Structural Stability of Linear Random Dynamical Systems

Structural stability of linear randomdynamical systemsNGUYEN DINH CONG�Institut f�ur Dynamische Systeme, Universit�at Bremen,Postfach 330 440, 28334 Bremen, GermanyAbstractIn this paper structural stability of discrete-time linear random dynam-ical systems is studied. A random dynamical system is called structurallystable with respect to a random norm if it is topologically conjugate to anyrandom dynamical system which is su�ciently close to it in this norm. Weprove that a discrete-time linear random dynamical system is structurallystable with respect to its Lyapunov norms if and only if it is hyperbolic.1 IntroductionThis paper is closely related to the author's work on classi�cation of linear hy-perbolic cocycles [C]. The problem of structural stability of dynamical systemsis classical and important in the theory of dynamical systems. In the determin-istic case, there are many di�erent kinds of structural stability and they are wellinvestigated, for a bibliography we refer to the works of Robbin [Rob], Irwin[Irw] and the references therein. Here we shall deal with discrete-time linearrandom dynamical systems (or, in other words, linear cocycles) and study theproblem of their structural stability. In some sense, random dynamical systemscan be viewed as dynamical systems which have a measure preserving dynami-cal system as a factor (or, in other words, are extensions of such a factor), see,e.g. Arnold and Crauel [AC], and Arnold [Arn]. Hence, a crucial role will beplayed by the interrelation of measurability and dynamics. Therefore, randomdynamical systems require not only more complicated (than in the deterministiccase) tools for investigation, but also new approaches to them.In this paper we obtain a result on structural stability of discrete-time lin-ear random dynamical systems, or, in other words, of linear cocycles, which is�On leave fromHanoi Institute of Mathematics. The research was supported by the Alexan-der von Humboldt Foundation, Germany. 1

analogous to Robbin's theorem on structural stability of discrete deterministiclinear hyperbolic dynamical systems (see [Rob], Theorem 2.4).In the deterministic case Robbin had proved the structural stability of linearhyperbolic automorphisms and then applied the result for solving the classi�ca-tion problem (see [Rob]). In the random case, in [C] we have proved a theoremon structural stability of linear cocycles generated by contracting or expand-ing linear random maps and have used this result in solving the classi�cationproblem. We note that in the deterministic case the weak theorem on structuralstability (only for contracting or expanding automorphisms) can be directly gen-eralized without di�culty to a theorem on structural stability of an arbitraryhyperbolic automorphism. In the random case, due to the interplay of mea-surability and dynamics it seems hopeless to obtain \directly" from the weaktheorem on structural stability its generalization. Therefore, we have to use thetheorem on classi�cation of linear hyperbolic cocycles to prove our theorem onstructural stability of linear cocycles which have exponential dichotomy. Weemphasize that although our theorem on structural stability looks like a naturalcorollary of the theorem on classi�cation of linear hyperbolic cocycles its proofrequires nontrivial techniques and gives us a deeper insight into the structureof random dynamical systems.The central object of our research are linear cocycles generated by linearrandom maps over a discrete metric dynamical system. We give here somenecessary de�nitions and assumptions.Let (;F ;P) be a probability space, � be an automorphism of (;F ;P)preserving probability measure P. In this paper we shall always assume that �is ergodic. This assumption is made only for simplicity of presentation. Theresults can be carried over without di�culty to the general case of a non-ergodicautomorphism. Consider a linear random map A(�) : ! Gl(d;R), i.e. A(�) isa measurable mapping from the probability space (;F ;P) to the topologicalspace Gl(d;R) of linear nonsingular operators of Rd equipped with its Borel�-algebra. It generates a linear cocycle (i.e. a random dynamical system, seeArnold and Crauel [AC]) over the dynamical system (; �) as�n(A;!) :=8<: A(�n�1!) � : : : �A(!); n > 0;id; n = 0;A�1(�n!) � : : : �A�1(��1!); n < 0:Further, we assume that A(�) satis�es the integrability conditionslog+ kA(�)�1k 2 L1 (P); (1)so that the Multiplicative Ergodic Theorem of Oseledets (see [Os]), which weshall abbreviate to MET, applies to the cocycle �n(A;!). According to theMET �n(A;!) has Lyapunov exponents �1; : : : ; �p with multiplicities d1; : : : ; dpand the phase space Rd is decomposed into the direct sum of invariant subspaces2

