STRUCTURAL STABILITY AND TOPOLOGICAL CLASSIFICATION OF

STRUCTURAL STABILITY AND TOPOLOGICAL CLASSIFICATION OF CONTINUOUS-TIME LINEAR HYPERBOLIC COCYCLES NGUYEN DINH CONG

Abstract. In this paper we study topological properties of continuoustime linear hyperbolic cocycles. Roughly speaking, two cocycles are called conjugate if there exists a random homeomorphism mapping their orbits into each other; a cocycle is called structurally stable if it is conjugate to every cocycle from a neighborhood of itself. We prove that any linear hyperbolic cocycle is structurally stable with respect to its Lyapunov norm for all suciently small values of the parameter a in the de nition of the Lyapunov norm. Concerning the classi cation problem, in the deterministic case it is well known that two linear hyperbolic ows are topologically equivalent if they have stable subspaces of the same dimension, and hence there are d + 1 topological classes of d-dimensional linear hyperbolic ows (see Irwin [11], p. 86). In this paper we prove that two cocycles are conjugate if and only if they have stable subspaces of the same dimension and their restrictions to their stable subspaces are both orientation preserving or orientation reversing with respect to a measurable choice of orientations on those stable subspaces. Our result is a generalization of the deterministic one: in case the base probability space consists of only one element, our theorem is reduced to the deterministic one. We give examples where there are four or in nitely many topological classes of two-dimensional cocycles. The analogous problems for discrete time were investigated in [15, 14].

1. Introduction In the theory of dynamical systems the problems of structural stability and classi cation are of crucial importance and have been investigated by many researchers. The problems of structural stability and topological classi cation of deterministic linear hyperbolic dynamical systems were well studied and the results are well known. We present here the classical result on the topological classi cation of linear hyperbolic ows (see Irwin [11], p. 86). 1991 Mathematics Subject Classi cation. Primary 58F10, 58F15. Secondary 28D10, 58F19. Key words and phrases. Linear hyperbolic cocycle, random homeomorphism, orientation, degree of a homeomorphism, topological classi cation, structural stability, topological conjugacy. Research was supported by the Alexander von Humboldt Foundation, Germany. 1

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Theorem. Two linear hyperbolic ows are topologically equivalent if and

only if they have stable subspaces of the same dimension. For more information on the deterministic case we refer to the works of Robbin [18], Kuiper [13], Irwin's monograph [11] and the references therein. In this paper we shall deal with continuous-time linear hyperbolic cocycles, or, in other words, random dynamical systems. Let ( ; F ; P) be a probability space, (t )t2R be a one-parameter group of automorphisms of ( ; F ; P) preserving probability measure P. We shall always assume that the ow (t)t2R is ergodic. A (continuous-time) linear random dynamical system over ( ; F ; P; (t )t2R ) (see Arnold [2], and Arnold and Crauel [3]), or, in other words, a (continuous-time) linear cocycle over ( ; F ; P; (t )t2R), is a mapping : R ! Gl(d; R) with the following properties (i) is B(R ) F ; B(Gl(d; R)) measurable, where B() denotes the Borel -algebra of topological spaces; (ii) the mappings (t; !) 2 Gl(d; R) form a cocycle over (t)t2R , i.e., they satisfy (0; !) = id for all ! 2 ; (t + s; !) = (t; s!) (s; !) for all t; s 2 R ; ! 2 ; (; !) : R ! Gl(d; R ) is continuous for each ! 2 : It is well known that if the mapping (; !) is continuous for each ! 2 and (t; ) is measurable for each t 2 R , then (; ) is measurable in the product measurable space. For a general de nition of random dynamical systems see Arnold's book [2]. An important notion of this paper is the notion of random homeomorphism h() of Rd which is a measurable map from the measurable space ( ; F ) (R d ; B(R d )) into the topological space (R d ; B(R d )), (!; x) 7! h(!)x, such that for each ! 2 the map h(!) is a homeomorphism of R d . Two cocycles (t; !) and (t; !) are called conjugate if there exists a random homeomorphism h() of R d and an invariant (with respect to (t )t2R ) set ~ 2 F of full P-measure such that for any ! 2 ~ the following relations hold h(!)0 = 0; (t; !) = h(t !)?1 (t; !) h(!) for all t 2 R : We see that the random homeomorphism h() maps trajectories of (t; !) into trajectories of (t; !). We emphasize that in the above de nition of topological conjugacy the assumption that ~ is invariant with respect to (t)t2R appears natural, because the relation h(!)0 = 0 might be required for all ! 2 , and the second relation itself contains the invariance with respect to

STRUCTURAL STABILITY AND CLASSIFICATION OF COCYCLES

3

(t )t2R due to the cocycle property, i.e., if ! satis es this relation then t !, t 2 R , also satis es it. We would like to say some words about the terminology. In the deterministic case, the ows (continuous-time dynamical systems) are classi ed by means of ow equivalence, which admits time reparametrization, but for the case of ows without periodic points the de nition of topological equivalence which does not admit time reparametrization, could be reasonable, as Robbin has noticed (see [18]). In our random case, continuous-time linear hyperbolic cocycles (in other words, continuous-time linear hyperbolic random dynamical systems) can be viewed as ows without periodic points, so we use the above de nition of topological equivalence and use the term conjugacy (as in the discrete-time case) to emphasize that we do not permit the time reparametrization. As it will be clear from the results of this paper, if we allow the time reparametrization then the topological classi cation of continuous-time linear hyperbolic cocycles remains unchanged. In this paper we obtain two main results. The rst concerns the structural stability of linear cocycles: Theorem 4.7. Every linear hyperbolic cocycle is structurally stable with respect to its Lyapunov norm k ka;! for all suciently small values of the parameter a > 0. The second main result concerns the topological classi cation problem: Theorem 7.2. Two continuous-time linear hyperbolic cocycles (t; !) and (t; !) are conjugate if and only if (i) dim Es (!) = dim E s (!); (ii) There exist measurable orientations on Es (!), E s (!) and a set 0 2 F which is invariant with respect to (t )t2R and has full P-measure such that deg s(t; !) = deg s(t; !) for all ! 2 0 ; t 2 R: Theorem 7.2 can be rephrased as follows: two linear hyperbolic cocycles are conjugate if and only if they have stable subspaces of the same dimension and their restrictions to their stable subspaces are both orientation preserving or orientation reversing with respect to a measurable choice of orientations on those stable subspaces. We remark that in our continuous-time case the continuity in t of linear cocycles plays an important role. In the structural stability problem it allows us to prove constructively the structural stability of any uniformly contracting (or expanding) cocycle. This result is a random version of the deterministic one (see Irwin [11], p. 85) and is presented in Section 2. For the main result on structural stability of linear hyperbolic cocycles in Section 4 we shall have to use random Lyapunov scalar products and norms. The notion of Lyapunov scalar products and norms and their properties are given in Section 3, where we also prove a theorem on canonical forms

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of (non-uniformly) contracting (or expanding) cocycles. The remaining part of the paper is devoted to the problem of topological classi cation. In Section 5 we give necessary conditions for topological conjugacy. In Section 6 we present some orientation properties of linear hyperbolic cocycles. In Section 7 we present the theorem on topological classi cation and give two examples showing that the number of topological classes depends on the underlying deterministic dynamical system over which the cocycles are de ned. The analogous problems for the discrete-time case were treated in the author's works [15, 14]. As it was noticed by Arnold [2], in the theory of random dynamical systems the interplay of measurability and dynamics is of crucial importance. Here, in our continuous-time case, the continuity in t of linear cocycles enters the game. It allows us to reduce the number of orientation conditions for a topological conjugacy from two in the discrete-time case (see [15]) to one. So, roughly speaking, in general, there are fewer topological classes of continuous-time linear hyperbolic cocycles than there are for discrete-time, but there are still more than for deterministic ows. On the other hand, the continuous-time case poses more diculties concerning measurability, because there are continuously many time-moments t, not just countable many as in the discrete-time case. Moreover, a discrete-time linear cocycle always has a generator, whereas a continuous-time one, in general, has not. Therefore, the methods in [15, 14] for the discrete-time case are hardly useful for our continuous-time case. In this paper, our approach is based on the detailed investigation of Lyapunov scalar products and orientation properties of linear cocycles. In our opinion, some intermediate results in Sections 3 and 6 are of independent interest. Finally, we remark that, while the deterministic result on topological classi cation of hyperbolic ows is rather simple (see Irwin [11], p. 86), the presence of the underlying deterministic dynamical system and probability space of linear cocycles makes the problem more complicated and gives rise to a much richer and more interesting structure of linear hyperbolic cocycles, which are considered as basic for the theory of random dynamical systems. 2. Canonical contracting and expanding cocycles In this section we prove a theorem on the canonical form of uniformly contracting and expanding cocycles.

De nition 2.1. A (possibly nonlinear) cocycle (t; !) is called uniformly contracting if there exists a positive number > 0 such that k(t; !)xk e?t kxk for all t 0; ! 2 ; x 2 R d : (t; !) is called uniformly expanding if there exists a positive number > 0 such that k(t; !)xk et kxk for all t 0; ! 2 ; x 2 R d :


5

Remark 2.2. We note that uniformly contracting and expanding cocycles

are particular classes of cocycles exhibiting an exponential dichotomy (see Palmer [17], Gundlach [10], and Section 4 below).

Theorem 2.3. Every uniformly contracting cocycle is conjugate to the

canonical contracting linear cocycle c(t; !) := e?t id. Every uniformly expanding cocycle is conjugate to the canonical expanding linear cocycle e(t; !) := et id.

Proof. Let (t; !) be an uniformly contracting cocycle. Consider the

function

de ned by the formula

f : R Rd ! R+

f (t; !; x) := k(t; !)xk: From the continuity of (; !), the condition (0; !) = id and De nition 2.1 it follows that for each pair of a xed ! 2 and a xed non-vanishing x 2 R d the map f (; !; x) : R ! (0; 1) is a bijection and monotone decreasing. Therefore, for every pair of a xed ! 2 and a xed non-vanishing x 2 R d there exists exactly one u(!; x) 2 R such that f (u(!; x); !; x) = 1: The so de ned function u(; ) : (R d nf0g) ! R is continuous with respect to x 2 R d nf0g and FB(R d nf0g); B(R)-measurable because of the continuity of f (t; !; x) with respect to x 2 R d and B(R ) F B(R d ); B(R + )-measurability of f . Now we construct a random homeomorphism, which will, and we shall prove that, furnish a topological conjugacy between (t; !) and the canonical contracting linear cocycle c(t; !) := e?t id. Set h(!)x := eu(!;x) (u(!; x); !)x for all ! 2 ; x 2 R d nf0g; (1) h(!)0 := 0: (2) First we show that h() is a random homeomorphism. 1. Fix an arbitrary ! 2 . Let x1 ; x2 be two dierent non-vanishing vectors of R d . If u(!; x1) 6= u(!; x2), then kh(!)x1 k = eu(!;x1) 6= eu(!;x2 ) = kh(!)x2k; which implies h(!)x1 6= h(!)x2. If u(!; x1) = u(!; x2) =: u then (u; !)x1 6= (u; !)x2, which implies h(!)x1 = eu(u; !)x1 6= eu(u; !)x2 = h(!)x2:

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If x 2 R d is an arbitrary non-vanishing vector, then it is easily seen that h(!)x 6= 0 = h(!)0. Thus, h(!) is injective. 2. Fix an arbitrary ! 2 . Let x 2 R d be an arbitrary non-vanishing vector. Set u := log kxk and y := (u; !)?1e?ux: It is easily seen that h(!)y = x. Furthermore, h(!)0 = 0. Therefore, h(!) is surjective. Taking into account its injectivity h(!) is bijective. 3. h(!)x depends continuously on x 2 R d because of the continuity of u(!; ) and (; !). The continuity of h(!)x at the point x = 0 is due to the contracting property of (t; !) and the de nition of h(!). Therefore, by Brouwer's theorem on the invariance of domain (see Dieudonne [6], p. 52, and Dold [7], p. 79), h(!) is a homeomorphism of R d . 4. h(!)x depends measurably on (!; x) 2 R d because of the measurability of u(!; x) and (t; !). So h(!) is a random homeomorphism of R d . Now, we prove that h() furnishes a topological conjugacy between (t; !) and c(t; !). Let t; s 2 R be arbitrary. By the cocycle property of , for arbitrary non-vanishing x 2 R d and arbitrary ! 2 we have (s + t; !)x = (t; s !) (s; !)x: This implies u(!; x) = u(s!; (s; !)x) + s: (3) By (1){(3), we have h(s !)((s; !)x) = eu(s !;(s;!)x)(u(s!; (s; !)x); s!)((s; !)x) = eu(!;x)?s (u(!; x) ? s; s!) (s; !)x = eu(!;x)?s (u(!; x); !)x: Consequently, h(s!) (s; !)x = e?seu(!;x) (u(!; x); !)x = e?sh(!)x: This is equivalent to h(!)x = esid h(s!) (s; !)x: For the vanishing vector of R d we have h(!)0 = esid h(s!) (s; !)0: Therefore, h(!) = esid h(s!) (s; !); or, which is equivalent, (s; !) = h(s!)?1 c(s; !) h(!); which means h() furnishes a topological conjugacy between (t; !) and c(t; !).


7

The case of an uniformly expanding cocycle can be easily treated analogously.

