure � of A such that for P-almost all ! 2 span(Z(�!)) = span(Z(�1!) [ Z(�2!)):Proof. (i) We haveM1 � Z(M1[M2) and Z(M1[M2) 2 bA, 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 4.7span(Z(M1[M2)) = span(M1[M2) � span(Z(M1)[Z(M2)) � span(Z(M1[M2)):(ii) For all ! 2 , put �!(�) := 12(�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 �maxwith the following properties:(i) supp�max! = Z(�max! ) 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 thereis an invariant measure � of A such that for any invariant measure �0 of A eitherd(Z(�0!)) < d(Z(�!)) =: r or d(Z(�0!)) = r and n(Z(�0!)) � n(Z(�!)) =: l. ByTheorem 4.23, the restriction of A to span(Z(�!)) is cohomologous to a cocycle B 2G(d) which is a direct sum of block-conformal subcocycles. Taking the Lebesguemeasures on the intersections of the unit sphere with the subspaces correspondingto the blocks of the subcocycles of B we see that B has an invariant measure �1with supp �1 = Z(�1), d(Z(�1)) = r, and n(Z(�1)) = l. Transforming �1 back to

32 ARNOLD et al.an invariant measure on A we �nd an invariant measure �max of A such that, foralmost all ! 2 ,supp�max! = Z(�max! ); d(Z(�max! )) = r; n(Z(�max! )) = l:We show that this �max is the maximal measure we are looking for. Part (i) isimmediate.Let � be an arbitrary invariant measure of A. Then �! := (�! + �max! )=2 isan invariant measure of A. If Z(�!) 6� span(Z(�max! )) then Lemma 4.24 impliesthat d(Z(�!)) > r which contradicts the choice of r. Therefore, span(Z(�max! )) =span(Z(�!)). Consequently, by the choice of r, l and � we have for almost all ! 2 d(Z(�!)) = d(Z(�max! )); n(Z(�!)) � n(Z(�max! ));Z(�!) � supp �! � supp�max! :Since Z(�!) 2 A and Z(�!) � supp�max! , Lemma 4.1 implies that n(Z(�!)) =n(Z(�max! )) and Z(�!) = Z(�max! ). This, by Lemma 4.24, implies that Z(�max! ) �Z(�!) � supp�!. Due to (i), �!(supp�max! ) = 1 for P-almost all ! 2 , hencepart (ii) is proved.Remark 4.26. A linear cocycle may have many maximal measures, but their sup-ports are the same. Thus any linear cocycle has a unique good maximal invariantset supp�max! on the span of which it has the form of a block conformal cocycle.5. The Jordan normal formIn this section we prove our �nal theorem on the Jordan normal form of a linearcocycle (Theorem 5.6) and give an algorithm of constructing the Jordan normalform via invariant measures.5.1. The Jordan form. We shall show that the structure of the algebraic hull ofa 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 propersubspace 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 equiv-alent to strong irreducibility. Recall De�nition 2.9 concerning irreducibility etc. ofa cocycle.We note that the de�nition of these particular classes of linear groups is borrowedfrom the theory of representations of groups. The following lemma is immediate.Lemma 5.2. (i) The notions of irreducibility and strong irreducibility are invariantwith respect to group conjugacy.(ii) Strong irreducibility implies irreducibility, the converse assertion is false.The following theorem establishes the equivalence of irreducibility of the cocycleand 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.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 33Proof. 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 invariantwith 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 invariantrandom proper subspace U(!) � Rd . Note that dimU(!) is constant. Take and �xa nonrandom proper subspace V � Rd of the same dimension. Choose a randombasis of U(!) and a nonrandom basis of V . Extend them to a random basis and anonrandom basis of Rd , respectively. Denote by C(!) the random linear operatorof Rd which map the constructed random basis to the nonrandom basis. ClearlyC 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 someH 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 invariantmeasure � of A we have Z(�!) = Sd�1, hence A is cohomologous to a conformalcocycle.Proof. Suppose that A is strongly irreducible. Let � be an ergodic invariant measureof A with linear cover Z(�!) = Sli=1[Vi(!)]. If dim span(Z(�!)) < d then A is notirreducible because the proper random subspace span(Z(�!)) is invariant. If l > 1then A is not strongly irreducible. Thus, dim span(Z(�!)) = d and l = 1, henceZ(�!) = Sd�1. Therefore, by Proposition 4.16, A is cohomologous to a conformalcocycle.Theorem 5.5. Let A 2 G(d). If A is irreducible, then for any ergodic invari-ant measure � of A we have Z(�!) = Sli=1[Vi(!)] with dim(V1(!)) = � � � =dim(Vl(!)) = k independent of ! and kl = d, hence A is cohomologous to a block-conformal cocycle.Proof. Let � be an ergodic invariant measure of A with linear cover Z(�!) =Sli=1[Vi(!)]. If dim span(Z(�!)) < d then A is not irreducible because the properrandom subspace span(Z(�!)) is invariant. By the ergodicity of � the subspacesVi(!) have the same dimension, hence, by Theorem 4.23, A is cohomologous to ablock-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 cocycleA = 0BBB@ A1(!) � � �0 A2(!) � �0 0 . . . �0 � � � 0 Ap(!) 1CCCA ;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. Thecohomology classes of the subcocycles Ai 2 G(di), up to their order, are uniquelydetermined by the cohomology class of A. Further, if V0 � � � � � Vp is an ar-bitrary maximal invariant ag of A, then, after a suitable reordering of indices,AVi=Vi�1 � Ai for all i = 1; : : : ; p.