Ei(!) of dimensions di which correspond to the Lyapunov exponents �i, i =1; : : : ; p, i.e A(!)Ei(!) = E(�!) andlimn!�1 n�1 log k�n(A;!)xk = �i () x 2 Ei(!)nf0g:This decomposition is called Oseledets splitting and the subspaces Ei(!) arecalled Oseledets subspaces of �n(A;!). In particular, Rd is decomposed into thedirect sum of the invariant stable, center and unstable subspaces, which dependmeasurably on ! 2 , Rd = EsA(!)�EcA(!)�EuA(!):De�nition 1.1. The linear cocycle �n(A;!) is said to be hyperbolic if itsLyapunov exponents are di�erent from 0.The next important notion in this paper is of a topological conjugacy pro-vided by a random homeomorphism h(�) of Rd , i.e. h(�) is a measurable mapfrom the probability space (;F ;P) into the topological space Homeo(Rd ) withthe compact-open topology.De�nition 1.2. Two linear cocycles �n(A;!) and �n(B;!) are calledconjugate if there exists a random homeomorphism h(�) of Rd such that foralmost all ! 2 the following relations hold� h(!)0 = 0;� B(!) = h(�!)�1 �A(!) � h(!):One of the e�ective tools for investigating linear cocycles are random normsand Lyapunov norms. Details about random norms, Lyapunov norms and thede�nition of structural stability of linear cocycles with respect to a random normare given in Section 2 below. The following theorem on structural stability oflinear hyperbolic cocycles with respect to their Lyapunov norms is the mainresult of this paper.Theorem 4.4. A linear cocycle �n(A;!) with A(�) satisfying (1) is struc-turally stable with respect to its Lyapunov norms k�ka;! for all su�ciently smallvalues of the parameter a > 0 if and only if it is hyperbolic.2 PreliminariesIn this paper we need a concept of random norms, Lyapunov scalar products andLyapunov norms. We give here necessary de�nitions and refer to Boxler [Box]for more details (see also Arnold's book [Arn] and [C]).3

De�nition 2.1. A random norm on Rd is a measurable function on � Rdwhich is a norm k � k! for each �xed ! 2 (in particular, the norm can begenerated by a random scalar product h�; �i!). In this case we say that Rd isequipped with the random norm k � k!.We note that Rd can have many di�erent random norms, among them the(nonrandom) standard Euclidean norm. The classical theory of Lyapunov expo-nents deals exclusively with the nonrandom Euclidean norm and it can be carriedover to the case of a random norm. We emphasize that linear random maps arede�ned as linear operators on Rd so they and the linear cocycles generated bythem are independent of the choice of the norm on Rd . The change of norms onRd a�ects only their quantitative properties which are de�ned via norms of mapsand vectors but does not a�ect the maps and cocycles themselves. The METis applicable also for the case of linear cocycles on Rd equipped with a randomnorm generated by a scalar product, and, in this case, in the integrability con-ditions (1) the Euclidean norm is substituted by the random norm. Actually, inthis paper we need the integrability conditions, which are su�cient conditions,only to assure the assertions of the MET. Our considerations are applicableas well for the cocycles which have the properties stated in the MET (thosecocycles are called to have spectral theory). We note also that the de�nitionof topological conjugacy is independent of the choice of the (possibly random)norm on Rd . An important subclass of the class of random homeomorphismsis the class of the so-called Lyapunov cohomologies, which is, by de�nition, theclass of those linear random homeomorphisms L(�) from Rd equipped with arandom norm k � k!;1 to Rd equipped with a random norm k � k!;2 such that forP-almost all ! 2 limn!�1 1n log supx 2 Rdkxk!;1 = 1 kL(�n!)xk�n!;2 == limn!�1 1n log supx 2 Rdkxk!;2 = 1 kL�1(�n!)xk�n!;1 = 0:The de�nition of Lyapunov cohomologies depends on the choice of norms onRd . If two linear cocycles are conjugate by a Lyapunov cohomology and one ofthem has spectral theory, then the other also has spectral theory, and they havethe same Lyapunov spectrum, i.e. the same set of Lyapunov exponents with thesame multiplicities (see Arnold [Arn]).Now we de�ne the Lyapunov norms and scalar products of a linear cocycle�n(A;!) with A(�) satisfying (1). Due to the MET, Rd is decomposed into thedirect sum of Oseledets subspaces Ei(!) of �n(A;!) for ! from an invariant set~ of full measure. Hence, every vector x 2 Rd can be represented uniquely in4