Remark 2.4. i) In Theorem 2.3 we did not use the linearity of (t; !). Thus it holds true also for the case of a nonlinear uniformly contracting (or expanding) continuous cocycle (t; !) which is a homeomorphism of R d for each pair (t; !). So, Theorem 2.3 is not only a theorem on classi cation of cocycles, it also is a theorem on linearization of continuous-time nonlinear topological cocycles. ii) In comparison with other methods in the works on this and related problems (see, e.g., Wanner [19, 20], and Nguyen [15, 14]) we emphasize that our method is constructive and we do not use any xed-point theorem. Our method is a random version of the proof of the ow equivalence of two deterministic linear contracting ows (see Irwin [11], p. 85). 3. The non-uniformly contracting and expanding linear cocycles

From now on, unless speci ed otherwise, we shall always assume that the linear cocycle (t; !) (and other linear cocycles also) satis es the following integrability conditions: sup log+ k1(t; !)k 2 L 1 (P); (4) 0t1

so the Multiplicative Ergodic Theorem of Oseledets, which we shall abbreviate by MET, is applicable to (t; !) (see Oseledets [16], and Arnold [2], Theorem 3.5.20). According to the MET (t; !) has Lyapunov exponents 1; : : : ; p with multiplicities d1; : : : ; dp and the phase space R d is decomposed into the direct sum of invariant subspaces Ei (!) of dimensions di which correspond to the Lyapunov exponents i, i = 1; : : : ; p, i.e (t; !)Ei(!) = Ei(t !) and lim t?1 log k(t; !)xk = i () x 2 Ei(!)nf0g: t!1 This decomposition is called Oseledets splitting and the subspaces Ei (!) are called Oseledets subspaces of the linear cocycle (t; !). In particular, the phase space R d is decomposed into the direct sum of the invariant stable, center and unstable subspaces, which depend measurably on ! 2 , R d = Es (! ) Ec (! ) Eu (! ): De nition 3.1. The linear cocycle (t; !) is said to be hyperbolic if its Lyapunov exponents are dierent from 0. It is said to be contracting (expanding) if all its Lyapunov exponents are negative (positive). In this paper we shall need a concept of random norms, Lyapunov scalar products and Lyapunov norms. We give here necessary de nitions and refer to Arnold's book [2] for more details (see also Nguyen [15, 14]).

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Due to the MET, for a linear cocycle (t; !) the phase space R d is decomposed into the direct sum of Oseledets subspaces Ei (!) for ! from an invariant set ~ of full measure. Hence, every vector x 2 R d can be represented uniquely in the form x = pi=1 xi with xi 2 Ei(!); ! 2 ~ ; i = 1; : : : ; p: De nition 3.2. For a positive number a, the Lyapunov scalar product of the linear cocycle (t; !) corresponding to a is de ned as follows: For ! 2 ~ and any x = pi=1 xi and y = pi=1 yi with xi ; yi 2 Ei (!) p X

hx; yia;! := hxi ; yiia;! ; where for u; v 2 Ei(!)

hu; via;! :=

i=1

Z +1

h(s; !)u; (s; !)vi ds

(5) e2(i s+ajsj) with h; i denoting the Euclidean scalar product. For ! 2= ~ put hx; yia;! := hx; yi. The Lyapunov norm of (t; !) corresponding to a is de ned by q kxka;! := hx; xia;! for all x 2 R d ; ! 2 : The key property of Lyapunov norms is stated in the following proposition (see Arnold [2], Theorem 3.9.6). ?1

Proposition 3.3. For all i = 1; : : : ; p; x 2 Ei (!); t 2 R ; a > 0 e t?ajtj kxka;! k(t; !)xka; ! e t+ajtj kxka;! : i

t

i

(6)

The selection theorem for multi-valued functions (see Deimling [5], Theorem 24) and the Gram-Schmidt orthogonalization procedure assure the existence of a (measurable) random orthonormal (with respect to the scalar product h; ia;! of the linear cocycle (t; !)) basis fe1;! ; : : : ; ed;! g such that e1;! ; : : : ; ed1 ;! belong to E1(!), ed1 +1;! ; : : : ; ed1 +d2 ;! belong to E2 (!), : : : , ed?dp +1;! ; : : : ; ed;! belong to Ep(!). For the Lyapunov scalar product h; ia;! we call such a random basis a Lyapunov random basis. We note that having a random basis we may consider matrix representations of linear maps with respect to that basis and, due to the orthonormality of the Lyapunov random basis, the matrix norm of the matrix representation of a linear random map with respect to the Lyapunov random basis is equal to the norm of that linear random map in the Lyapunov norm; in general the orthonormality of a basis implies the equality of the norm of a linear map and the matrix norm of its matrix representation. We remark that the above constructed Lyapunov


9

scalar product and Lyapunov norm depend on the constant a and the linear cocycle (t; !), so every linear cocycle has its family of Lyapunov scalar products and Lyapunov norms depending on a positive parameter. We give here some further properties of Lyapunov random norms, which will be needed later. First, we present the notion of Lyapunov cohomology, which is a special class of random homeomorphisms (see Arnold [2], x3.7). Two linear cocycles which are conjugate by a linear random homeomorphism, say L(!), are called cohomologous, and L(!) is called a cohomology. Now, let R d be equipped with two random norms k k1;! and k k2;! (i.e., k k1;! and k k2;! are norms on R d for each ! 2 and depend measurably on ! 2 ; in particular, they might be independent of !, i.e., nonrandom). L(!) is called a Lyapunov cohomology (with respect to the pair of norms kk1;! and kk2;! ) if for any ! from an invariant set of full P-measure 1 lim log sup kL(t !)xk2;t! = 0; t!1 t kxk1;t ! =1 1 lim log sup kL?1(t !)xk1;t! = 0: t!1 t kxk2;t ! =1 In this case, two cocycles cohomologous by L(!) are called Lyapunov cohomologous. The following properties of Lyapunov random norms allow us to use them for solving our problems (see Arnold [2], Corollary 3.9.9).

Lemma 3.4. A linear cocycle with respect to a Lyapunov norm is Lya-

punov cohomologous to itself with respect to the standard Euclidean norm. In other words, the identity map is a Lyapunov cohomology with respect to any pair of a Lyapunov norm and the standard Euclidean norm.

By the same arguments we have Lemma 3.5. Let fe1; : : : ; edg be the standard nonrandom Euclidean basis of R d , and ff1;! ; : : : ; fd;! g be a Lyapunov random basis corresponding to a Lyapunov scalar product. Then the coordinate change from fe1 ; : : : ; edg to ff1;! ; : : : ; fd;! g, i.e., the linear random homeomorphism L(!) of R d de ned by L(!)ei = fi;! , i = 1; : : : ; d, ! 2 , is a Lyapunov cohomology of linear cocycles on R d equipped with the standard Euclidean norm. Remark 3.6. If a linear cocycle (t; !) is conjugate to a linear cocycle (t; !) by the above linear random homeomorphism L(!) then the matrix representation of (t; !) with respect to the standard Euclidean basis fe1; : : : ; edg coincides with the matrix representation of (t; !) with respect to the random basis ff1;! ; : : : ; fd;! g, and vice versa. We emphasize that the above chosen Lyapunov basis is, in general, not continuous in t, it is only measurable in t, i.e., the mappings t 7! ei;t ! ; i =

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NGUYEN DINH CONG

1; : : : ; d; are not continuous (see Arnold [1]). Therefore, we can not apply directly the Lyapunov cohomology of the basis transformation from the standard Euclidean basis to a Lyapunov basis and then apply the results of Section 2, because after that transformation the new cocycle is no longer continuous in t and the results of Section 2 are not applicable to it. Our aim in this section is to nd a linear cocycle which is continuous in t, cohomologous to the linear cocycle under consideration and to which the results of Section 2 are applicable. The following theorem is an important step in our program and it also is of independent interest as an important property of Lyapunov random scalar products.

Theorem 3.7. For any a > 0; ! 2 the function (x; y; t) 7! hx; yia; ! ; (x; y; t) 2 R d R d R ; is continuous with respect to the Euclidean metrics in R d R d R and R . In particular, (x; t) 7! k(t; !)xka; ! is continuous if (; !) is. t

t

Proof. Let ! 2 be arbitrary. It suces to show that for any x; y 2 R d and " > 0 there exists > 0 such that for any x~; y~ 2 R d ; t 2 R such that kx~ ? xk ; ky~ ? yk ; jtj the following inequality holds jhx~; y~ia; ! ? hx; yia;! j ": (7) If ! 2= ~ , then by the de nition of the Lyapunov scalar product and the t

invariance of ~ we have jhx~; y~ia;t! ? hx; yia;! j = jhx~; y~i ? hx; yij 2(kxk + kyk + ): So one can easily nd which makes (7) true. Suppose ! 2 ~ , then t! 2 ~ for any t 2 R . We denote by Pi(!) the orthogonal projection onto the Oseledets subspace Ei (!), i = 1; : : : ; p, with respect to the Lyapunov scalar product h; ia;! of R d , or, what is equivalent, L the projection onto Ei(!) along j6=i Ej (!). Then by the de nition of the Lyapunov scalar product we have p Z +1 h(s; ! )P (! )x; (s; ! )P (! )y i X i i hx; yia;! = ?1 ds; (8) 2( s + a j s j ) i e i=1 and p Z +1 h(s; t ! )P (t ! )~ X x; (s; t!)Pi(t !)~yi ds: (9) i hx~; y~ia;t! = ?1 e2(i s+ajsj) i=1 Since the subspaces Ei(!) are invariant with respect to (t; !) we have Pi(t !)(t; !) = (t; !)Pi(!):


Setting

x^ := ?1(t; !)~x; y^ := ?1(t; !)~y;

11

we have Z +1 h(s; t ! )P ( t ! )~ x; (s; t !)Pi(t !)~yi ds = i e2(i s+ajsj) ?1 Z +1 h(s; t ! )(t; ! )P (! )^ yi ds i x; (s; t ! )(t; ! )Pi(! )^ = 2( s + a j s j ) i e ?1 Z +1 h(s + t; ! )P (! )^ x ; (s + t; !)Pi(!)^yi ds i = e2(i s+ajsj) ?1 Z +1 h(s; ! )P (! )^ yi e2i t e2a(jsj?js?tj) ds: i x; (s; ! )Pi(! )^ = 2( s + a j s j ) e i ?1 Therefore, when t is suciently small, Z +1 h(s; t ! )P ( t ! )~ x; (s; t !)Pi(t !)~yi ds i e2(i s+ajsj) ?1 is close to Z +1 h(s; !)Pi(!)^x; (s; !)Pi(!)^yi ds: e2(i s+ajsj) ?1 It remains to show that for x~ and y~ close to x and y the value Z +1 h(s; ! )P (! )^ yi ds = i x; (s; ! )Pi(! )^ e2(i s+ajsj) ?1 Z +1 h(s; ! )P (! )?1 (t; ! )~ x; (s; !)Pi(!)?1(t; !)~yi ds i = e2(i s+ajsj) ?1 is close to Z +1 h(s; ! )P (! )x; (s; ! )P (! )y i i i ds: 2( s + a j s j ) i e ?1 This follows immediately from the linearity of (s; !) and Pi(!) and the t!0 relation k?1(t; !) ? idk ?! 0. To summarize, we have proved that each integral in the sum of (9) is close to the corresponding integral in the sum of (8) if x~, y~, t are close to x, y, 0. Thus, hx~; y~ia;t! is close to hx; yia;! if x~, y~, t are close to x, y, 0. Next, we construct a random basis which is continuous in t, and in a particular case is a Lyapunov basis. Proposition 3.8. For any linear cocycle (t; !) and a > 0 there exists a random basis fe1(!); : : : ; ed (!)g which is orthonormal with respect to the Lyapunov scalar product h; ia;! and depends continuously on t in the sense that for any ! 2

t!0 kei(t !) ? ei(!)k ?! 0; i = 1; : : : ; d: (10)

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Proof. Let a > 0 be arbitrary and fb1; : : : ; bdg be a basis of R d which

is orthonormal with respect to the Euclidean scalar product. We construct inductively a basis of R d which satis es the conclusion of the proposition. Set e1 (!) := kb1k?a;!1 b1: Then ke1 (!)ka;! = 1 and spanfe1 (!)g = spanfb1 g. Suppose the vectors e1 (!); : : : ; em (!) have been constructed such that spanfe1(!); : : : ; ei(!)g = spanfb1 ; : : : ; big for i = 1; : : : ; m; and hei(!); ej (!)ia;! = ij ; i; j = 1; : : : ; m: Denote by e~(!) the orthogonal projection of bm+1 into R m := spanfe1 (! ); : : : ; em (! )g = spanfb1 ; : : : ; bm g with respect to the scalar product h; ia;! . Then the vector bm+1 ? e~(!) does not vanish, because the vectors b1 ; : : : ; bm+1 are linearly independent. Set em+1 (!) := kbm+1 ? e~(!)k?a;!1 (bm+1 ? e~(!)): Then, clearly kem+1 (!)ka;! = 1: Because of the construction of em+1 (!) it is easily seen that hei(!); em+1(!)ia;! = 0; i = 1; : : : ; m: Therefore, by induction we obtain a random basis fe1(!); : : : ; ed(!)g which depends measurably on ! 2 due to the measurability of the Lyapunov scalar product. The basis fe1 (!); : : : ; ed (!)g is orthonormal with respect to the Lyapunov scalar product h; ia;! due to its construction. It remains to prove relations (10), and we are going to do that by induction. Let ! 2 be arbitrary. By the construction of e1 (!) and Theorem 3.7 we have t!0 ke1(t !) ? e1 (!)k = j kb1k?a;1t! ? kb1k?a;!1 j ?! 0: Suppose we have proved t!0 kei (t!) ? ei (!)k ?! 0; i = 1; : : : ; m: (11) By the construction of the basis fe1 (!); : : : ; ed(!)g we have

e~(!) =

m X

hbm+1 ; ej (!)ia;! ej (!);

j =1

em+1(!) = kbm+1 ? e~(!)k?a;!1 (bm+1 ? e~(!));

(12) (13)


and

e~(t !) =

m X j =1

hbm+1 ; ej (t !)ia; ! ej (t !); t

13

(14)

em+1 (t !) = kbm+1 ? e~(t !)k?a;1t! (bm+1 ? e~(t!)): (15) By virtue of (11) and Theorem 3.7 we have t!0 jhbm+1; ej (t!)ia;t! ? hbm+1 ; ej (!)ia;! j ?! 0 for all j = 1; : : : ; m: Together with (11){(15) this implies t!0 kem+1 (t !) ? em+1 (!)k ?! 0: The induction proves (10) for all i = 1; : : : ; d. De nition 3.9. The random basis constructed in Proposition 3.8 is called a non-adapted continuous Lyapunov basis of (t; !) with respect to its Lyapunov scalar product h; ia;! . Remark 3.10. The price we pay for the continuity in t of a non-adapted continuous Lyapunov basis is its non-adaptedness, i.e., in general the subspaces of R d generated by groups of basis vectors are not invariant with respect to (t; !). Therefore, using a non-adapted Lyapunov basis we can not transform (t; !) into block-diagonal form. In case (t; !) has only one Lyapunov exponent a non-adapted continuous Lyapunov basis is also a Lyapunov basis of (t; !). Now we are able to classify (non-uniformly) contracting and expanding linear cocycles.