34 ARNOLD et al.Proof. Take and �x a maximal invariant ag U0 � � � � � Up of A, the exis-tence of which is ensured by Lemma 2.15. Put di := dimUi � dimUi�1, i =1; : : : ; p. Choose and �x a random basis f := ff1(!); : : : ; fd(!)g of Rd such thatff1(!); : : : ; fd1+���+di(!)g is a basis of Ui(!) for all ! 2 , i = 1; : : : ; p.Let e := fe1; : : : ; edg denote the standard Euclidean basis of Rd . De�ne arandom linear map C : ! Gl(d) by setting C(!)ei = fi(!) for all ! 2 andi = 1; : : : ; d. Put B(!) := C(�!)�1A(!)C(!) for all ! 2 :Obviously, B 2 G(d) and B � A. It is easily seen that the (nonrandom) spacesWi := spanfe1; : : : ; ed1+���+dig, i = 1; : : : ; p, are invariant with respect to B andthey constitute a maximal invariant ag of B. Therefore, B has the followingmatrix form: B = 0BBB@ B1(!) � � �0 B2(!) � �0 0 . . . �0 � � � 0 Bp(!) 1CCCA ;where Bi = BWi=Wi�1 2 G(di), i = 1; : : : ; p, are irreducible. By Theorem 5.5, thecocycles 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 totheir 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 inTheorem 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 simplerthan that for linear groups|as it will be clear from the considerations of Section 6dealing with orthogonal cocycles, hence Theorem 5.6 probably gives us the simplestpossible normal form for a general linear cocycle.(ii) Unlike the Jordan normal form of a matrix, there is no normal Jordan block forthe cocycle case|only the diagonal is determined by the cocycle, the elements abovethe diagonal are far from unique. As seen from the two-dimensional cocycles treatedin Section 7, the rule of change of the entry above the diagonal is a cohomologicalequation and there is no unique way of �nding the simplest form of this entry. Thehigher dimensional case is of course more complicated.Theorem 5.9. Let A 2 G(d) and �max be a maximal invariant measure of A. Ifspan(supp�max! ) = Rd for almost all ! 2 , then the Jordan normal form of A isblock-diagonal with block-conformal entries.Proof. Use the lifting operation and the fact that orthogonal cocycles are completelyreducible.As a by-product we derive the following Jordan form for amenable subgroups ofGl(d). Recall the classical de�nition of amenability: A locally compact group Gis called amenable if for every continuous G-action on a compact metrizable spaceX , there is a G-invariant probability measure on X (see, e.g., Zimmer [35]). Thenotion of the Jordan normal form for a linear group is the same as that for a linearrepresentation; see Kirillov [17, p. 116].

JORDAN NORMAL FORM FOR LINEAR COCYCLES 35Theorem 5.10. Any amenable Lie subgroup H of Gl(d) has (i.e. reduced to bymeans of group conjugacy) the following Jordan (matrix) formH = 0BBB@ H1 � � �0 H2 � �0 0 . . . �0 � � � 0 Hp 1CCCA ;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-productaction of A on � H is ergodic. Obviously, �H 2 H(A), where �H is the smallestalgebraic group containing H .Apply Theorem 5.6 to A and reduce it to the Jordan form A. Clearly a groupH 2 H(A) = H(A) satis�es the conclusion of the theorem. Since �H is conjugate toH we have proved the theorem.In a coordinate-free language, H has an invariant ag, where on each factor ofthe ag H preserves a �nite union of linearly independent subspaces of the samedimension together with a conformal structure (i.e. H acts on these subspaces byconformal 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 herean algorithm of reducing an arbitrary linear cocycle to the block-triangular formwith block-conformal subcocycles on the diagonal, hence in particular to the Jordannormal form.5.2.1. The algorithm. Let A 2 G(d). It is well known that A has at least oneinvariant ergodic measure. Let � be one such measure. Denote by e = fe1; : : : ; edgthe standard Euclidean basis of Rd .By Lemma 4.13, since � is ergodic its linear cover consists of random linearsubspaces of the same dimension, sayZ(�!) = l[i=1[Ui(!)]with dimUi(!) = k for all i = 1; : : : ; l, ! 2 . Put m := kl. Choose a randombasis f := ff1(!); : : : ; fd(!)g of Rd such that ff1(!); : : : ; fm(!)g is a basis ofspan(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(�!)�1A(!)R(!):Clearly B is cohomologous to A and has the following matrix representation in thebasis e B(!) = � B1(!) B12(!)0 B2(!) � ;

36 ARNOLD et al.where B1(!) is a (m �m)-matrix and has R(!)�! =: �! as its invariant measurewith Z(�!) = l[i=1[R(!)�1Ui(!)]; span(Z(�!)) = span(e1; : : : ; em):By Theorem 4.23, B1 is cohomologous to a block-conformal Gl(m)-cocycle. LetB1(!) = P (�!)�1C1(!)P (!) for a random m-dimensional matrix P and a randomblock-conformal (m�m)-matrix C1. Put, for all ! 2 ,~P (!) := � P (!) 00 Id�m � 2 Gl(d);where Ir denotes the unit (r � r)- matrix. Set for all ! 2 C(!) := ~P (�!)B(!) ~P (!)�1; C12(!) := P (�!)B12(!):Straightforward computations shows thatC(!) = � C1(!) C12(!)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 onRd�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)-matrixC2, 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(�!)�1B2(!)Q(!) = � C2(!) C23(!)0 C3(!) � :Put for all ! 2 D(!) := ~Q(�!)C(!) ~Q(!); where ~Q(!) := � Im 00 Q(!) � 2 Gl(d):Clearly, D is cohomologous to A and has the formD(!) = 0B@ C1(!) C12(!)Q(!)00 C2(!) C23(!)0 C3(!) 1CA :Continuing the process if m+ n < d. After � d steps we obtain the desired block-triangular cocycle which is cohomologous to A.5.2.2. Construction of the Jordan normal form. First we note that the linear spanof an ergodic invariant measure is not always a minimal invariant subspace of thecocycle, hence the above algorithm does not always give the Jordan normal formof A though the form it gives is block-triangular with block-conformal subcocycleson the diagonal. However, if in that algorithm we choose in each step an ergodicinvariant measure with the linear span of its support having the smallest possibledimension, then the quotient subcocycles on the diagonal will be irreducible andthe algorithm produces the Jordan normal form of A.Another way of constructing the Jordan normal form is to �rst apply the abovealgorithm with arbitrary (not necessarily ergodic) invariant measures, and thenapply Theorem 5.9 to the subcocycles on the diagonal.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 37In case A satis�es the integrability condition of the multiplicative ergodic the-orem (see Oseledets [23]), we can decompose A into a direct sum of subcocyclesaccording to the Oseledets splitting of A, and then combine it with the Jordannormal forms of the subcocycles.6. Orthogonal cocyclesThe results of Section 4 reduce the investigation of linear cocycles on the spanof the supports of their invariant measures to the investigation of block-conformalcocycles a �nite extension of which are conformal cocycles. Furthermore, sincewe can factorize the determinant the problem reduces to the study of orthogonalcocycles, which are the subject of this section.Criteria for a cocycle to be cohomologous to an orthogonal cocycle are givenin 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 thealgebraic hull. First we note that the de�nition of the algebraic hull of a G-cocycledepends 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 algebraichull because we have a bigger choice of cohomologies. However, we are able toprove 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)-cohomologousthen they are O(d)-cohomologous.Proof. Let C : ! Gl(d) be measurable and let for all ! 2 A(!) = C(�!)�1B(!)C(!):(11)We need to show that there is a measurable map C 0 : ! O(d) such that for all! 2 A(!) = C 0(�!)�1B(!)C 0(!):By the theorem on polar decomposition (see Gantmacher [11, p. 286]) each C(!) 2Gl(d) can be represented in the formC(!) = 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 canbe chosen to be measurable.It is easily seen thatx1(!) = kC(!)k; xk(!) = k^kC(!)k=k^k�1C(!)k for k = 2; : : : ; d:Taking the exterior product and then the norm on both sides of equation (11) andremembering that A and B are orthogonal we obtainxk(�!) = 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; : : : ; xdg =: D.