the form x = �pi=1xi with xi 2 Ei(!); ! 2 ~; i = 1; : : : ; p:De�nition 2.2. For a positive number a, the Lyapunov scalar product of�n(A;!) corresponding to a is de�ned as follows:For ! 2 ~ and any x = �pi=1xi and y = �pi=1yi with xi; yi 2 Ei(!)hx; yia;! := pXi=1hxi; yiia;!;where for u; v 2 Ei(!)hu; via;! :=Xn2Zh�n(A;!)u;�n(A;!)vie2(�in+ajnj) (2)with h�; �i denoting the Euclidean scalar product.For ! =2 ~ put hx; yia;! := hx; yi.The Lyapunov norm of �n(A;!) corresponding to a is de�ned bykxka;! :=qhx; xia;! for all x 2 Rd :The key property of Lyapunov norms is stated in the following proposition(see [Box] and [Arn], Chapter 3).Proposition 2.3. For all i = 1; : : : ; p; x 2 Ei(!); n 2 Z; a > 0e�in�ajnjkxka;! � k�n(A;!)xka;�n! � e�in+ajnjkxka;!: (3)We remark that the above constructed Lyapunov scalar products and Lya-punov norms depend on a and �n(A;!), so every linear cocycle has its familyof Lyapunov scalar products and Lyapunov norms depending on a positive pa-rameter a, and it is clear from inequality (3) that the most useful Lyapunovnorms are those corresponding to small a, especially such a that the intervals[�i�a; �i+a] (i = 1; : : : ; p) are disjoint. An useful property of Lyapunov normsis that the identity map of Rd is a Lyapunov cohomology with respect to themand the standard Euclidean norm, hence Lyapunov spectrum is invariant in theclass of Lyapunov norms of linear cocycles (see [Arn]).In this paper we shall consider only those random norms which are producedby scalar products and have the property that the identity map of Rd is aLyapunov cohomology with respect to them and the standard Euclidean norm(in particular, the Lyapunov norms of linear cocycles are of this class).De�nition 2.4. A linear cocycle �n(A;!) generated by a linear random mapA(�) on Rd equipped with a random norm k�k! is called structurally stable (with5

respect to the random norm k�k!) if there exists a positive number " such that forany linear random map B(�) satisfying the inequality kB(!)�A(!)k!;�! � " forP-almost all ! 2 the linear cocycle �n(B;!) generated by B(�) is conjugateto �n(A;!).Now we turn to the notion of an exponential dichotomy of a linear cocy-cle. The main source of ideas is the work of Palmer [Pal] dealing with thecase of deterministic dynamical systems. Here, in the random case, we followJohnson [Joh] and Gundlach [Gun].De�nition 2.5. We say that the linear cocycle �n(A;!) generated by a linearrandom map A(�) (not assumed to satisfy the integrability conditions of theMET) on Rd equipped with a random norm k � k! has an exponential dichotomy(with respect to the random norm k �k!) if there exist positive numbers K > 0,� > 0 and a family of projections P! of Rd depending measurably on ! 2 such that i) k�n(A;!)P!��1m (A;!)k�m!;�n! � K exp(��(n�m))for all n � m;! 2 ;ii) k�n(A;!)(id� P!)��1m (A;!)k�m!;�n! � K exp(�(n�m))for all n � m;! 2 :Remark 2.6. If the linear cocycle �n(A;!) has an exponential dichotomywith respect to the random norm k�k! with constantsK > 0; � > 0 and a familyof projections P! of Rd , then kP!k! � K for all ! 2 , but A(�) and A�1(�)might be unbounded. The property of a linear cocycle to have an exponentialdichotomy depends on the choice of the random norm.The following theorem is a random version of the roughness theorem forexponential dichotomies (see [Pal], Proposition 2.10).Theorem 2.7. Suppose we are given a random norm k � k! on Rd and alinear random map A(�) (not assumed to satisfy the integrability conditions ofthe MET) with bounded inverse (in the given random norm)kA�1(!)k�!;! �M for all ! 2 :Suppose further that the linear cocycle �n(A;!) generated by A(�) has an ex-ponential dichotomy with respect to k � k! with constants K;� and a family ofprojections P!, and � is a positive number less than �. Then for any linearrandom map D(�) (not assumed to be invertible) satisfying inequalitieskD(!)k!;�! < M�1;2K(1 + e��)(1� e��)�1kD(!)k!;�! � 1;2Ke�(e�� + 1)(e� � 1)�1kD(!)k!;�! � 1;6