Theorem 3.11. Every contracting linear cocycle is conjugate to the canonical contracting linear cocycle c(t; !). Every expanding linear cocycle is conjugate to the canonical expanding linear cocycle e(t; !). Proof. Let (t; !) be a contracting linear cocycle. Denote by and

its maximal and minimal Lyapunov exponent, respectively. Then < 0: Take and x a positive number a < ?. Consider a non-adapted continuous Lyapunov basis ff1 (!); : : : ; fd(!)g of (t; !) with respect to its Lyapunov scalar product h; ia;! . Denote by (t; !) the linear cocycle which has the same matrix representation in the standard Euclidean basis fe1; : : : ; edg as the matrix representation of (t; !) in the basis ff1 (!); : : : ; fd (!)g. By Remark 3.6, (t; !) and (t; !) are linearly conjugate by the linear random homeomorphism L(!) of R d de ned by L(!)ei = fi(!); i = 1; : : : ; d; ! 2 . Due to the continuity in t of the non-adapted continuous Lyapunov basis ff1(!); : : : ; fd(!)g the map L(t !) is continuous in t. Therefore, (t; !) is a

14

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(continuous in t) continuous-time linear cocycle. By Proposition 3.3, the fact noted above that the Euclidean norm of (t; !) coincides with the Lyapunov norm of (t; !) and the orthonormality of the non-adapted continuous Lyapunov basis ff1 (!); : : : ; fd(!)g with respect to the Lyapunov scalar product h; ia;! we obtain for all x 2 R d ; ! 2 ; t 2 R

et?ajtj kxk k (t; !)xk et+ajtjkxk: By the choice of a this implies that (t; !) is an uniformly contracting linear cocycle. Therefore, by Theorem 2.3, (t; !) is conjugate to c(t; !). This yields that (t; !) is conjugate to c(t; !). The case of an expanding linear cocycle can be easily treated analogously.

4. Structural stability In this section we present the rst main result of the paper | Theorem 4.7 on structural stability of a linear hyperbolic cocycle. We present here the notion of an exponential dichotomy of a linear cocycle. The main source of ideas is the work of Palmer [17] dealing with the case of deterministic dynamical systems. In our random case, we follow Gundlach [10] and Johnson [12] (see also Nguyen [14]). We notice that the MET can be reformulated for the linear cocycle in R d equipped with a random norm. In this section, unless speci ed otherwise, linear cocycles are assumed to be continuous in t and to satisfy the integrability conditions of the MET with respect to the random norm under consideration.

De nition 4.1. A cocycle (t; !) is said to have an exponential dichotomy with respect to a random norm k k! on R d (i.e., k k! is a norm on R d for each ! 2 and depends measurably on ! 2 ) if there exist positive numbers K > 0; > 0 and a family of projections P (!) of R d depending measurably on ! 2 such that i) (t; !) P (!) = P (t !) (t; !); for all t 2 R ; ! 2 ; ii) k (t; !)P (!)k!;t! K exp(?t); for all t 0; ! 2 ; iii) k (t; !)(id ? P (!))k!;t! K exp(t); for all t 0; ! 2 : Remark 4.2. The property of a cocycle to have an exponential dichotomy depends on the choice of the random norm. In the de nition of exponential dichotomy the cocycle (t; !) is not assumed to be linear and continuous in t. De nition 4.1 is applicable to any measurable cocycle on R d .


15

De nition 4.3.

We say that the linear cocycle (t; !) is structurally stable with respect to a random norm k k! on R d , if there exists a positive number " such that any linear cocycle (t; !) satisfying the inequality sup0t1 k (t; !) ? (t; !)k!;t! " for all ! 2 is conjugate to (t; !). The following proposition is a random continuous-time version of the roughness theorem for exponential dichotomies (see Palmer [17], Proposition 2.10, and Nguyen [14]).

Proposition 4.4. Suppose we are given a random norm k k! on R d and

a measurable linear cocycle (t; !) (not assumed to satisfy the integrability conditions of the MET and to be continuous in t) which has an exponential dichotomy with respect to kk! with constants K; and a family of projections P (!), and is a positive number less than . Then any measurable linear cocycle (t; !) (not assumed to satisfy the integrability conditions of the MET and to be continuous in t) satisfying the inequality

sup sup k (t; !) ? (t; !)k!;t! !2 0t1

minf(2K (1 + e?))?1(1 ? e? ); (2Ke(e? + 1))?1(e ? 1)g 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 holds

kQ(!) ? P (!)k!;! 2K 2 (1+ e?)(1 ? e? )?1 sup 0sup k (t; !) ? (t; !)k!; ! : t1 !2

t

Proof. This proposition is an easy analog of the theorem for the discrete-

time case by Gundlach [10], which in turn is basically the !-wise application of Palmer's theorem [17] (see also Wanner [20], Proposition 4.1).

Now, we prove the structural stability of linear cocycles having an exponential dichotomy with respect to a random norm.

Proposition 4.5. Suppose we are given a random norm k k! on R d which has the property that the identity map is a Lyapunov cohomology with respect to it and the standard Euclidean norm of R d and the map (x; t) 7! kxk ! is continuous for any ! 2 . Let (t; !) be a linear cocycle having an exponential dichotomy with respect to k k! with constants K 1; and a t

family of projections P (!), and be a positive number less than . Then any

16

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linear cocycle (t; !) satisfying the inequalities 2K (1 + e?)(1 ? e?)?1 sup sup k (t; !) ? (t; !)k!;t! 1; !2 0t1 ? ? 1 2Ke (e + 1)(e ? 1) sup sup k (t; !) ? (t; !)k!;t! !2 0t1 2 ? ? ? 1 2K (1 + e )(1 ? e ) sup sup k (t; !) ? (t; !)k!;t! !2 0t1

1; < 1

is conjugate to (t; !). In other words, the linear cocycle (t; !) is structurally stable with respect to k k! .

Proof. Put

c := 2K 2 (1 + e?)(1 ? e?)?1 sup sup k (t; !) ? (t; !)k!;t! < 1: !2 0t1

(16) By Proposition 4.4 the continuous-time linear cocycle (t; !) 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 holds kQ(!) ? P (!)k!;! c < 1: (17) Denote by Es (!); Eu (!) and E s (!); E u (!) the stable and unstable subspaces of (t; !) and (t; !), respectively. By the MET and the exponential dichotomy property of (t; !) and (t; !) we have the decomposition R d = Es (! ) Eu (! ) = E s (! ) E u (! ): It is easily seen that the projection P (!) is the projection onto Es (!) along Eu (!) and the projection Q(!) is the projection onto E s (!) along E u (!) for any ! from the invariant set ~ of full measure on which the MET holds for both (t; !) and (t; !). By the MET we have Es;u(t !) = (t; !)Es;u(!); E s;u(t !) = (t; !)E s;u(!); which implies that P (t!) and Q(t !) are continuous in t. Now we construct a random mapping h(!) : R d ! R d ; ! 2 : Fix an arbitrary ! 2 ~ . For each non-vanishing vector x 2 R d denote by y; z its projections onto Es (!); Eu (!) along Eu (!); Es (!), respectively. Since (t; !) exhibits an exponential dichotomy with constant K 1 the norms of its restrictions to its stable and unstable subspaces are strictly monotone, i.e., for any xed ! 2 the function t 7! k(t; !)ykt! is strictly decreasing (increasing) for any xed non-vanishing vector y from the stable (unstable) subspace of (t; !). By the arguments of Theorem 2.3, for non-vanishing y; z


17

there exist uniquely numbers u(!; y); v(!; z) which depend continuously on y; z, measurably on ! 2 such that k(u(!; y); !)yku(!;y)! = 1; k(u(!; z); !)zkv(!;z)! = 1: Since (t; !) has an exponential dichotomy with respect to k k! it is hyperbolic. Denote by > 0 the minimum of the absolute values of the Lyapunov exponents of . Take and x a positive number b < . Denote by k kb;! the Lyapunov norm of corresponding to the parameter b. By Proposition 3.3, Theorem 3.7 and the choice of b, for any non-vanishing vectors y^ 2 E s (!), z^ 2 E u (!) there exist uniquely numbers u~(!; y^), v~(!; z^) such that k (~u(!; y^); !)^ykb;u~(!;y^)! = 1; k (~v(!; z^); !)^zkb;v~(!;z^)! = 1: Set y := (u(!; y); !)y; z := (v(!; z); !)z; y~ := kQ(u(!;y) !)yk?b;1u(!;y)! ?1 (u(!; y); !)Q(u(!;y)!)y; z~ := k(id ? Q(v(!;z) !))zk?b;1v(!;z)! ?1(v(!; z); !)(id ? Q(v(!;z) !))z:

For any vector x 2 R d with non-vanishing y; z we de ne h(!)x := y~ + z~; i.e., h(!)x is the unique vector, which has projections y~ and z~ into E s (!) and E u (!) along E u (!) and E s (!). For the case y vanishes and z does not, i.e., x = z 2 Eu (!)nf0g, we set h(!)x := 0 + z~ = z~ 2 E u (!)nf0g: If z vanishes and y does not, i.e., x = y 2 Es (!)nf0g, we set h(!)x := y~ + 0 = y~ 2 E s (!)nf0g: If both y and z vanish, which is equivalent to x = 0, we set h(!)0 := 0: We show that h(!) is a random homeomorphism furnishing a topological conjugacy between (t; !) and (t; !). First of all, by the invariance of ~ with respect to (t )t2R and P( ~ ) = 1 we can restrict ourselves to considering only those ! 2 ~ . Let ! 2 ~ be arbitrary. 1. Let y1; y2 be arbitrary non-vanishing vectors from Es (!). Put y~1 := h(!)y1 and y~2 := h(!)y2. If y~1 = y~2 =: y~, then, by the de nition of h(!) it is easily seen that u~(!; y~) = u(!; y1) = u(!; y2) =: u:

18

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Furthermore, Q(u!)(u; !)y1 = Q(u!)(u; !)y2 = (u; !)~y: kQ(u!)(u; !)y1kb;u! kQ(u!)(u; !)y2kb;u! This implies ! k Q (u !)(u; !)y1kb;u! u Q( !) (u; !)y1 ? kQ(u !)(u; !)y k u (u; !)y2 = 0: 2 b; ! Taking into account (17) and the inclusions (u; !)y1; (u; !)y2 2 Es (u!) = im P (u!) we obtain (see also Nguyen [14], Lemma 3.1) u u (u; !)y1 = kQ(u!)(u; !)y1kb;u! (u; !)y2: kQ( !)(u; !)y2kb; ! Due to the equality k(u; !)y1ku! = k(u; !)y2ku! = 1 this implies (u; !)y1 = (u; !)y2: Hence y1 = y2. By the construction of h(!) we have h(!)y 6= 0 = h(!)0 for any nonvanishing y 2 Es (!). Thus, h(!)y1 6= h(!)y2 for any two dierent y1; y2 2 Es (!). 2. Analogously, for any two dierent z1 ; z2 2 Eu (!) we have h(!)z1 6= h(!)z2. Consequently, by virtue of its construction, h(!) is an injective mapping. 3. Let y~ 2 E s (!) be arbitrary non-vanishing. Put u~ := u~(!; y~), y0 := (~u; !)~y 2 E s (u~ !)nf0g. Because of (17) there exists a unique non-vanishing vector y00 2 im P (u~ !) = Es (u~ !) such that Q(u~ !)y00 = y0. Set y := ky00k?u~1! y00 and y := ?1 (~u; !)y. It is easily seen that h(!)y = y~. Analogously, for any non-vanishing z~ 2 E u (!) there exists a non-vanishing z 2 Eu (!) such that h(!)z = z~. Consequently, by virtue of its construction, h(!) is surjective. Therefore, taking into account the result of Step 2, h(!) is a bijective mapping. 4. The function x 7! h(!)x; x 2 R d , is continuous because of (17), the continuity in the second variable of u(!; y), v(!; z), u~(!; y^), v~(!; z^) and the continuity in t of (t; !), (t; !). We notice that the continuity of the mapping x 7! h(!)x at the points x 2 Eu (!)[Es (!) is due to the exponential dichotomies of (t; !) and (t; !). Therefore, by Brouwer's theorem on the invariance of domain, h(!) is a homeomorphism of R d for each ! 2 ~ . 5. By virtue of the measurability of u(!; y), v(!; z), u~(!; y^), v~(!; z^), (t; !), (t; !) and P (!), Q(!) the mapping (!; x) 7! h(!)x; ! 2 ~ ; x 2 R d , is measurable. Thus, h(!) is a random homeomorphism. Now we prove that h(!) furnishes a topological conjugacy between (t; !) and (t; !). Let ! 2 ~ , x 2 R d be arbitrary. Set y := P (!)x 2 Es (!), z := (id ? P (!))x 2 Eu (!). Suppose y and z are non-vanishing. Denote u := u(!; y) and v := v(!; z).