38 ARNOLD et al.Now (11) takes the formU(�!)DV (�!)A(!) = B(!)U(!)DV (!);which is equivalent toDV (�!)A(!)V (!)�1D�1 = U(�!)�1B(!)U(!):(12)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 andcorrespond to the subspace E1 of the �rst l1 basis vectors e1; : : : ; el1 , the next l2entries 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 lastlm basis vectors, y1 > y2 > � � � > ym > 0. Now we show that the spaces E1; : : : ; Emare invariant with respect to X . Let Xe1 =Pdj=1 �jej , then1 = ke1k = kDXD�1e1k =vuut dXj=1 �2jx2jx�21 ;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 preservesalso 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 areinvariant. Since the restrictions of D to Ek are scalars we see that X commuteswith 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 thespecial case d = 2 of Proposition 6.1.It immediately follows from Proposition 6.1 that the algebraic hull of an or-thogonal cocycle A in the space of O(d)-cocycles is the intersection with O(d) ofits algebraic hull in the space of Gl(d)-cocycles. Furthermore, A is minimal as aGl(d)-cocycle if and only if A is a minimal O(d)-cocycle. In this section, we shallrestrict ourselves to orthogonal cocycles and H(A) will stand for the algebraic hullof A in the space of O(d)-cocycles.6.2. Invariant measures of orthogonal cocycles. Utilizing the compactnessof O(d) we can obtain more information about invariant measures of orthogonalcocycles.Proposition 6.2. Assume that A is a minimal orthogonal cocycle with range H 2H(A). Then the ergodic invariant measures of A are exactly the normalized Lebesguemeasures of the orbits of H in Sd�1. In particular, all ergodic invariant measuresof A are deterministic and the union of their supports equals Sd�1.Proof. Immediate from Theorem 3.14 and the fact that any orbit of H on Sd�1 isclosed due to the compactness of H .Theorem 6.3. Assume that A is a minimal orthogonal cocycle with range H 2H(A). Let L;G � O(d) be closed subgroups, such that G � H;L. Put X := G=Land assume that G acts naturally on X. Then the ergodic invariant measures on

JORDAN NORMAL FORM FOR LINEAR COCYCLES 39X of A are exactly (deterministic) normalized Lebesgue measures (i.e. the uniqueH-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 anyx 2 X the set Hx is minimal. Now, the same argument as the one for the proof ofTheorem 3.14 also proves this theorem.6.3. Equivalence of orthogonal cocycles. We show that two invariants of or-thogonal cocycles (algebraic hull and invariant measures) are equivalent in somesense.De�nition 6.4. Let H 2 Gl(d) be a Lie subgroup, X an H-space, A and B twoH-cocycles. We say that A is equivalent to B with respect to invariant measures on Xif 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 andthose of B.In the case of classical measures (on X = Sd�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 equiv-alence relation.(ii) If A is cohomologous to B then they are equivalent with respect to invariantmeasures on X for any X.(iii) A is equivalent to B with respect to invariant measures on X if and only ifthere is a H-cocycle C which is cohomologous to A as H-cocycles and has the samecollection of invariant measures on X as B does.Proposition 6.6. Let A and B be orthogonal cocycles. If H(A) = H(B) then thereis an orthogonal cocycle C cohomologous to A such that C has the same collectionof ergodic invariant measures as B, hence A is equivalent to B with respect toinvariant measures.Proof. Take and �x K 2 H(A) = H(B), K � O(d). Then there are K-cocyclesA0 and B0 cohomologous to A and B. Clearly A0 and B0 are orthogonal minimalcocycles, hence the ergodic invariant measures of A0 and B0 are exactly the de-terministic normalized Lebesgue measures of the orbits of K on Sd�1, hence theyhave the same collection of ergodic invariant measures. The proposition followsnow from Lemma 6.5.Remark 6.7. The conditionH(A) = H(B) is strictly stronger then the equivalencewith respect to invariant measures of A and B. For, if H(A) 6= H(B) but both ofthem act transitively on Sd�1 (e.g., O(d) 2 H(A) and SO(d) 2 H(B)) then bothA and B have a unique invariant measure, namely the Lebesgue measure of Sd�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) ifand only if A and B are equivalent with respect to invariant measures on X for anyhomogeneous space X = O(d)=L with L being a closed subgroup of O(d) and theaction of O(d) on X being natural.Proof. Let H(A) = H(B). Take and �x H 2 H(A) = H(B). Then there existminimal H-cocycles A0 and B0 which are cohomologous to A and B, respectively.