the linear random map B(!) := A(!)+D(!) is invertible, and the linear cocycle�n(B;!) generated by B(�) has an exponential dichotomy with constants 2K(1+e�)(1 � e��)�1, � � � and a family of projections Q! of the same rank as P!.Moreover, for all ! 2 , the following inequality holdskQ! � P!k � 2K2(1 + e��)(1� e��)�1 sup!2 kD(!)k!;�!: (4)Proof. The proof of this theorem is completely analogous to the proof ofProposition 1.12 in [Gun]. It is basically the !-wise application of Palmer'stheorem (see [Pal], Proposition 2.10). �Next, we turn to the notion of random orientations on Rd and on its sub-spaces, which depend measurably on ! 2 . In Rd one always has the standard(nonrandom) Euclidean basis fe1; : : : ; edg. It is well known that an Euclideanspace is orientable and a choice of an orientation on an Euclidean space is de-termined by a choice of an ordered basis which is considered to be positivelyoriented. If one considers the standard Euclidean basis fe1; : : : ; edg (in thisorder of basis' vectors) to be positively oriented, then one obtains the standardorientation on Rd , which is independent of ! 2 . In general, in our set-up, wecan consider orientations on Rd which depend measurably on ! 2 . Analo-gously, on any random linear subspace E(!) of Rd , which depends measurablyon ! 2 , we can consider orientations which depend measurably on ! 2 .Now, suppose we have two linear hyperbolic cocycles �n(A;!) and �n(B;!)generated by two linear random maps A(�) and B(�) on Rd satisfying the inte-grability conditions of the MET. Denote by As(!) and Au(!) the restrictionsof A(!) to the subspaces EsA(!) and EuA(!), respectively. By the MET, EsA(!)and EuA(!) depend measurably on ! 2 . Choose measurably and �x orienta-tions on EsA(!) and EuA(!). Denote by degAs(!) and degAu(!) the degrees(with respect to the chosen orientations) of the maps As(!) and Au(!), re-spectively. It is well known that the degree of a linear bijective map from a�nite-dimensional Euclidean space to a �nite-dimensional Euclidean space isequal to the sign of the determinant of its matrix representation with respectto positively oriented bases in those spaces. We note that As(!) maps EsA(!)to EsA(�!), hence degAs(!) depends on the orientations of EsA(!) and EsA(�!).For simplicity of notation, in case one of the subspaces EsA(!) and EuA(!) istrivial we set the corresponding restricted linear map equal to id. The similarobjects for �n(B;!) are de�ned analogously and we denote them similarly. PutCsAB := f! 2 j degAs(!) � degBs(!) = �1g;CuAB := f! 2 j degAu(!) � degBu(!) = �1g:De�nition 2.8. A measurable set E 2 F is called a coboundary if thereexists a measurable set C 2 F such that E = C4�C (mod 0), where C4�C =(Cn�C) [ (�CnC) is the symmetric di�erence of the sets C; �C.7

The following classi�cation theorem was proved in [C].Theorem 2.9. Let A(�) and B(�) be linear random maps satisfying theintegrability conditions of the MET with respect to a random norm generatedby a random scalar product, and �n(A;!) and �n(B;!) be hyperbolic. Then�n(A;!) and �n(B;!) are conjugate if and only if the following conditions hold� dimEs;uA (!) = dimEs;uB (!); (mod 0) (5)� Cs;uAB are coboundaries: (6)Remark 2.10. In Theorem 2.9 we have (measurably) chosen and �xed ori-entations on fEsA(!)g!2, fEuA(!)g!2 and fEsB(!)g!2, fEuB(!)g!2, butstatements of that theorem are independent of the choice of those orientations,i.e. if condition (6) holds for a measurable choice of orientations on those sub-spaces then it holds for any other measurable choice of orientations (see [C]).Actually the theorem in [C] is proved for the case of standard Euclidean norm,but it can be carried over to the case of a random norm generated by a randomscalar product, as formulated in Theorem 2.9, without any di�culty.3 Structural stability of a linear cocycle havingan exponential dichotomyFirst, we prove a simple property of projections on an Euclidean space. For aprojection P on an Euclidean space we denote by imP and kerP its image andnull space, respectively. Denote by rankP := dim imP its rank.Lemma 3.1. Assume that P and Q are two projections of Rd satisfyinginequality kP �Qk < 1. ThenrankP = rankQ =: rand, furthermore, if fe1; : : : ; erg and fer+1; : : : ; edg are bases of imP and kerP ,respectively, then fQe1; : : : ; Qerg and f(id�Q)er+1; : : : ; (id�Q)edg are basesof imQ and kerQ, respectively.Proof. Suppose fe1; : : : ; erg is a basis of imP . We prove that the vectorsQe1; : : : ; Qer are linearly independent. Suppose, conversely, they are linearlydependent, i.e. there exist real numbers �1; : : : ; �r such that �21 + : : :+ �2r = 1and �1Qe1 + : : : + �rQer = 0. Then Q(�1e1 + : : : + �rer) = 0. Therefore,the vector e := �1e1 + : : : + �rer belongs to the space kerQ. Consequently,(P �Q)e = Pe = e. This yields kP �Qk � 1, which contradicts the assumptionof the lemma. Therefore, the vectors Qe1; : : : ; Qer are linearly independent.8