19

Let t 2 R be arbitrary. Denote xt := (t; !)x, yt := (t; !)y, zt := (t; !)z. Then, by the de nition of the projections P (!) we have P (t!)xt = P (t !)(t; !)x = (t; !)y = yt 2 Es (t !): (18) This implies k(u ? t; t !)ytku! = 1: (19) Analogously, zt = (id ? P (t !))xt = (t; !)z 2 Eu (t!); k(v ? t; t !)ztkv ! = 1: (20) Set y := (u; !)y; z := (v; !)z: By the construction of h(!) we have u ! ) y ; h(!)y = ?1(u; !) kQ(Qu(!) ykb;u! v ! )) z : h(!)z = ?1(v; !) k(id(id? ?Q(Q(v!)) zkb;v ! This implies u ! ) Q ( y ; ? 1 (21) (t; !) h(!)y = (t; !) (u; !) u kQ( !)ykb;u! (id ? Q(v !))z (t; !) h(!)z = (t; !) ?1(v; !) k(id ? Q(v !))zkb;v ! : (22) By virtue of (18){(20) and the cocycle property of (t; !) we have u ! ) y ; (23) h(t !)yt = h(t !)((t; !)y) = ?1(u ? t; t!) kQ(Qu(!) ykb;u! v ! )) z (24) h(t !)zt = h(t !)((t; !)z) = ?1(v ? t; t !) k(id(id? ?Q(Q(v!)) zkb;v ! : From (21){(24), the construction of h(!) and the cocycle property of (t; !) it follows that h(t !) (t; !)x = (t; !) h(!)x: (25) Analogously, (25) holds true also in case one of y and z or both of them vanish. Thus, we obtain the relation (t; !) = h?1(t !) (t; !) h(!) for all t 2 R ; ! 2 ~ ; which proves that h(!) furnishes a topological conjugacy between (t; !) and (t; !).

20

NGUYEN DINH CONG

Remark 4.6. In Proposition 4.5 the Lyapunov norm is used essentially.

It enables us to de ne the functions u~, v~, hence to construct the conjugating homeomorphism h. Any Lyapunov norm of a linear cocycle satis es the assumptions imposed on k k! in Proposition 4.5. Theorem 4.7. Every linear hyperbolic cocycle is structurally stable with respect to its Lyapunov norm k ka;! for all suciently small values of the parameter a > 0. Proof. Let (t; !) be a linear hyperbolic cocycle and 1; : : : ; p be its Lyapunov exponents. Put := minfjij j i = 1; : : : pg. Let a 2 (0; ). By Proposition 3.3, has an exponential dichotomy with respect to its Lyapunov norm kka;! with constant 1; ? a. Furthermore, by Lemma 3.4, the identity map is a Lyapunov cohomology with respect to k ka;! and the standard Euclidean norm of R d . The continuity of the map (x; t) 7! kxkt! follows from Theorem 3.7. Therefore, by Proposition 4.5, is structurally stable with respect to k ka;! . 5. Necessary conditions for topological conjugacy Let (t; !) and (t; !) be two linear hyperbolic cocycles. Theorem 5.1. If (t; !) is conjugate to (t; !), then dim Es (!) = dim E s (!); and hence dim Eu (!) = dim E u (!):

Proof. It is easily seen that if (t; !) is conjugate to (t; !) then the

discrete-time linear cocycles generated by (1; !) and (1; !) are conjugate. Therefore, by Theorem 3.3 of [15], the dimensions of the stable and unstable subspaces of those discrete-time linear cocycles, which are the same as the stable and unstable subspaces of (t; !) and (t; !), coincide. Now, in order to use some further discrete-time results of [15] on necessary conditions for topological conjugacy, we are going to make discretization of our continuous-time cocycles. Let U 2 F with P(U ) > 0, 0 < 2 R . Denote by k(!) the return time of the discrete dynamical system f( )ngn2Z with respect to U (see Cornfeld et al. [4], p. 20), i.e., k(!) := min fn 2 N j ( )n! 2 U g: n1 Corresponding to the set U we have the induced measure space (U; FU ; PU ), where FU := fA 2 F j A U g; P(A) PU (A) := for all A 2 FU : P(U )


21

The automorphism induces a measure preserving automorphism of (U; FU ; PU ), which we shall denote by U , by the following formula (see Cornfeld et al. [4], p. 20) U ! = ( )k(!) !; for ! 2 U: Set A(!) := (k(!); !); for ! 2 U: Then A(!) generates a discrete-time linear cocycle over (U ; U ) by the formula 8 n?1 > n > 0; < A((U ) ! ) : : : A(! ); n = 0; ;U (n; !) := > id; : A?1 (( )n ! ) : : : A?1 (( )?1 ! ); n < 0: U U By the above construction, it is easily seen that 8 n?1 > n > 0; < ( (k((U ) ! ) + + k(! )); ! ); n = 0; ;U (n; !) = > (0; !); : (? (k(( )n ! ) + + k(( )?1 ! )); ! ); n < 0: U U Now, repeat the above procedure with respect to (t; !): B (!) := (k(!); !); for ! 2 U; and 8 n?1 > n > 0; < B ((U ) ! ) : : : B (! ); n = 0; ;U (n; !) := > id; : B ?1 (( )n ! ) : : : B ?1 (( )?1 ! ); n < 0: U U We also have 8 n?1 > n > 0; < ( (k((U ) ! ) + + k(! )); ! ); n = 0; ;U (n; !) = > (0; !); : ( (k(( )n ! ) + + k(( )?1 ! )); ! ); n < 0: U U In what follows we shall need the notion of degrees of mappings of oriented manifolds. We recall brie y here some properties of degrees of homeomorphisms of oriented nite-dimensional Euclidean spaces and refer to Dold's book for more details and more general cases (see [7], Chapters 4, 8): Those degrees take values from the set f1; ?1g, the degree of the identity map is equal to 1, and the degree of the composition of two homeomorphisms is equal to the product of their degrees. For a mapping f we denote by deg f its degree with respect to the chosen orientations on its domain and image. Fix arbitrary measurable orientations on each the stable and unstable subspaces Es (!), Eu (!) and E s (!), E u (!) of (t; !) and (t; !), respectively. We note that if we are given an orientation on an Euclidean space R l then every basis of R l is either positively oriented or negatively oriented. Every choice of a basis of R l which is considered to be positively oriented de nes uniquely an orientation of R l . Speaking of a measurable orientation on Es (!) (or Eu (!), E s (!), E u (!), etc) we mean that we are given a basis of Es (!)

22

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which depends measurably on ! 2 and is considered to be positively oriented for every ! 2 , or, in other words, there exists a measurable basis representing that orientation. The measurability here is the measurability of basis vectors as mappings from the probability space ( ; F ; P) into the Euclidean space Rd (all the linear spaces under consideration are subspaces of R d ). Denote by As(!) and B s(!) the restrictions of A(!) and B (!) to Es (!) and E s (!), respectively. We introduce the notation s s s C;U; ; := f! 2 U j deg A (! ) deg B (! ) = ?1g; s s s i.e., C;U; ; is the set of all ! 2 U such that A (! ) and B (! ) have dierent u degrees. The set C;U;; is de ned similarly for the unstable part. A set V 2 FU is called a coboundary with respect to the discrete-time dynamical system (f(U )ngn2Z; U; FU ; PU ), or, in short, with respect to U , if there exists a set W 2 FU such that V = W 4W (mod 0), where 4 denotes the symmetric dierence of sets and mod 0 is understood with respect to PU (see also Nguyen [15]).

Proposition 5.2. If (t; !) is conjugate to (t; !), then for any > 0, s u U 2 F with P(U ) > 0, the sets C;U; ; and C;U;; are coboundaries with respect to the discrete-time dynamical system (f(U )n gn2Z; U; FU ; PU ). Proof. It is easily seen that if (t; !) is conjugate to (t; !) then for any > 0, U 2 F with P(U ) > 0, the discrete-time linear cocycle ;U (n; !) is conjugate to the discrete-time linear cocycle ;U (n; !). Therefore, Theorem 3.14 of [15] applies and gives the desired result. The following lemma, which is an immediate application of Lemma 3.15 of [15], shows that the statement of Proposition 5.2 is independent of the measurable choice of orientations on Es;u(!) and E s;u(!). s;u respect to U ) with respect Lemma 5.3. If C;U; ; are coboundaries (with s;u s;u

to one choice of measurable orientations on E (!) and E (!), then they are coboundaries with respect to any other choice of measurable orientations on Es;u (!) and E s;u(!).

Denote by s(t; !), u (t; !) and s(t; !), u(t; !) the restrictions of (t; !) and (t; !) to their stable and unstable subspaces, respectively.

Theorem 5.4. If (t; !) is conjugate to (t; !), then for any choice of measurable orientations on Es (!) and Eu (!) there exist measurable orientations on E s (!) and E u (!) and a set 0 2 F which is invariant with respect


23

to (t )t2R and has full P-measure such that with respect to those orientations for any ! 2 0 and t 2 R the following equalities hold deg s(t; !) = deg s(t; !); deg u(t; !) = deg u(t; !):

Proof. Denote by h(!) the random homeomorphism furnishing the topological conjugacy between (t; !) and (t; !), i.e., for any ! from an invariant set 1 of full P-measure (t; !) = h?1(t !) (t; !) h(!) for all t 2 R : (26) In [15] it was proved that for all ! 2 1 \ ~ , where ~ is the invariant set of full P-measure on which the MET holds for both (t; !) and (t; !), E s;u(!) = h(!)Es;u(!); (27) and the restrictions of h(!) to Es;u(!) are homeomorphisms between Es;u(!) and E s;u(!). In fact, in [15] this statement was proved for the case of discretetime linear cocycles, but the arguments for the continuous-time case are the same. Besides, one can consider the discretization of the continuous-time linear cocycles to get the above result by noting that the stable and unstable subspaces of the continuous-time linear cocycles and their discretizations are the same. Denote by hs;u(!) the restrictions of h(!) to Es;u(!), respectively. Suppose we are given measurable orientations on Es;u(!). Choose orientations on E s;u(!) such that hs;u(!) have degree 1, in other words, the orientations on E s;u(!) are generated by the orientations on Es;u(!) and the homeomorphism h(!). It is clear that the chosen orientations are measurable. By (26) the following relations hold with respect to the above chosen orientations deg s;u(t; !) = (deg hs;u(t !))?1deg s;u(t; !)deg hs;u(!) = deg s;u(t; !): 6. Some orientation properties of linear hyperbolic cocycles We shall present some properties of linear hyperbolic cocycles concerning orientation. These results are necessary for our aim of classifying linear hyperbolic cocycles. Theorem 6.7 is the key step toward that aim. In this section bases of R d or of its subspaces are always understood to be tuples of ordered linearly independent vectors so that we can always speak about orientations of bases as the orientation of the space was chosen. An l-tuple fe1 ; : : : ; el g of vectors of R d is called orthonormal if hei; ej i = ij for all i; j = 1; : : : ; l. We say that the sequence fen1 ; : : : ; enlgn2N of l-tuples of vectors !1 of R d tends to (or converges to) the l-tuple fe1; : : : ; elg if keni ? eik n?! 0; i = 1; : : : ; l. This is the standard convergence on the Stiefel manifold of orthonormal l-tuples of R d if the above tuples are orthonormal. We call that the choice of an orientation on a nite-dimensional Euclidean space is

24

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equivalent to the choice of a basis which is considered to be positively oriented. The scalar product is the standard Euclidean product of R d and the norm is the Euclidean norm. The operator norm of linear operators of R d and of linear mappings between its linear subspaces are de ned in the standard way via Euclidean norms of vectors. Now let (t; !) and (t; !) be linear cocycles. From Theorem 5.4 it follows that assuming hyperbolicity of (t; !), (t; !) and topological conjugacy between (t; !) and (t; !) one can choose according to any measurable orientations on Es;u(!) (or E s;u(!)) measurable orientations on E s;u(!) (or Es;u(!)) such that the conclusion of Theorem 5.4 holds. We show in the following lemma that without the assumption of a topological conjugacy between (t; !) and (t; !) the conclusion of Theorem 5.4 itself is symmetrical with respect to (t; !) and (t; !).

Lemma 6.1. If there exist measurable orientations on Es (!) and E s (!) and an invariant set 0 of full P-measure such that with respect to those orientations

deg s(t; !) = deg s(t; !) for all ! 2 0 ; t 2 R; (28) then for any measurable orientation on Es (!) there exists a measurable orientation on E s (!) such that with respect to these new orientations the equality (28) holds. An analogous statement holds true for the unstable subspaces Eu (!) and E u (!).

Proof. Suppose we are given measurable orientations ori1(Es (!)) and

ori1(E s (!)) such that (28) holds. Let ori2 (Es (!)) be an arbitrary measurable orientation on Es (!). Set ( s if ori2(Es (!)) = ori1(Es (!)); 1 (E (! )) ori2 (E s (!)) := ori s ?ori1 (E (!)) if ori2(Es (!)) = ?ori1(Es (!)): Then ori2(E s (!)) is a new measurable orientation on E s (!). By this construction (28) holds with respect to ori2(Es (!)) and ori2 (E s (!)). The case of unstable subspaces is analogous.