40 ARNOLD et al.Let L � O(d) be an arbitrary closed subgroup, X = O(d)=L, and O(d) actnaturally on X . By Theorem 6.3, A0 and B0 have the same collection of ergodicinvariant measures on X , which by Lemma 6.5 implies that A and B are equivalentwith respect to invariant measures on X .Now, assume that A and B are equivalent with respect to invariant measureson X for any homogeneous space X = O(d)=L with L being a closed subgroup ofO(d) and the action of O(d) on X being natural. Take and �x H1 2 H(A) andH2 2 H(B). Clearly, there are a minimal H1-cocycle C and a minimal H2-cocycleD which are cohomologous to A and B respectively. Since H1 2 H(C) = H(A) andH2 2 H(D) = H(B) we can apply Theorem 6.3 to C and D with X = O(d)=H2and obtain that the ergodic invariant measures on X of C and D are exactly thenormalized 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 unityelement of O(d), consists of the single point eH2, the cocycleD has the deterministicinvariant Dirac measure �eH2 on X . Now, C and D are equivalent with respect toinvariant measures on X by the assumption on A, B and Lemma 6.5, hence Chas a (possibly random) invariant Dirac measure. Since all the ergodic invariantmeasures of C are deterministic, C has a deterministic invariant Dirac measure. ByTheorem 6.3, this deterministic invariant Dirac measure is supported by an orbitof H1 on X . Therefore, there is x = gH2 2 X , g 2 O(d), such that H1x = x.This implies that for any h 2 H1 we have hgH2 = gH2, hence g�1hg 2 H2, whenceg�1H1g � H2. Changing the role of H1 and H2 we �nd g1 2 O(d) such thatg�11 H2g1 � H1. Therefore,g�1g�11 H2g1g � g�1H1g � H2;which implies that g�1g�11 H2g1g = g�1H1g = H2:Thus H1 is conjugate to H2 implying H(A) = H(B).6.4. About classi�cation of orthogonal cocycles. The remarks at the begin-ning of this section should convince the reader that it is of crucial importance toclassify orthogonal cocycles as completely as possible.We have found two invariants of cohomology, namely the equivalence of invariantmeasure and the algebraic hull, which turned out to be equivalent in the orthogonalcase. We hence can �rst classify their algebraic hulls which are (conjugacy classesof) 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 followingtheorem.Theorem 6.9. Let A be an orthogonal cocycle. The following statements are equiv-alent:(i) A is elementary in the sense that the Lebesgue measure Leb of Sd�1 is the onlyinvariant measure of A;(ii) The Lebesgue measure Leb of Sd�1 is an ergodic invariant measure of A;(iii) Any group H 2 H(A) acts transitively on Sd�1.Proof. Looking at cohomologous cocycles we see that it su�ces to prove the theoremfor 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 in-variant measure of A, hence the unique invariant measure of A. Also from Propo-sition 6.2, if Leb is an ergodic invariant measure of A then Sd�1 is an orbit of H ,

JORDAN NORMAL FORM FOR LINEAR COCYCLES 41hence H acts transitively on Sd�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 onlyone invariant measure. However, for a given algebraic hull there are, in general,in�nitely many cohomology classes of orthogonal cocycles corresponding to thishull, as the following example shows.Example 6.10. Let d = 2 and let be the trivial one-point space (hence thedeterministic case). For each irrational � 2 R we setA� := � cos(2��) sin(2��)� sin(2��) cos(2��) � :Then SO(2) 2 H(A�) for any �, hence A� is elementary. However, there arein�nitely (continuously) many cohomology classes of such cocycles.7. Classification of low-dimensional cocyclesTo 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 G-coboundary (or simply a coboundary if it is clear which group G is meant) if it isG-cohomologous to the trivial G-cocycle, i.e. the G-cocycle taking the value I|theidentity of G|for all ! 2 .7.1. Classi�cation of one-dimensional linear cocycles. Note that Gl(1) is theAbelian multiplicative group R� of non-vanishing real numbers. Clearly,Gl(1) �= Z2 � R+� ;and for any a 2 Gl(1) we have the decompositiona = sign(a) � jaj; sign(a) 2 Z2; jaj 2 R+� :(13)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 con-dition (note that it is exactly condition (4.8.33) of Theorem 4.8.1 of [21]), whereasthe condition (ii) is a radial (growth rate) one.(ii) Since the only compact subgroup of R+� is the trivial subgroup, the results ofSubsection 3.4 readily give criteria for an R+� -cocycle to be an R+� -coboundary (seealso Schmidt [25] and Zimmer [32]).7.2. Classi�cation of two-dimensional linear cocycles. We now give a com-plete classi�cation of the two-dimensional linear cocycles. In particular, we willimprove the results of Thieullen [28] and Oseledets [24]. We mention that Thieullenand Oseledets have used the method of barycenters to classify two-dimensional co-cycles in terms of invariant measures, as we also do below in de�ning the classesIIi, i = 0; 1; 2. They dealt with cocycles satisfying the integrability conditionsof the multiplicative ergodic theorem, but their results hold verbatim for general