This implies rankP � rankQ. By symmetry we obtain also rankQ � rankP .Therefore, rankP = rankQ. Consequently, the vectors Qe1; : : : ; Qer form abasis of imQ. Applying the above arguments to the projections id � P andid�Q we have the lemma proved. �Now, we prove a theorem on structural stability of linear cocycles having anexponential dichotomy with respect to a random norm.Theorem 3.2. Suppose that we are given a random norm k � k! on Rdwhich is generated by a random scalar product and a linear random map A(�)with bounded inverse (in the given random norm)kA�1(!)k�!;! �M for all ! 2 : (7)Suppose further that A(�) satis�es the integrability conditions of the MET (withrespect to the given random norm) and the linear cocycle �n(A;!) generated byA(�) has an exponential dichotomy with respect to k �k! with constants K;� anda family of projections P!, and � is a positive number less than �. Then for anylinear random map D(�) (not assumed to be invertible) satisfying for all ! 2 inequalities kD(!)k!;�! � (2M)�1; (8)2K(1 + e��)(1� e��)�1kD(!)k!;�! � 1; (9)2Ke�(e�� + 1)(e� � 1)�1kD(!)k!;�! � 1; (10)2K2(1 + e��)(1� e��)�1 sup!2 kD(!)k!;�! < 1 (11)the linear random map B(!) := A(!) + D(!) is invertible, satis�es the inte-grability conditions of the MET with respect to the random norm k � k! and thelinear cocycle �n(B;!) generated by B(�) is conjugate to �n(A;!).Proof. Suppose that the linear random maps A(�); D(�); B(�) satisfy theconditions this theorem. By Theorem 2.7, B(�) is invertible. Moreover, by (7){(8), we have the estimations12kA(!)k!;�! � kB(!)k!;�! = kA(!)(id +A�1(!)D(!))k!;�! � 32kA(!)k!;�!:Therefore, log+ kB(�)k!;�! 2 L1 (P). Analogously, we have23kA�1(!)k�!;! � kB�1(!)k�!;! � 2kA�1(!)k�!;!: (12)Therefore, log+ kB�1(�)k�!;! 2 L1 (P). Thus, the linear random map B(�) sat-is�es the integrability conditions of the MET with respect to the random normk � k!. 9

Now, we consider the following family of linear random maps depending ona real parameter � 2 [0; 1] ~B(�; !) := A(!) + �D(!): (13)It is easily seen that ~B(0; !) = A(!) and ~B(1; !) = B(!). From the as-sumptions of the theorem it follows that for any � 2 [0; 1] the linear randommap ~B(�; �) is invertible and satis�es the integrability conditions of the METwith respect to the random norm k � k!. By Theorem 2.7, for any � 2 [0; 1],the linear cocycle �n( ~B(�; �);!) has an exponential dichotomy with constants2K(1+e�)(1�e��)�1, ��� and a family of projections Q�;! of the same rank asP! . This implies that �n( ~B(�; �);!) are hyperbolic and their stable and unsta-ble subspaces coincide (mod 0) with imQ�;! and kerQ�;!, respectively. Fromthe assumptions of our theorem �n(A;!) is hyperbolic and its stable and un-stable subspaces coincide (mod 0) with imP! and kerP!, respectively. By (11),Theorem 2.7 and Lemma 3.1 we havedim imP! = dim imQ�;! =: r for all � 2 [0; 1]; ! 2 :Therefore, for any � 2 [0; 1]dimEsA(!) = dimEs~B(�;�)(!) (mod 0);dimEuA(!) = dimEu~B(�;�)(!) (mod 0):In particular, dimEsA(!) = dimEsB(!) (mod 0);dimEuA(!) = dimEuB(!) (mod 0):Thus condition (5) of Theorem 2.9 is ful�lled for the pair �n(A;!); �n(B;!).Now we take and �x a random basis fe1;!; : : : ; er;!g of imP! and a ran-dom basis fer+1;!; : : : ; ed;!g of kerP! . Fix the orientations on linear subspacesfimP!g!2 and fkerP!g!2 corresponding to the random bases fe1;!; : : : ; er;!gand fer+1;!; : : : ; ed;!g, respectively, i.e. the random bases fe1;!; : : : ; er;!g andfer+1;!; : : : ; ed;!g are considered to be positively oriented. By Lemma 3.1,for any � 2 [0; 1], ! 2 , the vectors Q�;!e1;!; : : : ; Q�;!er;! form a basis ofimQ�;! and the vectors (id � Q�;!)er+1;!; : : : ; (id � Q�;!)ed;! form a basis ofkerQ�;!. The measurability of the bases fQ�;!e1;!; : : : ; Q�;!er;!g and f(id �Q�;!)er+1;!; : : : ; (id � Q�;!)ed;!g follows from the measurability of the basesfe1;!; : : : ; er;!g and fer+1;!; : : : ; ed;!g and the measurability of the family ofprojections Q�;!. Fix the orientations on fimQ�;!g!2 and fkerQ�;!g!2 cor-responding to the bases fQ�;!e1;!; : : : ; Q�;!er;!g and f(id�Q�;!)er+1;!; : : : ; (id�Q�;!)ed;!g, respectively, i.e. the bases fQ�;!e1;!; : : : ; Q�;!er;!g and f(id �Q�;!)er+1;!; : : : ; (id�Q�;!)ed;!g are considered to be positively oriented.10