Lemma 6.2. If (t; !) is hyperbolic, then for any measurable orientation on Es (!) there exists a measurable orientation on Eu (!) such that deg u(t; !) = deg s(t; !) for all t 2 R ; ! 2 ~ ; where ~ denotes the invariant set of full P-measure on which the MET holds for (t; !). The analogous statement with Es (!) and Eu (!) changing their roles also holds true.


25

Proof. Suppose we are given a measurable orientation on Es (!). Take

the measurable orientation on Eu (!) which is compatible with the given orientation on Es (!) and the standard xed orientation on R d , i.e., the basis of R d consisting of the vectors of a positively oriented basis of Es (! ) and of a positively oriented basis of Eu (!) (in this order) is positively oriented in R d with the given standard orientation. By this choice of orientations and the direct decomposition of (t; !) into its stable and unstable components we have deg (t; !) = deg s(t; !) deg u (t; !): Due to the continuity in t of (t; !) and the equality (0; !) = id we have deg (t; !) = deg (0; !) = 1: Therefore, deg u (t; !) = deg s (t; !): The second part of the lemma can be treated analogously. For our later construction we shall de ne here a process of orthonormalization of l-tuples of linearly independent vectors of Rd (l = 1; : : : ; d) and study its properties. Let fe1; : : : ; elg be an l-tuple of ordered linearly independent vectors of Rd . We denote by O(e1 ; : : : ; el ) := ff1; : : : ; fl g the l-tuple of ordered linearly independent vectors of R d obtained from fe1; : : : ; el g by the following process of orthonormalization: f1 := kee1 k ; 1 Pm?1 e m ? i=1 hem ; fi ifi fm := ke ? Pm?1 he ; f if k for m = 2; : : : ; l: m i=1 m i i The l-tuple ff1; : : : ; fl g = O(e1 ; : : : ; el ) has the following two properties: 1. hfi; fj i = ij for i; j = 1; : : : ; l; 2. spanff1; : : : ; fmg = spanfe1; : : : ; em g for m = 1; : : : ; l. Lemma 6.3. Let R l be an oriented subspace of R d and fe1; : : : ; el g be a basis of R l . Then O(e1 ; : : : ; el ) is an orthonormal basis of R l and has the same orientation as fe1 ; : : : ; el g.

Proof. Denote by ff1 ; : : : ; fl g the l-tuple O(e1 ; : : : ; el). From the definition of O(e1 ; : : : ; el ) it follows that the subspace spanned by f1 ; : : : ; fl coincides with the subspace spanned by e1 ; : : : ; el which is Rl . Therefore, ff1; : : : ; fl g is a basis of R l . Its orthonormality is due to its construction. Choose and x a positively oriented orthonormal basis of R l and denote it by fg1; : : : ; gl g. Denote by U1 and U2 the linear operators of R l which map gi into ei and fi , i = 1; : : : ; l, respectively. It is easily seen that the orientations

26

NGUYEN DINH CONG

of fe1 ; : : : ; el g and ff1; : : : ; flg are de ned by the signs of det U1 and det U2 , respectively. From the construction of f1 ; : : : ; fl it follows that l Y

!?1

mX ?1

kem ? hem; fiifik det U1; m=1 i=1 Pm?1 where for m = 1 the sum i=1 hem; fiifi is not present. Therefore, det U1 and det U2 have the same sign, which implies that fe1; : : : ; el g and ff1; : : : ; fl g det U2 =

have the same orientation. The following lemma shows that O(e1 ; : : : ; el) depends continuously on fe1 ; : : : ; el g.

Lemma 6.4. Let fej1; : : : ; ejl g, j 2 N , be a sequence of l-tuples of ordered !1 linearly independent vectors of R d such that kejm ? em k j?! 0; m = 1; : : : ; l, where fe1 ; : : : ; el g is an l-tuple of ordered linearly independent vectors of R d . Then the sequence O(ej1 ; : : : ; ejl ), j 2 N , tends to the l-tuple O(e1 ; : : : ; el ) as j tends to 1. Proof. We prove the lemma by induction, using arguments similar to

those in Proposition 3.8. Denote ff1j ; : : : ; flj g := O(ej1 ; : : : ; ejl ), j 2 N , and ff1; : : : ; flg := O(e1 ; : : : ; el). By de nition j f1j = e1j ; ke1k f1 = kee1 k : 1 !1 !1 Due to the assumption we have ej1 j?! e1 6= 0, which implies f1j j?! f1 . Suppose we have proved !1 fnj j?! fn for n = 1; : : : ; m: Set m X j j ~ fm+1 := em+1 ? hejm+1 ; fij ifij ;

f~m+1 := em+1 ?

i=1 m X i=1

hem+1 ; fiifi:

By the assumptions of the lemma and of the induction we have !1 ~ f~mj +1 j?! fm+1 6= 0: Therefore, ~mj +1 j!1 f~m+1 f j fm+1 = ~j ?! ~ =f : kfm+1 k kfm+1 k m+1


27

Now, we present some orientation properties of orthogonal projections of and of linear cocycles, which are needed for the proof of the main result of this section | Theorem 6.7. Let E and F be two linear subspaces of R d . We denote by PE the orthogonal projection onto E in R d , and by PEjF its restriction to F considered as a linear mapping from F to E .

Rd

Lemma 6.5. Let E and F be two linear l-dimensional oriented subspaces

of R d such that

kPE ? PF k < 1:

Then PEjF is a linear homeomorphism from F to E and PF jE is a linear homeomorphism from E to F . Furthermore, the following equality holds deg PEjF = deg PF jE :

Proof.

By Lemma 3.1 of [14] PEjF and PF jE are linear homeomorphisms between E and F . Further, let ff1 ; : : : ; fl g be a basis of F , then fPE f1 ; : : : ; PE fl g is a basis of E . Denote gi := fi ? PE fi 2 E ?; for i = 1; : : : ; l; where E ? denotes the orthogonal complement of E in R d . Set fi(t) := PE fi + tgi = tfi + (1 ? t)PE fi; for t 2 [0; 1]; i = 1; : : : ; l: Then ff1(t); : : : ; fl (t)g are linearly independent, because fPE f1 ; : : : ; PE fl g is a basis of E and gi 2 E ?; i = 1; : : : ; l. Set Mt := spanff1 (t); : : : ; fl (t)g; for t 2 [0; 1]: Clearly, M0 = E; M1 = F . By virtue of the construction of vectors fi (t); i = 1; : : : ; l, the spaces Mt ; t 2 [0; 1], contain no vectors from E ?. Therefore, PMt R d \ (id ? PE )R d = (id ? PMt )R d \ PE R d = f0g, because dim PMt R d = dim PE R d ; t 2 [0; 1]. Consequently, for any non-vanishing x 2 R d and t 2 [0; 1] k(PMt ? PE )xk2 = k(id ? PE )PMt x ? PE (id ? PMt )xk2 = k(id ? PE )PMt xk2 + kPE (id ? PMt )xk2 < kPMt xk2 + k(id ? PMt )xk2 = kxk2: Thus, kPMt ? PE k < 1 for any t 2 [0; 1]. Therefore, PEjMt PMtjE , t 2 [0; 1], is a continuous family of homeomorphisms of E furnishing a homotopy from idE to PEjF PF jE . Consequently, by the homotopic invariance of the degree we have deg (PEjF PF jE ) = deg idE = 1. This implies deg PEjF = deg PF jE .

28

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We denote by k the dimension of the stable subspace of (t; !). Consider the Grassmann manifold Gr(k; d) of all k-dimensional linear subspaces of R d and de ne a metric on Gr(k; d) by the formula (E; F ) := kPE ? PF k: The space Gr(k; d) is a complete compact metric space. The map (t; !) 7! Es (t!) takes values in Gr(k; d), is continuous in t and measurable in !. Fix an arbitrary F 2 Gr(k; d). Take and x an orientation on F . Let 0 < a < 1 be arbitrary. Denote Da := f! 2 j (Es (!); F ) < ag which is the inverse image in of the open ball fE 2 Gr(k; d) j (E; F ) < ag in Gr(k; d) with center in F and radius a with respect to the measurable mapping ! 7! Es (!). Set D~ a := f! 2 j (Es (!); F ) ag:

Lemma 6.6. Let 0 < a < 1 be arbitrary. Assume that for an ! 2 we have ! 2 D~ a for all 2 [0; t] and we are given an orientation on Es (!). Then the equality deg s(t; !) = 1 is equivalent to the equality deg PEs (!)jF = deg PEs (t!)jF : Proof. Consider the map

h(; !) := PF jEs ( !) s(; !) PEs (!)jF : By the assumption of the lemma, h(; !) is a family of linear homeomorphisms of F depending continuously on 2 [0; t]. Therefore, since the degree is a homotopic invariant, we have deg h(t; !) = deg h(0; !): By Lemma 6.5 and the equality (0; !) = id, we have deg h(0; !) = deg PF jEs (!) deg PEs (!)jF = 1; deg h(t; !) = deg PEs (t!)jF deg s (t; !) deg PEs (!)jF : Therefore, deg s(t; !) = deg PEs (t!)jF deg PEs (!)jF ; which is equivalent to the conclusion of the lemma, because degrees take values from the set f1; ?1g.


29

We remark that the analogous result for the unstable subspace of (t; !) also holds true. Next we formulate and prove the main result of this section.

Theorem 6.7. Suppose we are given two linear hyperbolic cocycles (t; !) and (t; !) with dim Es (!) = dim E s (!). If there exist measurable orientations on Es (!), E s (!) and an invariant set 0 2 F of full P-measure such that

deg s(t; !) = deg s(t; !) for all ! 2 0 ; t 2 R; then there exist a random linear isometric mapping U (!) from Es (!) to E s (!), i.e., a linear homeomorphism depending measurably on ! 2 and preserving Euclidean norms of vectors, and an invariant set ~ 0 0 of full P-measure such that lim kU (t !) PEs (t!) ? U (!) PEs (!) k = 0; for all ! 2 ~ 0 \ ~ ; t!0 (29) where ~ is the invariant set of full measure on which the MET holds for both (t; !) and (t; !), and U (t !) PEs (t !), U (!) PEs (!) are considered as linear operators of R d .

Proof. Since 0 \ ~ is invariant and of full P-measure we can restrict

ourselves to considering only those ! from it. Therefore, for simplicity of notations we assume that 0 = ~ = . Denote k := dim Es (!) = dim E s (!). The theorem holds trivially in case k = 0 or k = d. Let 1 k < d. We introduce in Gr(k; d) Gr(k; d) the metric ~((E1 ; E2); (F1; F2 )) := maxf(E1 ; F1); (E2 ; F2)g: Then Gr(k; d) Gr(k; d) is a complete compact metric space. The mapping ! 7! (Es (!); E s (!)) is measurable by the MET. Take and x a point (F1 ; F2) 2 Gr(k; d) Gr(k; d) such that the set G := f! 2 j (Es (!); F1) < 21 ; (E s (!); F2) < 21 g; which is the inverse image of the open ball V := f(E1 ; E2) 2 Gr(k; d) Gr(k; d) j ~((E1; E2 ); (F1; F2)) < 21 g in Gr(k; d) Gr(k; d) with center in (F1; F2 ) and radius 12 , has positive Pmeasure. Choose and x orthonormal bases ff11; : : : ; fk1g in F1 and ff12; : : : ; fk2g in F2. Choose and x the orientations on F1 and F2 such that the chosen bases

30

NGUYEN DINH CONG

are positively oriented. Set G~ := f! 2 j (Es (!); F1) 21 ; (E s (!); F2) 21 g; G^ := f! 2 j (Es (!); F1) 32 ; (E s (!); F2) 32 g: By the assumption of the theorem there exist measurable orientations on Es (!) and E s (!) such that deg s(t; !) = deg s(t; !) for all ! 2 ; t 2 R : (30) By Lemma 6.1 we can assume that the orientation on E s (!) has the property that ^ deg PF2jE s (!) = 1; for all ! 2 G: (31) For ! 2 nG we set t1 (!) := supft < 0 j t ! 2 Gg = supft < 0 j (Es (t !); E s (t !)) 2 V g; t2 (!) := inf ft > 0 j t ! 2 Gg = inf ft > 0 j (Es (t !); E s (t !)) 2 V g; and for ! 2 G t3 (!) := supft < 0 j t ! 2= Gg = supft < 0 j (Es (t !); E s (t !)) 2= V g; t4 (!) := inf ft > 0 j t ! 2= Gg = inf ft > 0 j (Es (t !); E s (t !)) 2= V g: Since the map (t; !) 7! (Es (t !); E s (t !)) is a ow on Gr(k; d) Gr(k; d), which is continuous in t for each xed ! 2 and measurable in ! for each xed t 2 R due to the MET and the continuity of (; !) and (; !), or, in other words, a continuous Markov process on the metric space (Gr(k; d) Gr(k; d); ~), the functions t1(!), t2(!), t3 (!), t4 (!) are measurable (see Dynkin [8], Chapter 4). Set S1 := f! 2 nG j t1 (!) = ?1g; S10 := f! 2 j 91 (!); 2(!) 2 R such that 1 (!) ! 2 S1; 2 (!) ! 2= S1 g: S10 is the set of all ! 2 such that ft !gt2R belongs to S1 for large negative t and sometime exits S1 . Denote S1;n := f! 2 S1 j t2 (!) < ng; n 2 N ; 1 [ S1;n: S~1 := n=1