42 ARNOLD et al.two-dimensional linear cocycles. We also note that the results of Thieullen and Os-eledets 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-dimensionalrandom subspaces.Proposition 7.5. (i) The classes IIi, i = 0; 1; 2, are disjoint and invariant withrespect 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 onthe projective space RP 1 has, respectively, no, exactly one or at least two invariantrandom Dirac measures.Proof. Obvious.Theorem 7.6. (i) A cocycle A 2 G(2) is of class II2 if and only if A is cohomolo-gous 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 anda2 � b1 in G(1).Proof. Part (i) and the \if" part of (ii) are easy. The \only if" part of (ii) is animmediate 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 � a1(!) a12(!)0 a2(!) � and A is not Gl(2)-coho-mologous to a diagonal cocycle, which is equivalent to the condition that the func-tional equation f(�!) = a1(!)a2(!)f(!) + a12(!)a2(!)(14)has no measurable solution.(ii) Two cocyclesA = � a1(!) a12(!)0 a2(!) � and B = � b1(!) b12(!)0 b2(!) �of class II1 are Gl(2)-cohomologous if and only if ai is Gl(1)-cohomologous to bi bysome cohomology ci, i = 1; 2, and the functional equationf(�!) = b1(!)a2(!)f(!) + b12(!)c2(!)a2(!) � b1(!)c1(!)a12(!)a1(!)a2(!)(15)has a measurable solution.Proof. Let A 2 G(2) be of class II1. Then it has a one-dimensional invariant randomsubspace U(!) = spanff1(!)g. Extend f1 to a random basis f = ff1(!); f2(!)gof R2 . It is easily seen that in the basis f the cocycle A has the matrix form� a1(!) a12(!)0 a2(!) �. Since A is not of the class II2, it is not Gl(2)-cohomologousto a diagonal cocycle.Now, if A is Gl(2)-cohomologous to a diagonal cocycle B by a random linearmap C, then B has two one-dimensional invariant random subspaces. ObviouslyC(!)f1(!) := g1(!) is also a one-dimensional invariant random subspace of B.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 43Clearly, we can �nd another one-dimensional random subspace spanfg2(!)g of Bsuch that fg1(!); g2(!)g is a basis of R2 for all ! 2 . Therefore, by an appropri-ate choice of a random basis, the triangular cocycle A(!) = � a1(!) a12(!)0 a2(!) �is Gl(2)-cohomologous to a diagonal cocycle if and only if there is a measurablemap f : ! R such that the random matrix C(!) = � 1 f(!)0 1 � furnishes aGl(2)-cohomology between A and the diagonal cocycle � a1(!) 00 a2(!) �. This isequivalent to the condition that f is a measurable solution of the equationf(�!)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 andB of the class II1 must transform their unique one-dimensional invariant ran-dom subspaces into one another. Therefore, the cohomology C has a triangularform � c1(!) f(!)0 c2(!) � in an appropriate random basis. Furthermore, clearly cifurnishes a Gl(1)-cohomology between ai and bi, i = 1; 2. >From the cohomol-ogy 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 admitsan invariant random measure which is equivalent to the Lebesgue measure of S1;otherwise A is said to be of class II10Theorem 7.9. (i) The classes IIi0, i = 0; 1, are disjoint and invariant with respectto 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 cohomologousto a conformal cocycle.(iii) Let A be of class II0. Then A is of class II10 if and only if the action of A onthe projective space RP 1 admits exactly one invariant random measure supportedon 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)-cohomologousto a block-conformal cocycle with one-dimensional blocks, nontrivial law of permu-tation 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 theirlaws of permutation are cohomologous (as cocycles taking values in the group of per-mutations �(2) ' Z2) and the Gl(1)-cocycles generated by them over the extendedprobability 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 II0;n0if some, hence any H 2 H([A]) has exactly n elements, n 2 N. If H 2 H([A]) hasin�nitely many elements then we say that A is of class II0;10 .Lemma 7.12. The classes II0;n0 , n = 1; 2; : : : ;1, are disjoint and invariant withrespect to cohomology. Their union is the whole of the class II00.

44 ARNOLD et al.Proof. Obvious.Theorem 7.13. (i) Let A be a cocycle of class II0;n0 , n = 1; 2; : : : ;1. Then~A(!) := j detA(!)j�1=2A(!) 2 G(2) is Gl(2)-cohomologous to an orthogonal co-cycle of class II0;n0 . If 1 < n < 1 then the cocycle ~A is Gl(2)-cohomologous to acocycle taking values in the Abelian group Gn := fexp(k 2�in ) j k = 0; : : : ; n� 1g ofrotations on angles k 2�n of S1, and Gn 2 H( ~A). The class II0;10 is empty.(ii) Two cocycles A and B of class II0;n0 , n < 1, are Gl(2)-cohomologous if andonly if the R+� -cocycles j detA(!)j and j detB(!)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 II0;10 if and only if ~A is cohomologous to anelementary cocycle.Proof. (i) By Theorem 7.9, A is cohomologous to a conformal cocycle, hence ~Ais cohomologous to an orthogonal cocycle. For simplicity we assume that A isconformal, hence [A(!)] = j detA(!)j�1=2A(!) = ~A(!) for all ! 2 and [A] isorthogonal. Clearly, [A] is of class II0;n0 . If n < 1, since H([A]) 3 H � O(2) hasn elements, it coincides with the group Gn. This implies that the cocycle [A] iscohomologous to a cocycle taking values in Gn.The class II0;10 is empty because any A 2 II0;10 would leave any one-dimensionalsubspace 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 Ais elementary.8. Relation to the multiplicative ergodic theoremLet A 2 G(d). Assume now that A satis�es the following integrability conditions:log+ kA(�)�1k 2 L1 (P):(16)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 areindependent of ! due to the ergodicity of �. Furthermore, the phase space Rdis decomposed into the direct sum of Oseledets subspaces Ei(!) of dimensions dicorresponding to the Lyapunov exponents �i, i = 1; : : : ; p, i.e.limn!�1 n�1 log k�A(n; !)xk = �i(!)() x 2 Ei(!)nf0g;where k � k denotes the standard Euclidean norm of Rd . The subspaces Ei(!) aremeasurable and invariant with respect to A, i.e., A(!)Ei(!) = Ei(�!). We notethat the statements of the MET hold on an invariant set of full P-measure. TheLyapunov 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 integra-bility 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 METhold true for B. Denote by GSP (d) the space of all Gl(d)-cocycles which have aspectral theory. Clearly, GIC(d) � GSP (d) � G(d).