Let ~Bi(�; !) and ~Bk(�; !) denote the restrictions of ~B(�; !) to imQ�;!and kerQ�;!, ! 2 , respectively. Denote by bi(�; !) the matrix represen-tation of ~Bi(�; !) with respect to the bases fQ�;!e1;!; : : : ; Q�;!er;!g andfQ�;�!e1;�!; : : : ; Q�;�!er;�!g. Similarly for the unstable direction. From thechoice of the orientations on fimQ�;!g!2 and fkerQ�;!g!2 it follows thatdeg ~Bi;k(�; !) = sign det bi;k(�; !) for all � 2 [0; 1]; ! 2 :Since for each �xed � 2 [0; 1] �n( ~B(�; �);!) has an exponential dichotomy asshown above and k ~B�1(�; !)k�!;! � 2M for all ! 2 due to (7){(8) we canapply Theorem 2.7 with A(�) substituted by ~B(�; �) and get that Q�;! dependscontinuously on � 2 [0; 1] for any �xed ! 2 . This implies that the vectorsQ�;!ej;! (j = 1; : : : ; d) depend continuously on � 2 [0; 1]. Further, by (13),~B(�; !) is continuous in � 2 [0; 1], which together with the continuity in �of Q�;! implies that ~Bi(�; !) and ~Bk(�; !) depend continuously on � 2 [0; 1].Therefore, the matrix-functions bi;k(�; !) depend continuously on � 2 [0; 1].Hence, the functions sign det bi;k(�; !) are continuous. This implies that for any�xed ! 2 the functions sign det bi;k(�; !) are constants. In particular, wehave sign det bi;k(0; !) = sign det bi;k(1; !) for all ! 2 . On the other hand,from the fact that imQ�;! and kerQ�;! coincide (mod 0) with Es~B(�;�)(!) andEu~B(�;�)(!) (� 2 [0; 1]), respectively, and ~B(0; !) = A(!), ~B(1; !) = B(!) itfollows that degAs;u(!) = sign det bi;k(0; !) (mod 0);degBs;u(!) = sign det bi;k(1; !) (mod 0):This implies degAs;u(!) = degBs;u(!) (mod 0):Therefore, the sets Cs;uAB have null measure. Thus, they are coboundaries, socondition (6) of Theorem 2.9 is ful�lled for the pair �n(A;!), �n(B;!). There-fore, �n(A;!) and �n(B;!) are conjugate. �In the roughness theorem for exponential dichotomies (see [Pal], and [Gun])conditions (7){(8) are needed only to assure invertibility of B(�). But for atopological conjugacy between �n(A;!) and �n(B;!) conditions (7){(8) areessential as the following proposition shows.Proposition 3.3. There exists a linear cocycle having an exponential di-chotomy but being not structurally stable (with respect to the standard Euclideannorm).Proof. Take for the role of the probability space (;F ;P) the half-openinterval [0,1) with the Borel �-algebra and the Lebesgue measure dx. Choose for11