Then for any xed n 2 N the sets f?k(n+1)S1;ngk2N are pairwise disjoint, because they are characterized by t2 (!) 2 [k(n + 1); k(n + 1) + n); k 2 N . Furthermore, those sets have the same P-measure due to the de nition of the

ow (t )t2R . Therefore, all they have null P-measure. This implies P(S~1 ) = 0. It is easily seen that 1 [ S10 = S~1 m S1;2: m=1


Consequently,

31

P(S10 ) = 0:

By virtue of their de nition the sets S1nS10 ; S1 [ S10 ; S10 are invariant with respect to the ow (t )t2R . The sets S2; S20 ; S3; S30 ; S4; S40 are de ned similarly. Set 4 [

^ := n (Si [ Si0): i=1

Then ^ is invariant with respect to the ow (t)t2R . Hence, by the ergodicity of (t)t2R , there are two possibilities, namely P( ^ ) = 1 or P( ^ ) = 0. We remark that for any ! 2 ^ the corresponding two functions ti(!) (i = 1; 2 or i = 3; 4) are nite. I) The case P( ^ ) = 1. For simplicity of notations we assume that ^ = , i.e., for all ! 2 the above functions are nite. Since the set V is open in Gr(k; d) Gr(k; d) and the ow (t; !) 7! (Es (t !); E s (t !)) is continuous in t we have t3 (!) < 0 < t4 (!); for all ! 2 G; (32) and t1 (!) !; t2 (!) ! 2 G~ for ! 2 nG; t3 (!) !; t4 (!) ! 2 G~ for ! 2 G: Now, we shall construct, in Steps 1{2, a random isometric linear mapping U (!) from Es (!) to E s (!) and prove, in Steps 3{8, that it satis es the conclusion of the theorem. 1. Let ! 2 nG be arbitrary. We set !1 := t1 (!) !; e1i (!1) := PEs (!1 ) fi1 for i = 1; : : : ; k; e2i (!1) := PE s (!1 ) fi2 for i = 1; : : : ; k: Then, by Lemma 3.1 of [14], fe11 (!1); : : : ; e1k (!1)g is a basis of Es (!1) and fe21(!1 ); : : : ; e2k (!1 )g is a basis of E s (!1), because kPEs (!1 ) ? PF1 k 21 ; kPE s (!1 ) ? PF2 k 21 due to the relation !1 2 G~ . Denote r1 := deg PEs (!1 )jF1 2 f1; ?1g;

32

NGUYEN DINH CONG

and Set

e31 (!1) := r1e11 (!1); e3i (!1) := e1i (!1) for i = 2; : : : ; k:

e3i (!) := (?t1 (!); !1)e3i (!1) for i = 1; : : : ; k; e2i (!) := (?t1 (!); !1)e2i (!1) for i = 1; : : : ; k: Denote by fe41(!); : : : ; e4k (!)g, and fe51(!); : : : ; e5k (!)g the k-tuples O(e31 (!); : : : ; e3k (!)) and O(e21 (!); : : : ; e2k (!)), respectively. Set U (!)e4i (!) := e5i (!) for i = 1; : : : ; k; and extend U (!) to the whole space Es (!) by linearity. Then U (!) is an isometric mapping from Es (!) to E s (!), because the bases fe41 (!); : : : ; e4k (!)g and fe51 (!); : : : ; e5k (!)g are orthonormal in Es (!) and E s (!), respectively, due to Lemma 6.3. By the above construction, Lemma 6.5, the choice of orientations on F1; F2 and (30){(31) we have deg U (!) = (deg (?t1 (!); !1))?1 deg (?t1 (!); !1) = 1: (33) 2. Now, let ! 2 G be arbitrary. Set !3 := t3 (!) !; !4 := t4 (!) !: We note that, by virtue of openness of V , continuity of (; !), and their de nition !3 and !4 belong to G~ nG. Therefore, by Step 1 the orthonormal bases fe41 (!3); : : : ; e4k (!3)g of Es (!3) and fe51 (!3); : : : ; e5k (!3)g of E s (!3) are de ned. Further, the orthonormal bases fe41(!4); : : : ; e4k (!4)g of Es (!4) and fe51(!4 ); : : : ; e5k (!4 )g of E s (!4) are also de ned and they are O(deg PEs (!4 )jF1 PEs (!4 ) f11; PEs (!4) f21; : : : ; PEs (!4 ) fk1) and O(PE s (!4 ) f12; : : : ; PE s (!4) fk2), respectively, because t1 (!4) = 0. We note that by the de nition of t3 (!); t4(!) the following relation holds !3 2 G~ for all 2 [0; t4(!) ? t3(!)]: Set r2 := 1 if the basis fe41 (!3); : : : ; e4k (!3)g of Es (!3) is positively oriented and r2 := ?1 otherwise. Denote g11(!3) := r2 e41(!3); gi1(!3) := e4i (!3) for i = 2; : : : ; k; g12(!3) := r2 e51(!3); gi2(!3) := e5i (!3) for i = 2; : : : ; k:


33

By (33) and their construction fg11(!3); : : : ; gk1(!3)g is a positively oriented basis of Es (!3) and fg12(!3); : : : ; gk2(!3)g is a positively oriented basis of E s (!3). They are orthonormal, because fe41(!3 ); : : : ; e4k (!3)g and fe51(!3); : : : ; e5k (!3)g are orthonormal. Denote fg11(!); : : : ; gk1(!)g := O(s(?t3 (!); !3)g11(!3); : : : ; s(?t3 (!); !3)gk1(!3)); fg12(!); : : : ; gk2(!)g := O( s(?t3 (!); !3)g12(!3); : : : ; s(?t3 (!); !3)gk2(!3)): Then fg11(!); : : : ; gk1(!)g and fg12(!); : : : ; gk2(!)g are orthonormal positively oriented bases of Es (!) and E s (!), respectively. By virtue of the equality t1(!4 ) = 0 and the construction in Step 1, the bases fe41 (!4); : : : ; e4k (!4)g of Es (!4) and fe51 (!4); : : : ; e5k (!4)g of E s (!4) are positively oriented and orthonormal. Denote fg13(!); : : : ; gk3(!)g := O(s(?t4 (!); !4)e41 (!4); : : : ; s(?t4 (!); !4)e4k (!4)); fg14(!); : : : ; gk4(!)g := O( s(?t4 (!); !4)e51(!4); : : : ; s(?t4 (!); !4)e5k (!4)): Then fg13(!); : : : ; gk3(!)g and fg14(!); : : : ; gk4(!)g are orthonormal positively oriented bases of Es (!) and E s (!), respectively. Denote by U1(!) the linear orthogonal operator of Es (!) which maps the basis fg11(!); : : : ; gk1(!)g into the basis fg13(!); : : : ; gk3(!)g and by U2(!) the linear orthogonal operator of E s (!) which maps the basis fg12(!); : : : ; gk2(!)g into the basis fg14(!); : : : ; gk4(!)g, i.e., U1 (!)gi1(!) = gi3(!) and U2(!)gi2(!) = gi4(!) for i = 1; : : : ; k. Since the above bases are positively oriented, we have det U1 (!) = det U2 (!) = 1: By the theorem on the canonical form of an orthogonal matrix (see Gantmacher [9], p. 285) any orthogonal (k k)-matrix R with det R = 1 is orthogonally similar to a canonical orthogonal matrix ! ! ( cos ' sin ' cos ' sin ' R = R1 diag ? sin '1 cos '1 ; ? sin '2 cos '2 ; : : : 2 2 1 1 ! ) cos ' sin ' q q : : : ; ? sin ' cos ' ; 1; : : : ; 1 R1?1; q q where R1 2 O(k; R ), '1 ; : : : ; 'q 2 [0; 2), and the numbers cos '1 i sin '1, : : : , cos 'q i sin 'q , 1; : : : ; 1 are the eigenvalues of the matrix R, and so, they depend continuously on the matrix R. A theorem on the structure of the transforming matrix (see Gantmacher [9], p. 148, supplement to Theorem 7) assures us the choice of the transforming matrix R1 in continuous dependence of the matrix R (R1 = P (R), where P () is a polynomial). Therefore, we can assume that the transforming matrix R1 depends continuously on R. We apply this result to operators U1 (!) and U2(!). We shall identify the linear operators of Es (!) and of E s (!) with their matrix representations in the orthonormal bases fg13(!); : : : ; gk3(!)g of Es (!) and fg14(!); : : : ; gk4(!)g of

34

NGUYEN DINH CONG

E s (!), respectively, and use the same notation for the linear operators and for their matrix representations. Since det U1(!) = det U2 (!) = 1, we have ( 1 (! ) sin '1 (! ) ! 1 (! ) sin '1 (! ) ! cos ' cos ' U1 (!) = R1(!)diag ? sin '1 1(!) cos '11 (!) ; ? sin '2 1(!) cos '21 (!) ; 1 1 2 2 ! ) 1 1 'q (!) sin 'q (!) ; 1; : : : ; 1 R?1 (!); : : : ; ?cos 1 sin '1q (!) cos '1q (!) and ( 2 (! ) sin '2 (! ) ! 2 (! ) sin '2 (! ) ! cos ' cos ' 1 1 U2 (!) = R2(!)diag ? sin '2(!) cos '2 (!) ; ? sin '2 2(!) cos '22 (!) ; 2 2 1 1 ! ) 2 2 'q^(!) sin 'q^(!) ; 1; : : : ; 1 R?1 (!); : : : ; ?cos 2 sin '2q^(!) cos '2q^(!) where cos '11 i sin '11 ; : : : ; cos '1q i sin '1q , 1; : : : ; 1 are the eigenvalues of U1 (!), so they depend continuously on U1 (!), and cos '21i sin '21; : : : ; cos '2q^ i sin '2q^, 1; : : : ; 1 are the eigenvalues of U2(!), so they depend continuously on U2 (!). The orthogonal matrix R1 (!) is de ned by U1 (!) and depends continuously on U1(!), and the orthogonal matrix R2 (!) is de ned by U2(!) and depends continuously on U2(!). Set 3 (! ) t(!) := t (!?)t? t3 ((!) > 0; 4 1 (! ) sin t(! )'1 (! ) ! cos t ( ! ) ' U3 (!) := R1 (!)diag ? sin t(!)'1 1(!) cos t(!)'11(!) ; 1 1 ! 1 1 cos t(!)'2(!) sin t(!)'2(!) ; : : : ? sin t(!)'12(!) cos t(!)'12(!) ) 1 (! ) sin t(! )'1(! ) ! cos t ( ! ) ' q q : : : ; ? sin t(!)'1(!) cos t(!)'1(!) ; 1; : : : ; 1 R1?1(!); q q ! ( 2 2 cos t ( ! ) ' ( ! ) sin t ( ! ) ' ( ! ) 1 1 U4 (!) := R2 (!)diag ? sin t(!)'2(!) cos t(!)'2(!) ; 1 1 ! cos t(!)'22(!) sin t(!)'22(!) ; : : : ? sin t(!)'22(!) cos t(!)'22(!) ) 2 (! ) sin t(! )'2(! ) ! cos t ( ! ) ' q ^ q ^ : : : ; ? sin t(!)'2(!) cos t(!)'2(!) ; 1; : : : ; 1 R2?1(!): q^ q^ Then U3(!) and U4 (!) are orthogonal operators of Es (!) and of E s (!), respectively. Denote by U5 (!) the linear isometric mapping from Es (!) to E s (!), which maps the orthonormal basis fg11(!); : : : ; gk1(!)g of Es (!) into


35

the orthonormal basis fg12(!); : : : ; gk2(!)g of E s (!). Set U (!) := U4 (!) U5 (!) U3?1 (!): (34) Then U (!) is an isometric mapping from Es (!) to E s (!). 3. We have constructed in Steps 1 and 2 an isometric mapping U (!) from s E(!) to E s (!) for all ! 2 . Its measurability with respect to ! 2

follows from its construction: U (!) depends measurably on ! on each of the measurable sets G and nG, every step of the construction of U (!) respects its measurability in !. It remains to prove (29) what we shall do by considering 5 possible cases. 4. The case ! 2 G. By (32), for t 2 (t3(!); t4(!)), t ! 2 G. Let t 2 (t3(!); t4(!)) be arbitrary. It is easily seen that t3 (t !) = t3(!) ? t; (35) t4 (t !) = t4(!) ? t; and the points !3; !4 for the construction of U (!) in Step 2 coincide with the corresponding points for the construction of U (t !). We have U (t !) = U4 (t !) U5 (t!) U3?1 (t!): (36) By (35), the continuity in t of (t; !), (t; !), and Lemma 6.4 the orthonormal k-tuples fg1i (t !); : : : ; gki (t!)g tend to the orthonormal k-tuples fg1i (!); : : : ; gki (!)g, i = 1; 2; 3; 4, respectively, as t tends to 0. Therefore, taking into account (35) and the construction of U3 (t !), U4 (t !) the k-tuples fU3(t !)g11(t !); : : : ; U3(t !)gk1(t!)g, fU4 (t !)g12(t!); : : : ; U4(t !)gk2(t !)g are orthonormal and tend to the orthonormal k-tuples fU3(!)g11(!); : : : ; U3(!)gk1(!)g, fU4 (!)g12(!); : : : ; U4 (!)gk2(!)g, respectively. By (34) and (36), U (t !) and U (!) are the linear isometric mappings from Es (t !) to E s (t!) and from Es (!) to E s (!) which map the orthonormal basis fU3(t !)g11(t !); : : : ; U3 (t !)gk1(t!)g of Es (t !) to the orthonormal basis fU4(t !)g12(t!); : : : ; U4 (t!)gk2(t !)g of E s (t !) and the orthonormal basis fU3(!)g11(!); : : : ; U3(!)gk1(!)g of Es (!) to the orthonormal basis fU4(!)g12(!); : : : ; U4(!)gk2(!)g of E s (!), respectively. From the above proved convergence of k-tuples is follows that lim kU (t !) PEs (t!) ? U (!) PEs (!) k = 0: t!0 5. The case ! 2 nG with t1 (!) < 0 < t2 (!). For any t 2 [t1(!); t2(!)] we have t ! 2 nG. Let t 2 [t1(!); t2(!)] be arbitrary, we have t1 (t!) = t1 (!) ? t; t2 (t!) = t2 (!) ? t:

36

NGUYEN DINH CONG

Using the arguments analogous to those of Step 4 we have that the k-tuples fe41(t !); : : : ; e4k (t !)g and fe51(t !); : : : ; e5k (t !)g are orthonormal and they tend to the orthonormal k-tuples fe41 (!); : : : ; e4k (!)g and fe51 (!); : : : ; e5k (!)g, respectively. By de nition U (t !) and U (!) are linear isometric mappings which map fe41 (t !); : : : ; e4k (t !)g to fe51(t !); : : : ; e5k (t !)g and fe41(!); : : : ; e4k (!)g to fe51 (!); : : : ; e5k (!)g, respectively. This implies lim kU (t !) PEs (t!) ? U (!) PEs (!) k = 0: t!0 6. The case ! 2 nG with t1 (!) = 0 < t2 (!). In this case U (!) is the linear isometric mapping from Es (!) to E s (!) which maps the orthonormal basis fe41 (!); : : : ; e4k (!)g of Es (!) into the orthonormal basis fe51(!); : : : ; e5k (!)g of E s (!). By virtue of the inequality t2 (!) > 0 the arguments of Step 5 assure us the right continuity of U (t !) PEs (t !) at t = 0: lim kU (t !) PEs (t!) ? U (!) PEs (!) k = 0: t!+0 It remains to prove its left continuity. There are two possibilities. a) There exists a t0 < 0 such that for all t 2 [t0 ; 0) t! 2 G. Let t 2 (t0; 0) be arbitrary. We have t4(t !) = ?t; t3(t !) t0 ? t: Therefore, by the construction of Step 2, the orthonormal k-tuples fU3(t !)g11(t !); : : : ; U3(t !)gk1(t!)g, fU4 (t !)g12(t!); : : : ; U4(t !)gk2(t !)g tend to the orthonormal k-tuples fe41 (!); : : : ; e4k (!)g, fe51(!); : : : ; e5k (!)g, respectively, as t tends to ?0. (In fact, U3(!) and U4 (!) in Step 2 were constructed to assure this property.) This implies lim kU (t !) PEs (t!) ? U (!) PEs (!) k = 0: t!?0 b) For any 0 < 0 there exists 0 < < 0 such that ! 2 nG. Because t1 (!) = 0 for any 1 < 0 there exists 1 < ~ < 0 such that ~! 2 G. Take ?2 small enough to assure that (t; !) and (t; !) are close to the identity mapping for 2 t 0 and 2 ! 2 G, and t ! 2 G^ for all 2 t 0. Let 2 < t < 0 and !~ := t ! 2 nG. Then t1(~!) 2 ? t > 2 : Set ! := t1 (~!)!~ : Taking into account the continuity of (; !), (; !), Lemma 6.4 and the choice of 2 , by the construction in Step 1, the orthonormal k-tuples fe41(~!); : : : ; e4k (~!)g and fe51(~!); : : : ; e5k (~!)g are close to the orthonormal ktuples fe41 (!); : : : ; e4k (!)g and fe51 (!); : : : ; e5k (!)g, respectively. The orthonormal k-tuples fe41(!); : : : ; e4k (!)g and fe51(!); : : : ; e5k (!)g coincide with


37

O(deg PE(!)jF1 PE(!) f11; PE(!)f21; : : : ; PE(!) fk1) and O(PE (!)f12; : : : ; PE (!) fk2), s

s

s

s

s

s

respectively, because t1 (!) = 0. We note that since t1(!) = 0 the bases fe41(!); : : : ; e4k (!)g and fe51(!); : : : ; e5k (!)g coincide with 2 2 1 1 O(deg PEs (!)jF1 PEs (!) f1 ; : : : ; PEs (!) fk ) and O(PE s (!) f1 ; : : : ; PE s (!) fk ), respectively. By the choice of 2 , (31) and Lemma 6.6, we have deg s(t + t1 (~!); !) = 1: By (30), this implies deg s(t + t1 (~!); !) = 1: Hence, by the choice of 2 and Lemma 6.6, we have deg PEs (!)jF1 = deg PEs (!)jF1 : Therefore, by the choice of 2 and Lemma 6.4, the orthonormal k-tuples fe41(!); : : : ; e4k (!)g and fe51(!); : : : ; e5k (!)g are close to the orthonormal ktuples fe41 (!); : : : ; e4k (!)g and fe51(!); : : : ; e5k (!)g, respectively. Consequently, U (~!) PEs (~!) is close to U (!) PEs (!) . Now, take 2 < 3 < 0 such that 3 ! 2 nG. Suppose 3 < < 0 and !^ := ! 2 G. Then t3 (^!) 3 ? ; t4 (^!) ?: Set !^ 1 := t3 (^!)!^ ; !^ 2 := t4 (^!)!^ : By the just proved result on closeness of fe41(!); : : : ; e4k (!)g, fe51 (!); : : : ; e5k (!)g to fe41(!); : : : ; e4k (!)g, fe51(!); : : : ; e5k (!)g, and the choice of 3 the orthonormal k-tuples fe41 (^!1); : : : ; e4k (^!1)g and fe51 (^!1); : : : ; e5k (^!1)g are close to the orthonormal k-tuples fe41(^!2 ); : : : ; e4k (^!2 )g and fe51(^!2); : : : ; e5k (^!2)g. Therefore, by the construction of Step 2, the linear orthogonal operators U1 (^!) and U2 (^!) are close to the identity. This implies that U3 (^!) and U4 (^!) are close to the identity, too. Consequently, the orthonormal k-tuples fU3(^!)g11(^!); : : : ; U3(^!)gk1(^!)g and fU4 (^!)g12(^!); : : : ; U4(^!)gk2(^!)g are close to the orthonormal k-tuples fg11(^!); : : : ; gk1(^!)g and fg12(^!); : : : ; gk2(^!)g, respectively. These k-tuples in turn are close to the orthonormal k-tuples fg11(^!1); : : : ; gk1(^!1 )g and fg12(^!1); : : : ; gk2(^!1)g, respectively, due to the continuity of (t; !), (t; !) and the choice of 3. We notice that the linear mapping from Es (^!1) to E s (^!1) which maps the orthonormal basis fg11(^!1); : : : ; gk1(^!1 )g of Es (^!1 ) to the orthonormal basis fg12(^!1); : : : ; gk2(^!1)g of E s (^!1) coincides with U (^!1), because by the de nition of the bases fg11(^!1); : : : ; gk1(^!1 )g and fg12(^!1); : : : ; gk2(^!1)g, since U (^!1) maps the basis fe41(^!1 ); : : : ; e4k (^!1 )g into the basis fe51 (^!1); : : : ; e5k (^!1)g, U (^!1 ) maps fg11(^!1); : : : ; gk1(^!1 )g into fg12(^!1); : : : ; gk2(^!1)g. Therefore, U (^!) PEs (^!) is

38

NGUYEN DINH CONG

close to U (^!1 ) PEs (^!1 ) . By the choice of 2 ; 3 , U (^!1 ) PEs (^!1 ) is close to U (!) PEs (!) . Consequently, U (^!) PEs (^!) is close to U (!) PEs (!) . All the above arguments can be easily made explicit to get quantitative estimations for the values 2 ; 3 which are needed to assure closeness of U (t !) PEs (t!) to U (!) PEs (!) for all 3 < t < 0. Therefore, t 7! U (t !) PEs (t!) is left continuous at 0. 7. The case ! 2 nG with t1 (!) < 0 = t2 (!). This case is analogous to the case treated in Step 6. The main diculty in this case is the proof of the right continuity of U (t !) PEs (t!) at t = 0, which can be tackled by using the arguments analogous to those of Step 6b. 8. The case ! 2 nG with t1 (!) = 0 = t2 (!). By the result of Step 6 U (t !) PEs (t!) is left continuous at t = 0, and it is right continuous at t = 0 by Step 7. Therefore, it is continuous at t = 0. II) The case P( ^ ) = 0. By the de nition of ^ we have P

4 [

i=1

(Si

[ S 0) i

!

= 1:

It is easily seen that S1 nS10 = S2 nS20 nG is the set of all ! 2 nG with t1 (!) = ?1; t2 (!) = +1, and S3 nS30 = S4nS40 G is the set of all ! 2 G with t3 (!) = ?1; t4 (!) = +1. Since Si0 are invariant null sets, Si nSi0 are invariant, i = 1; : : : ; 4, and P(G) > 0 we have P(S3 nS30 ) = 1. So, the set

0 := S3nS30 which is the set of all ! 2 G with t3 (!) = ?1, t4 (!) = +1 is invariant with respect to the ow (t )t2R and has full P-measure. Again, for simplicity of notations we assume that 0 = . In this case we de ne U (!) to be the linear isometric mapping from Es (!) to E s (!) which maps the orthonormal basis O(PEs (!) f11; : : : ; PEs (!) fk1) of Es (!) into the orthonormal basis O(PE s (!) f12; : : : ; PE s (!) fk2) of E s (!). Then, by virtue of the continuity of (t; !), (t; !) and Lemmas 6.3, 6.4, U (!) is a random linear isometric mapping from Es (!) to E s (!) which satis es (29). By the way, we remark that, in this case, there always exist measurable orientations on Es (!) and E s (!) such that for all t 2 R ; ! 2

deg s(t; !) = deg s(t; !): More concretely, one can choose orientations on Es (!) and E s (!) such that the basis fPEs (!) f11; : : : ; PEs (!) fk1g of Es (!) and the basis fPE s (!) f12 ; : : : ; PE s (!) fk2g of E s (!) are positively oriented. The desired equalities of degrees follow from Lemma 6.6.

Remark 6.8. Theorem 6.7 remains true if Es (!) and E s (!) are replaced

by other invariant subspaces of (t; !) and (t; !). In particular, it holds true for the unstable subspaces of (t; !) and (t; !).


39

7. Topological classification In this section we shall present the second main result of this paper | Theorem 7.2 on topological classi cation of continuous-time linear hyperbolic cocycles.

Lemma 7.1.

Every linear hyperbolic cocycle is conjugate to a linear hyperbolic cocycle which exhibits an exponential dichotomy with constant K = 1 with respect to the standard Euclidean norm and has its stable invariant subspace orthogonal with the unstable invariant subspace.

Proof. Let (t; !) be a linear hyperbolic cocycle and 1; : : : ; p be its Lyapunov exponents. Set a := 21 minfj1j; : : : ; jpjg > 0. Let ff1 (!); : : : ; fd(!)g be a non-adapted continuous Lyapunov basis of (t; !) with respect to its Lyapunov scalar product h; ia;! . Denote by L(!) the basis transformation from the standard Euclidean basis fe1 ; : : : ; ed g of R d to the basis ff1(!); : : : ; fd(!)g, i.e., L(!)ei = fi(!); i = 1; : : : ; d; ! 2 . It is easily seen that L(t !) is continuous in t. Set

~ (t; !) := L?1 (t!) (t; !) L(!): Then (t; !) and ~ (t; !) are conjugate by L(!). By its de nition and Proposition 3.3, ~ (t; !) is a linear hyperbolic cocycle and exhibits an exponential dichotomy with constant K = 1 with respect to the standard Euclidean norm. Moreover, since Es (!) and Eu (!) are orthogonal with respect to the Lyapunov scalar product h; ia;! (see Arnold [2], Theorem 3.9.6), Es~ (!) and Eu~ (!) are orthogonal with respect to the standard Euclidean product by virtue of the construction of ~ (t; !).