JORDAN NORMAL FORM FOR LINEAR COCYCLES 45Obviously, if A 2 GSP (d) has more than one Lyapunov exponent, then A is de-composable. Therefore, if A 2 GSP (d) is irreducible then A has only one Lyapunovexponent.De�nition 8.1. Two cocycles A;B 2 G(d) are called Lyapunov cohomologous ifthey are cohomologous by a random linear operator C : ! Gl(d) which satis�esthe following conditions for almost all ! 2 :limn!�1 1n log kC�1(�n!)k = 0: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, thenB 2 GSP (d). Therefore, the space GSP (d) is invariant with respect to Lyapunovcohomology.Proposition 8.2. If two linear cocycles A and B from GSP (d) are cohomologousby a random linear map C : ! Gl(d), then C is a Lyapunov cohomology, andhence A and B are Lyapunov cohomologous, whence they have the same Lyapunovspectrum.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 byx and y, i.e., � := \(x; y) is the unique number from the closed interval [0; �=2]such that cos� = jhx;yijkxk kyk , where k � k and h�; �i denote the standard Euclidean normand 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 ) := inff\(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 indepen-dent 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 andall ! 2 ~, limn!�1n�1 log\�Ui(�n!);[j 6=iUj(�n!)� = 0:Proof. First we prove the proposition for the case n = 2. Let U; V � Rd be linearlyindependent invariant random subspaces of A. We will show that, for almost all! 2 , limn!�1n�1 log\(U(�n!); V (�n!)) = 0:(17)A little elementary geometry and trigonometry show that, for any two non-vanishing vectors x and y of Rd and for any M 2 Gl(d),\(x; y)2kMk kM�1k � \(Mx;My) � 2kMk kM�1k\(x; y):(18)For all ! 2 , put �(!) := \(U(!); V (!)):

46 ARNOLD et al.>From (18) it follows that, for almost all ! 2 ,�(!)2kA(!)k kA(!)�1k � �(�!) � 2kA(!)k kA(!)�1k �(!):Therefore, for almost all ! 2 and all n 2 N,�(!)2k�A(n; !)k k�A(n; !)�1k � �(�n!) � 2k�A(n; !)k k�A(n; !)�1k �(!):Since A 2 GSP (d), this implies that, for almost all ! 2 ,lim supn!1 log�(�n!)n � lim supn!1 1n� log k�A(n; !)k+ log k�A(n; !)�1k� <1:Therefore, a theorem of Tanny [27] implies that, for almost all ! 2 ,lim supn!1 log�(�n!)n = 0:Similarly, lim infn!1 n�1 log�(�n!) = 0 for almost all ! 2 . The estimates forthe 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 , putf(!) := min1�i�n\�Ui(!);[j 6=iUj(!)�:>From the assumption it follows that, for any ! 2 there is a permutationfk1; : : : ; kng of f1; : : : ; ng such thatA(!)Ui = Uki(�!) for all i = 1; : : : ; n:This, together with (18), implies that, for almost all ! 2 ,f(!)2kA(!)k kA(!)�1k � f(�!) � 2kA(!)k kA(!)�1k f(!):The same argument as above for the case n = 2 implies that almost surelylimn!�1 n�1 log f(�n!) = 0.Proposition 8.5. Let A 2 GSP (d) be irreducible. Then there exists a block-con-formal cocycle B 2 GSP (d) which is Lyapunov cohomologous to A.Proof. Since A is irreducible, it is cohomologous to a block-conformal cocycle, hencethere 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 isa random basis of Rd . Denote by e = fe1; : : : ; edg the standard Euclidean basisof Rd . By Proposition 8.4, the linear random map C : ! Gl(d) which mapsthe basis f to the basis e for all ! 2 is a Lyapunov cohomology. Clearly, Af iscohomologous to a conformal Gl(r)-cocycle. It is easily seen that we can �nd anr-dimensional Lyapunov cohomology which transforms Af into a conformal matrixGl(r)-cocycle. Combining this Lyapunov cohomology with C we obtain a Lyapunovcohomology which transforms A into a block-conformal matrix cocycle, which is ofcourse in GSP (d).We next address the question whether GSP (d) is closed under cohomology.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 47Lemma 8.6. Let (;F ;P) be a standard probability space and � be ergodic andaperiodic. Then there exists a measurable map f : ! R+ such that, for almostall ! 2 , lim supn!1 log f(�n!)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 Nthere is a set En 2 F such that the sets En; �En; : : : ; �n�1En are disjoint andP�Sn�1i=0 �iEn) > 3=4, which implies that, for all k = 0; 1; : : : ; n� 1,34n � P(�kEn) � 1n:(19)Now, by induction, we shall construct two sequences of measurable sets fFig andfGig satisfying the following conditions:(a) For any for any i 2 N there is ki 2 f0; 1; : : : ; 2i+2� 1g such that Fi = �kiE2i+2 ;(b) Gi = Fi nSi�1j=1 Fi;(c) P(Gi) � 2�i�3.Put F1 = G1 = E23 = E8. By (19), F1 and G1 satisfy the conditions (a) to (c).Suppose that we have constructed F1; : : : ; Fi�1 and G1; : : : ; Gi�1 satisfying theconditions (a) to (c). Clearly, from the disjoint sets E2i+2 ; : : : ; �2i+2�1E2i+2 thereis a set �kiE2i+2 , ki 2 f0; 1; : : : ; 2i+2 � 1g, such that P��kiE2i+2 \ �Si�1j=1 Fj�� �2�i�2P�Si�1j=1 Fj�. Put Fi := �kiE2i+2 and Gi := Fi n Si�1j=1 Fj . By (19) and thechoice of Fi; Gi we haveP(Gi) = P(Fi)� P�Fi \ � i�1[j=1Fj�� � 32i+4 � 2�i�2 i�1Xj=1 P(Fi)� 32i+4 � 2�i�2 i�1Xj=1 2�j�2 � 32i+4 � 2�i�22�2 = 2�i�3:Thus, the sequences fFig and fGig satisfy the conditions (a) to (c). Obviously, thesets Gi, i 2 N, are disjoint.Construct a function g : ! R+ by settingg(!) := ( 2i+3 for ! 2 Gi; i 2 N;1 for ! 2 nS1i=1Gi:Obviously, g(!) � 1 for all ! 2 , and g is measurable and is not integrable.We claim that, for almost all ! 2 ,lim supn!1 n�1g(�n!) =1:(20)For, suppose the opposite is true. Then, by a theorem of Tanny [27], there is ~ 2 Fsuch that P(~) = 1 and for all ! 2 ~limn!1n�1g(�n!) = 0:For all K 2 N, putU(K) := f! 2 ~ j n�1g(�n!) < 1=100 for all n � Kg:

48 ARNOLD et al.Clearly, the sets U(K),K 2 N, are measurable, U(K) � U(K+1) andS1i=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 haveg(�n!) < n=100 < L:(21)Clearly, there is 3 < i 2 N such that 2i � L < 2i+1. By (21) and the de�nitionof g, for any ! 2 U(L) and n 2 fL;L + 1; : : : ; 81Lg, �n! 62 Gi, equivalently�nU(L)\Gi = ;. Put V := �40LU(L). Then �mV \Gi = ; for all �40L < m < 40,equivalently V \ �lGi = ; for all �40L < l < 40L. Thus,P(V ) > 3=4; and V \ �lGi = ; for all � 40L < l < 40L:(22)By the de�nition of Gi we have Gi � Fi = �kiE2i+2 , hence the sets��kiGi; ��ki+1Gi; : : : ; �2i+2�ki�1Giare disjoint, so that for the set G := 2i+2�ki�1[j=�ki �jGiwe have P(G) = 2i+2P(Gi) � 2i+22�i�3 = 1=2:Furthermore, from (22), from the de�nition of G and the inequality 40L > 2i+2 itfollows that V \G = ;, which implies P(V [G) = P(V )+P(G) > 1. Thus we arriveat a contradiction, and the claim is proved. Put f(!) := exp(g(!)), and we haveproved 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 andkB(!)k < M for all ! 2 .(iii) There is an A 2 G(d) such that the entire cohomology class of A lies outsideGSP (d).Proof. (i) Take the function f from Lemma 8.6. For all ! 2 , setB(!) = f(�!)A(!)f(!)�1:Clearly B 2 G(d) and B is cohomologous to A. Suppose that B 2 GSP (d). Then, byProposition 8.2, the random linear map C(!) := f(!)I is a Lyapunov cohomology,hence, for almost all ! 2 , limn!1 log kC(�n!)kn = 0:This implies, for almost all ! 2 ,limn!1 log f(�n!)n = 0;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 almostsurely limn!1 1n n�1Xi=0 log g(�i!) = +1:

JORDAN NORMAL FORM FOR LINEAR COCYCLES 49Put A(!) := g(!) id for all ! 2 . Suppose that there is a measurable mapC : ! Gl(d) such that B(!) := C(�!)�1A(!)C(!) 2 GSP (d). Thenlim infn!1 1n n�1Xi=0 log g(�i!) = lim infn!1 1n n�1Xi=0 log k�A(n; !)k= lim infn!1 1n n�1Xi=0 log kC(�n�1!)�B(n; !)C(!)k= limn!1 1n n�1Xi=0 log k�B(n; !)k+ lim infn!1 1n n�1Xi=0 log kC(�n�1!)k < +1: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 withrespect to Lyapunov cohomology.Proof. We use a construction of Gerstenhaber (see Halmos [14, p. 32]). Leta2n�1 = a2n = (2a)�1n�3=2; n 2 N; where a = 1Xi=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 Xng, with the Borel �-algebra and Lebesgue measure.Clearly is a probability space. Let T be an arbitrary ergodic transformation ofX1 preserving the Lebesgue measure. De�ne � to be the (induced) transformationof acting by the formula�(n; x) := � (n+ 1; x) if x < an+1;(1; Tx) 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 andf(!) = �n1=2 for ! 2 X2n�1. Then it is easily seen that f is not integrable and,for all ! 2 , limn!1 1nf(�n!) = 0:(23)Put g(!) := exp(f(!)). Let A 2 GIC(d) be arbitrary, putB(!) := g(�!)g(!)�1A(!):Then B is Lyapunov cohomologous to A due to (23), hence B 2 GSP (d). Supposethat 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 dynami-cal system (; �) in Proposition 8.8 can be an arbitrary aperiodic ergodic dynamicalsystem on a Lebesgue space.We shall use the following fact about quotient cocycles.

50 ARNOLD et al.Lemma 8.10. Let A 2 GSP (d) and f = ff1(!); : : : ; fd(!)g be a random orthonor-mal basis of Rd . Assume that A has the following matrix representation with respectto f A(!) = � A1(!) A12(!)0 A2(!) �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 unionof 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 ofA. Then there is a Lyapunov cohomology C : ! Gl(d) such that, for all ! 2 ,C(�!)�1A(!)C(!) = � AU (!) �0 ARd=U (!) � :Proof. Take a random orthonormal basis f := ff1(!); : : : ; fd(!)g of Rd such thatff1(!); : : : ; fr(!)g is a basis of U(!) for all ! 2 and de�ne C to be the basischange 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 maximalinvariant ag of A. Then apply Proposition 8.5 to the irreducible subcocycles onthe 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), andA = 0BBB@ A1(!) � � �0 A2(!) � �0 0 . . . �0 � � � 0 Ap(!) 1CCCAis a Jordan normal form of A which belongs to GSP (d). Then Ai 2 GSP (di),di := dimAi, i = 1; : : : ; p. Moreover, Ai has one-point Lyapunov spectrum, andLyapunov exponent �i is a Lyapunov exponent of A, i = 1; : : : ; p; the Lyapunovspectrum 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.