the role of the automorphism � the map x 7! x+� (mod 1) with an irrational�. Decompose the interval = [0; 1) into a countable sum of disjoint half-openintervals = 1[i=1Ci;where Ci = [ci; ci+1), c1 = 0, ci+1 � ci = a�1i�2, a = P1n=1 n�2. Construct aone-dimensional linear random map A(�) : ! Rnf0g as followsA(!) := 1i+ 1 for ! 2 Ci; i 2 NIt is easily seen that A(�) satis�es the integrability conditions of the MET,because the sum 1Xi=1 log(i+ 1)i2converges. The linear cocycle �n(A;!) generated by A(�) has an exponen-tial dichotomy with respect to the standard Euclidean norm with constantsK = 1; � = 12 , projections P! = id and has Lyapunov exponent �A < 0. ByCorollary 3.5 of Knill [Kn] there exist measurable sets ~Ci � Ci which are notcoboundaries. Construct for i = 1; 2; 3; : : : linear random mapsBi(!) := � �A(!) if ! 2 ~Ci;A(!) otherwise:For any i 2 N the linear random map Bi(�) is invertible and satis�es the integra-bility conditions of the MET. It is easily seen that the linear cocycles �n(Bi;!)generated by Bi(�), i 2 N, have an exponential dichotomy with respect to thestandard Euclidean norm with constants K = 1; � = 12 . Further, for all i 2 N,the linear cocycles �n(Bi;!) have the same Lyapunov exponent as the linearcocycle �n(A;!). By the construction of Bi(�), for any i 2 N the set CsABicoincides with ~Ci, and, therefore, is not a coboundary. Consequently, by virtueof Theorem 2.9, for any i 2 N, the linear cocycle �n(Bi;!) is not conjugate tothe linear cocycle �n(A;!). Moreover, we havesup!2 kBi(!)�A(!)k = 2i+ 1 i!+1�! 0:Therefore, the linear cocycle �n(A;!) is not structurally stable with respect tothe standard Euclidean norm. �Remark 3.4. If we do not require conditions (7){(8) in Theorem 3.2, then itcan happen that in any neighborhood of the linear random map A(�) there existsa linear random map which is not invertible. If instead of conditions (7){(8) inTheorem 3.2 we require the invertibility of B(�) as in [Gun], then we obtain12

an exponential dichotomy of the linear cocycle �n(B;!) but the linear randommap B(�) might not satisfy the integrability conditions of the MET. Moreover,in that case, as Proposition 3.3 shows, even if we restrict ourselves to the spaceof linear random maps which satisfy the integrability conditions of the METthe linear cocycle �n(A;!) might not be structurally stable.Since the linear cocycle �n(A;!) constructed in Proposition 3.3 has an expo-nential dichotomy with respect to the standard Euclidean norm it is hyperbolic.Therefore, while it is not structurally stable with respect to the standard Eu-clidean norm, it is structurally stable with respect to its Lyapunov norms aswill be proved in Theorem 4.4. This shows a nice property of Lyapunov norms.However, the price we have to pay for this is that Lyapunov norms are, in gen-eral, hard to compute. As seen from their de�nition, we need to know long-termbehavior of cocycles in order to compute their Lyapunov norms.4 Structural stability of linear hyperbolic cocy-clesWe give here a necessary condition for structural stability of a linear cocycle.Theorem 4.1. Suppose that we are given a random norm k � k! on Rdgenerated by a random scalar product and A(�) is a linear random map satisfyingthe integrability conditions of the MET with respect to the given random norm.If the linear cocycle �n(A;!) is structurally stable with respect to k � k!, then itis hyperbolic.Proof. Set Ck := f!j kA(!)k!;�! < kg:Then, 1[k=1Ck = ; Ck+1 � Ck:Therefore, there exists an integer m 2 N such thatP(Cm) � 1=2: (14)Fix such an m and introduce linear random mapsBk(!) = � (1 + �k)A(!) for ! 2 Cm;A(!) for ! 2 nCm:where �k are real numbers with small absolute values, k 2 N. It is easilyseen that the MET applies to the linear cocycles �n(Bk;!) generated by linearrandom maps Bk(!). From the de�nitions of Cm; Bk(�) we havekBk(!)�A(!)k!;�! � j�kjm for all ! 2 : (15)13

For any k 2 N, �n(Bk;!) can be represented as follows�n(Bk;!) = (1 + �k)l(n;!)�n(A;!);where l(n; !) = Pn�1i=0 1Cm(�i!) with 1Cm(�) denoting the characteristic func-tion of the set Cm 2 .From the MET and Birkho�'s ergodic theorem it follows that, for any k 2 N,�n(Bk;!) has Lyapunov exponents�i(Bk) = �i(A) + P(Cm) log(1 + �k) (16)with the same multiplicities di as the Lyapunov exponents �i(A) of �n(A;!).We denotes by ds(A); dc(A); du(A) the dimensions of the stable, center andunstable subspaces of the cocycle �n(A;!), respectively. More precisely,ds(A) = X�i(A)<0 di;dc(A) = X�i(A)=0 di;du(A) = X�i(A)>0 di;with di denoting the multiplicity of the Lyapunov exponent �i(A) of cocycle�n(A;!).Suppose that �n(A;!) is not hyperbolic, then dc(A) > 0. Consequently,by virtue of (14){(16) in any neighborhood of A(�) (with respect to the givenrandom norm) we can choose hyperbolic linear random maps B+k (�); B�k (�) (bychoosing small positive and negative �k) such thatds(B�k ) = ds(A) + dc(A);du(B�k ) = du(A);ds(B+k ) = ds(A);du(B+k ) = du(A) + dc(A):Therefore, by Theorem 2.9, �n(B+k ;!) is not conjugate to �n(B�k ;!). Thisimplies that �n(A;!) is not structurally stable with respect to the given randomnorm. �Remark 4.2. Theorem 4.1 gives a necessary condition for structural stabilitywith respect to a random norm in terms of Lyapunov exponents which areinvariant in the class of those random norms which have the property that theidentity map is a Lyapunov cohomology with respect to them and the standardnonrandom Euclidean norm of Rd . 14