Theorem 7.2. Two continuous-time linear hyperbolic cocycles (t; !)

and (t; !) are conjugate if and only if (i) dim Es (!) = dim E s (!); (ii) There exist measurable orientations on Es (!), E s (!) and a set 0 2 F which is invariant with respect to (t )t2R and has full P-measure such that deg s(t; !) = deg s(t; !) for all ! 2 0 ; t 2 R:

Proof. The \only if" part follows from Theorems 5.1 and 5.4. Denote k := dim Es (!) = dim E s (!). If k = 0 or k = d, then the theorem follows from Theorem 3.11. Let 1 k < d. By Lemma 7.1, there exist linear

hyperbolic cocycles 1 (t; !) and 1(t; !) which are conjugate to (t; !) and (t; !), respectively, and have the following properties dim Es 1 (!) = dim E s 1 (!) = k;

40

NGUYEN DINH CONG

1 (t; !) and 1 (t; !) exhibit exponential dichotomies with constant K =

1 with respect to the standard Euclidean norm, hence their restrictions to their stable (unstable) subspaces are uniformly contracting (expanding); with respect to the Euclidean scalar product Es 1 (!) is orthogonal to Eu1 (!) and E s 1 (!) is orthogonal to E u 1 (!). By Theorem 5.4, Lemma 6.1 and the assumption of the theorem, there exist measurable orientations on Es 1 (!), E s 1 (!) and an invariant set 1 0 of full P-measure such that deg s1 (t; !) = deg s1(t; !) for all ! 2 1 ; t 2 R : By Lemma 6.2 there exist measurable orientations on Eu1 (!), E u 1 (!) and an invariant set 2 1 of full P-measure such that deg u1 (t; !) = deg u1 (t; !) for all ! 2 2 ; t 2 R: From Theorem 6.7 it follows that there exist random linear isometric mappings U1 (!) from Es 1 (!) to E s 1 (!) and U2 (!) from Eu1 (!) to E u 1 (!) and an invariant set 3 2 of full P-measure such that for any ! 2 3 (37) lim kU (t !) PEs 1 (t!) ? U1 (!) PEs 1 (!) k = 0; t!0 1 lim kU (t !) PEu1 (t!) ? U2 (!) PEu1 (!) k = 0: (38) t!0 2

De ne a random linear operator of R d by U (!)x := U1 (!) PEs 1 (!) x + U2(!) PEu1 (!) x for all ! 2 3 ; x 2 R d : Since Es 1 (!) is orthogonal to Eu1 (!), E s 1 (!) is orthogonal to E u 1 (!), and U1 (!), U2(!) are random linear isometric mappings between them, U (!) is a linear orthogonal operator of R d , which maps Es 1 (!) and Eu1 (!) to E s 1 (!) and E u 1 (!), respectively. By (37){(38), the map t 7! U (t !) is continuous. Set 2(t; !) := U ?1 (t!) 1(t; !) U (!): Then 2(t; !) is the linear hyperbolic cocycle which is conjugate to 1(t; !) by U (!) and for any ! 2 3 E s 2 (!) = Es 1 (!); (39) E u 2 (!) = Eu1 (!): Moreover, s2(t; !) is uniformly contracting and u2 (t; !) is uniformly expanding, because U (!) is orthogonal and 1(t; !) exhibits an exponential dichotomy with constant K = 1 with respect to the standard Euclidean norm. Therefore, by (39), using a slight modi cation of Theorem 2.3, we have s2(t; !) and u2 (t; !) conjugate to s1 (t; !) and u1 (t; !), respectively. Taking the direct sum we have 2(t; !) conjugate to 1 (t; !) (the procedure of taking direct sum was done in detail in the proof of Proposition 4.5). Consequently, (t; !) is conjugate to (t; !).


41

Remark 7.3. The condition of Theorem 7.2 concerning degrees of cocy-

cles can be interpreted as an orientation condition for topological conjugacy of continuous-time linear hyperbolic cocycles. It can also be considered as a \coboundary" condition in the continuous-time case, because the set of points ! 2 at which s(t; !) and s(t; !) have dierent degrees is a null set, hence a coboundary with respect to any automorphism of ( ; F ; P). In the trivial case, when consists of only one element, the orientation condition holds and Theorem 7.2 is equivalent to the deterministic one (see Irwin [11], p. 86). We note that there are d +1 topological classes in this case.

Example 7.4.

Consider the case d = 2, = [0; 1) ' S 1, F is the Borel -algebra of , P is its Lebesgue measure, t ! = ! + t (mod 1). The dynamical system ((t )t2R ; ) is ergodic and periodic with period 1. Let fe1 ; e2g be the standard Euclidean basis of R 2 . For z 2 R we introduce the notations esz := e1 cos z + e2 sin z; euz := ?e1 sin z + e2 cos z: We de ne two linear hyperbolic cocycles by the following formulae: for all ! 2 [0; 1) = ; x 2 R 2 ; t 2 R 1 (t; !)x := e?t hx; es! ies!+t + et hx; eu! ieu!+t; 1 (t; !)x := e?t hx; e1 ie1 + et hx; e2ie2 : It is easily seen that for all ! 2 [0; 1) =

Es 1 (!) = spanfes! g; Eu1 (!) = spanfeu! g; E s 1 (!) = spanfe1 g; E u 1 (!) = spanfe2 g: Applying the construction of Proposition 5.2 with = 1, U = and the orientations on Es 1 (!), Eu1 (!), E s 1 (!), E u 1 (!) chosen such that the bases fes! g, feu! g (! 2 [0; 1)s;u= ), fe1 g, fe2 g are positively oriented, respectively, we have U = id and C;U;1; 1 = (for the de nition see the paragraph before Proposition 5.2) which are not coboundaries with respect to U . Therefore, by Proposition 5.2, 1 (t; !) is not conjugate to 1(t; !). This example shows that we can have more topological classes of linear hyperbolic random dynamical systems than of deterministic ones. Moreover, for this concrete case we can fully classify the linear hyperbolic cocycles.

Proposition 7.5. In the case of Example 7.4 there are exactly four topological classes of continuous-time linear hyperbolic cocycles.

42

NGUYEN DINH CONG

Proof. There is one class of contracting cocycles, and a second class of

expanding cocycles. We show that there are exactly two classes of the linear hyperbolic cocycles with one-dimensional stable subspaces over the above ((t )t2R; ), by proving that they are characterized by deg s (1; 0) = 1 and deg s (1; 0) = ?1, respectively. To prove this, rst we note that s(1; 0) is a nonsingular linear operator of Es (0), hence its degree does not depend on the choice of orientation on Es (!). Now, let (t; !) and (t; !) be two linear hyperbolic cocycles with dim Es (!) = dim E s (!) = 1, and es0 be a non-vanishing vector of Es (0). Set es! := (!; 0)es0 2 Es (!) for all ! 2 [0; 1) = : Analogously, let f0s be a non-vanishing vector of E s (0). Set f!s := (!; 0)f0s 2 E s (!) for all ! 2 [0; 1) = : Take the measurable orientations on Es (!) and E s (!) with respect to which the bases fes! g and ff!s g are positively oriented. It is not dicult to see that for any t > 0 deg s(t; 0) = (deg s(1; 0))[t] ; where [t] denotes the greatest integer not exceeding t. And, for any ! 2 [0; 1) = , t > 0 deg s(t; !) = deg s(t + !; 0) = (deg s (1; 0))[t+!] : For t < 0, ! 2 [0; 1) = we have ?1 deg s(t; !) = deg s (?t; t !) = deg s (?t; ft + !g) = (deg s(1; 0))[?t+ft+!g] = (deg s(1; 0))?[t+!] ; where fzg := z ? [z] 2 [0; 1) for any z 2 R . To summarize, for any 0 6= t 2 R , ! 2 [0; 1) =

deg s(t; !) = (deg s(1; 0))(sign t)[t+!] : Analogously, for any 0 6= t 2 R, ! 2 [0; 1) =

deg s(t; !) = (deg s(1; 0))(sign t)[t+!] : Therefore, if deg s(1; 0) = deg s(1; 0) then deg s (t; !) = deg s(t; !) for all t 2 R ; ! 2 . Thus, by Theorem 7.2 (t; !) is conjugate to (t; !). Now, suppose that deg s(1; 0) = ?deg s(1; 0). Applying the construction of Proposition 5.2 with = 1, U = and the above chosen orientations we s have C;U; ; = , which is not a coboundary with respect to U . Therefore, by Proposition 5.2, (t; !) is not conjugate to (t; !). Now we construct an example of a deterministic dynamical system ((t)t2R ; ), over which there are in nitely many topological classes of random dynamical systems.


43

Let := [0; 1) [0; 1) ' T 2 , F be its Borel -algebra, P be its Lebesgue p measure, (t )t2R be the quasi-periodic motion with frequencies (1; 2), i.e., for x := (x1 ; x2) 2 t x =: y = (y1; y2) is de ned by y1 := x1 + tp (mod 1); y2 := x2 + 2 t (mod 1): This dynamical system is ergodic.

Proposition 7.6. There exist in nitely many topological classes of two-

dimensional continuous-time linear hyperbolic cocycles over the above ((t )t2R ; ).

Proof. Denote by fe1; e2 g the standard Euclidean basis of R 2 . Again, for z 2 R we set

esz := e1 cos z + e2 sin z; euz := ?e1 sin z + e2 cos z: For ! := (!1; !2) 2 [0; 1) [0; 1) = we denote p t(!) := !2 ? 2 !1 (mod 1): Denote " ! i i + 1 Gi;n := (2n)! ; (2n)! ; i = 0; 1; : : : ; (2n)! ? 1; n 2 N : We construct linear cocycles by the following formulae: for all ! := (!1; !2) 2 [0; 1) [0; 1) = ; x 2 R 2 ; t 2 R ( ?t es ies + et hx; eu ieu if t(!) 2 Gi;n; !1 !1 +t !1 !1 +t i;n(t; !) := ee?t hhx; t x; e1ie1 + e hx; e2ie2 if t(!) 2= Gi;n: (40) Take and x a number 0 < " < 1. Set Ui;n := (0; ") Gi;n : We note that the automorphism 1 on is decomposed into the direct product of two automorphisms on the components of the product space = [0; 1) [0; 1): it acts p as the identity on the rst coordinate and as the rotation by the angle 2 on the second one. Further, by Lemma 8.4 of [15] there are in nitely many sets from the collection fGi;ng, i = 0; 1; : : : ; (2n)! ? 1; n 2 N , such that the symmetric dierence of any two p from them is not a coboundary with respect to the rotation by the angle 2 on [0; 1). Therefore, we have in nitely many sets from the collection fUi;n j i = 0; 1; : : : ; (2n)! ? 1; n 2 N g, say fUj;m j (j; m) 2 Ig, with the property that the symmetric dierence of

44

NGUYEN DINH CONG

any two sets from them is not a coboundary with respect to 1 . Now, take the orientation on Es i;n (!) such that its basis ff! g de ned by ( s for t(!) 2 Gi;n; ! = (!1; !2) 2 [0; 1) [0; 1) = ; f! := ee!1 for t(!) 2= Gi;n 1 is positively oriented (i = 0; 1; : : : ; (2n)! ? 1; n 2 N ). Consider the construction of Proposition 5.2 with := 1; U := (0; ") [0; 1) . By (40), we have s C;U; i;n ;i ;n = Ui;n 4Ui ;n : Further, U is invariant with respect to , hence U x = x for all x 2 U . Therefore, for any two dierent pairs (j; m) and (j 0; m0 ) from I the set Uj;m4Uj ;m is not a coboundary with respect to U . Consequently, by Proposition 5.2, any two cocycles from the in nite collection fj;m j (j; m) 2 Ig are not conjugate. 0

0

0

0

0

0

Acknowledgments

I am greatly indebted to Professor L. Arnold for his fruitful discussion, valuable comments, constant help and encouragement during the course of this work. I would also like to thank Doctors V. M. Gundlach and T. Wanner for their careful reading of the manuscript and valuable comments and suggestion for improving the rst draft of this paper. I thank the referee for a comment on Proposition 4.5. References [1] L. Arnold. Anticipative problems in the theory of random dynamical systems. In M. Cranston and M. Pinsky, editors, Stochastic Analysis, AMS Proceeding of Symposia in Pure Mathematics, pages 529{541, Providence, Rhode Island, 1995. [2] L. Arnold. Random dynamical systems. Preliminary version 2, Bremen, 1994. To be published. [3] L. Arnold and H. Crauel. Random dynamical systems. In L. Arnold, H. Crauel, and J.-P. Eckmann, editors, Lyapunov Exponents, Proceedings, Oberwolfach 1990, volume 1486 of Lecture Notes in Math., pages 1{22. Springer, Berlin Heidelberg New York, 1991. [4] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinai. Ergodic theory. Springer, Berlin Heidelberg New York, 1982. [5] K. Deimling. Nonlinear functional analysis. Springer, Berlin Heidelberg New York, 1985. [6] J. Dieudonne. Grundzuge der modernen Analysis, Volume 9. VEB Deutscher Verlag der Wissenschaften, Berlin, 1987. [7] A. Dold. Lectures on algebraic topology. Springer, Berlin Heidelberg New York, 1972. [8] E. Dynkin. Markov processes. Volume 1. Springer-Verlag, Berlin Gottingen Heidelberg, 1965. [9] F. R. Gantmacher. The theory of matrices, Vol. 1. Chelsea, New York, 1977.


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[10] V. M. Gundlach. Random homoclinic orbits. Random & Computational Dynamics, 3:1{33, 1995. [11] M. C. Irwin. Smooth dynamical systems. Academic Press, New York, 1980. [12] R. Johnson. Remarks on linear dierential systems with measurable coecients. Proc. Amer. Math. Soc., 100:491{504, 1987. [13] N. H. Kuiper. The topology of the solutions of a linear dierential equation on Rn. In A. Hattori, editor, Manifolds-Tokyo 1973, pages 195{203. Univ. Tokyo Press, Tokyo, 1975. [14] Nguyen Dinh Cong. Structural stability of linear random dynamical systems. Ergodic Theory and Dynamical Systems, 16, 1996. [15] Nguyen Dinh Cong. Topological classi cation of linear hyperbolic cocycles. Journal of Dynamics and Dierential Equations, 8:427{467, 1996. [16] V. I. Oseledets. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19:197{231, 1968. [17] K. J. Palmer. Exponential dichotomies, the shadowing lemma and transversal homoclinic points. Dynam. Report. Ser. Dynam. Syst. Appl., 1:265{306, 1988. [18] J. W. Robbin. Topological conjugacy and structural stability for discrete dynamical systems. Bull. Amer. Math. Soc., 78:923{952, 1972. [19] T. Wanner. Zur Linearisierung zufalliger dynamischer Systeme. PhD thesis, Universitat Augsburg, 1993. For English version see Dynamics Reported 4. [20] T. Wanner. Qualitative behavior of random dierential equations. Report 304, Universitat Augsburg, 1994. Hanoi Institute of Mathematics, P.O.Box 631 Bo Ho, 10000 Hanoi, Vietnam

Current address : Institut fur Dynamische Systeme, Universitat Bremen, Postfach 330 440, 28334 Bremen, Germany E-mail address : [email protected]

STRUCTURAL STABILITY AND TOPOLOGICAL CLASSIFICATION OF

STRUCTURAL STABILITY AND TOPOLOGICAL CLASSIFICATION OF

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