JORDAN NORMAL FORM FOR LINEAR COCYCLES 51AcknowledgmentThe work of the second-named author is supported by the Deutsche Forschungs-gemeinschaft, Germany and the work of the third-named author was partially sup-ported by the Volkwagen-Stiftung, Germany, the University of Bremen, Germany,and the Cariplo Foundation for Scienti�c Research, Italy. The authors would liketo thank H. Crauel for providing the proof of Lemma 4.8 and helpful conversationson closed random sets, A. L. Onishchik, E. B. Vinberg and Le Hong Van for helpfuldiscussions 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 andA. Furman for helpful conversations on algebraic ergodic theory and group actions.References[1] L. Arnold. Random Dynamical Systems. Springer, Berlin Heidelberg New York, 1998. Toappear.[2] L. Arnold and Nguyen Dinh Cong. Linear cocycles with simple Lyapunov spectrum are densein L1. Report Nr. 410, Institut f�ur Dynamische Systeme, Universit�at Bremen, Aug. 1997.[3] C. Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions, volume 580of Lecture Notes in Math. Springer, Berlin Heidelberg New York, 1977.[4] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinai. Ergodic Theory. Springer, Berlin HeidelbergNew York, 1982.[5] H. Crauel. Random probability measures on polish spaces. Habilitationsschrift, University ofBremen, 1995.[6] C. Dellacherie and P.-A. Meyer. Probabilities and Potential. North-Holland, Amsterdam,1978.[7] R. M. Dudley. Real Analysis and Probability. Wadsworth & Brooks/Cole, Belmont, California,1989.[8] J. Feldman and C. C. Moore. Ergodic equivalence relations, cohomology, and von Neumannalgebras. I, II. Trans. Amer. Math. Soc., 234:289{359, 1977.[9] H. Furstenberg. A Poisson formula for semisimple Lie groups. Annals of Math, 77:335{383,1963.[10] H. Furstenberg. Rigidity and cocycles for ergodic actions of semisimple Lie groups. In Semi-naire Bourbaki, Nr 559, 1979/1980, volume 842 of Lecture Notes in Math., pages 273{292.Springer, Berlin Heidelberg New York, 1982.[11] F. R. Gantmacher. The Theory of Matrices, Vol. 1. Chelsea, New York, 1977.[12] V. Y. Golodets and S. D. Sinelshchikov. Locally compact groups appearing as ranges ofcocycles of ergodic Z-actions. Ergodic Theory and Dynamical Systems, 5:47{57, 1985.[13] Y. Guivarc'h and A. Raugi. Propri�et�es de contraction d'un semi-groupe de matices inversibles.Coe�cients de Liapuno� d'un produit de matrices al�eatoires independantes. Israel J. Math.,65:165{196, 1989.[14] P. R. Halmos. Lectures on Ergodic Theory. Chelsea publishing company, NY, 1956.[15] E. Hewitt and K. Ross. Abstract Harmonic Analysis. Springer, Berlin Heidelberg New York,1963.[16] J. E. Humphreys. Linear Algebraic Groups. Springer, Berlin Heidelberg New York, 1975.[17] A. A. Kirillov. Elements of the Theory of Representations. Springer, Berlin Heidelberg NewYork, 1976.[18] O. Knill. Spectral, ergodic and cohomological problems in dynamical systems. PhD thesis,ETH, Z�urich, 1993.[19] G. W. Mackey. Ergodic theory and virtual groups. Math. Annalen, 166:187{207, 1966.[20] G. A. Margulis. Discrete Subgroups of Semisimple Lie Groups. Springer, Berlin HeidelbergNew York, 1991.[21] Nguyen Dinh Cong. Topological Dynamics of Random Dynamical Systems. Clarendon Press,Oxford, 1997.[22] A. L. Onishchik and E. B. Vinberg. Lie Groups and Algebraic Groups. Springer, BerlinHeidelberg New York, 1990.

52 ARNOLD et al.[23] V. I. Oseledets. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dy-namical systems. Trans. Moscow Math. Soc., 19:197{231, 1968.[24] V. I. Oseledets. Classi�cation of GL(2,R)-valued cocycles of dynamical systems. Report Nr.360, Institut f�ur Dynamische Systeme, Universit�at Bremen, 1995.[25] K. Schmidt. Amenability, Kazhdan's property T, strong ergodicity and invariant means forergodic group actions. Ergodic Theory and Dynamical Systems, 1:223{236, 1981.[26] K. Schmidt. Algebraic Ideas in Ergodic Theory. Regional Conference Series in Mathematics,number 76. Amer. Math. Soc., Providence (Rhode Island), 1990.[27] D. Tanny. A zero-one law for stationary sequences. Z. Wahrscheinlichkeitstheorie verw. Ge-biete, 30:139{148, 1974.[28] P. Thieullen. Oseledets theorem in the elliptic case for two by two matrices. Abstract of a talkgiven at Workshop on Ergodic Theory and Dynamical Systems, 28 August { 2 September,Toru�n, Poland, 1994.[29] E. B. Vinberg. Linear Representations of Groups. Birkh�auser, Boston, 1989.[30] P. Walters. A dynamical proof of the multiplicative ergodic theorem. Trans. Amer. Math.Soc, 335:245{257, 1993.[31] R. J. Zimmer. Extensions of ergodic group actions. Illinois J. Math., 20:373{409, 1976.[32] R. J. Zimmer. Compactness condition on cocycles of ergodic transformation groups. J. LondonMath. Soc., 15:155{163, 1977.[33] R. J. Zimmer. Induced and amenable ergodic actions of Lie groups. Ann. Sci. Ec. Norm.Sup., 11:407{428, 1978.[34] R. J. Zimmer. On the cohomology of ergodic group actions. Israel J. Math., 35:289{300, 1980.[35] R. J. Zimmer. Ergodic Theory and Semisimple Groups. Birkh�auser, Boston Basel Stuttgart,1984.Institute for Dynamical Systems, University of Bremen, P. O. Box 330 440, 28334Bremen, GermanyE-mail address: arnold@mathematik.uni-bremen.deInstitute for Dynamical Systems, University of Bremen, P. O. Box 330 440, 28334Bremen, GermanyandInstitute for Applied Mathematics, University of Heidelberg, GermanyE-mail address: cong@mathematik.uni-bremen.deChair of Probability, Department of Mechanics and Mathematics, Moscow StateUniversity, Moscow 119 899, RussiaE-mail address: oseled@mech.math.msu.su