Lemma 4.3. Let A(�) be a linear random map satisfying (1), and �n(A;!)be hyperbolic. Then there exists a positive number aA such that for any a 2(0; aA) the cocycle �n(A;!) is structurally stable with respect to its Lyapunovnorm k � ka;!.Proof. ChooseaA := minf min�i(A)>0�i(A); min�i(A)<0(��i(A))g:Let a 2 (0; aA) be arbitrary. Consider the restriction of the automorphism �to the invariant set ~ of full measure on which the assertions of the MET for�n(A;!) hold. By Proposition 2.3, the restriction of �n(A;!) to the dynamicalsystem (~; �) has an exponential dichotomy with respect to the Lyapunov normk � ka;! of �n(A;!) with constants 1; aA � a and with the family of orthogonalprojections onto EsA(!) (in Rd with respect to the Lyapunov scalar producth�; �ia;!). Furthermore, by (3) we have that for all ! 2 ~kA�1(!)ka;�!;! � ea�aA :Therefore, by Theorem 3.2 the restriction of �n(A;!) to (~; �) is structurallystable with respect to the Lyapunov norm k�ka;!. Hence, �n(A;!) is structurallystable with respect to the Lyapunov norm k � ka;!. �Combining Lemma 4.3 with Theorem 4.1 we obtain immediately the formu-lated in Introduction Theorem 4.4 on structural stability of linear hyperboliccocycles. Theorem 4.4 is a generalization of the well-known deterministic the-orem of Robbin on structural stability of a hyperbolic automorphism of Rd([Rob], Theorem 2.4). In the trivial case, when consists of only one element,Theorem 4.4 is equivalent to that of Robbin, because any Lyapunov norm isthen equivalent to the standard Euclidean norm.AcknowledgmentI would like to thank Professor L. Arnold for fruitful discussion, for his constanthelp and encouragement in the course of this work. I would also like to thankthe referee for many helpful comments.References[Arn] L. Arnold. Random dynamical systems. Preliminary version 2, Bremen,1994. 15

[AC] L. Arnold and H. Crauel. Random dynamical systems. In: L. Arnold,H. Crauel and J.-P. Eckmann (Eds) Lyapunov exponents, Oberwolfach1990, Vol. 1486 of Lecture Notes in Mathematics, 1{22. Springer-Verlag:Berlin, 1991.[Box] P. Boxler. A stochastic version of center manifold theory. Probab. Th.Rel. Fields, 83(1989), 509{545.[C] Nguyen Dinh Cong. Topological classi�cation of linear hyperbolic cocy-cles. Submitted to J. Dyn. Di�erential Equations.[Gun] V. M. Gundlach. Random homoclinic orbits. Random & ComputationalDynamics 3(1995), 1{33.[Irw] M. C. Irwin. Smooth dynamical systems. Academic Press: London, 1980.[Joh] R. A. Johnson. Remarks on linear di�erential systems with measurablecoe�cients. Proc. Amer. Math. Soc. 100(1987), # 3, 491{504.[Kn] O. Knill. The upper Lyapunov exponent of Sl(2,R) cocycles: Disconti-nuity and the problem of positivity. In: L. Arnold, H. Crauel and J.-P.Eckmann (Eds) Lyapunov exponents, Oberwolfach 1990, Vol. 1486 of Lec-ture Notes in Mathematics, 86{97. Springer-Verlag: Berlin, 1991.[Os] V. I. Oseledets. A multiplicative ergodic theorem. Lyapunov characteris-tic numbers for dynamical systems. Trans. Moscow Math. Soc. 19(1968),197{231.[Pal] K. J. Palmer. Exponential dichotomies, the shadowing lemma andtransversal homoclinic points. In: Dynamics reported, Volume 1, 1988,265{306.[Rob] J. W. Robbin. Topological conjugacy and structural stability for discretedynamical systems. Bull. Amer. Math. Soc., 78(1972), # 6, 923{952.

16