Topological classification of linear hyperbolic cocycles

Topological classification of linear

hyperbolic cocycles

Nguyen Dinh Cong∗

Abstract

In this paper linear hyperbolic cocycles are classified by the relation oftopological conjugacy. Roughly speaking, two linear cocycles are conju-gate if there exists a homeomorphism which maps their trajectories intoeach other. The problem of classification of discrete-time deterministichyperbolic dynamical systems was investigated by Robbin (1972). Heproved that there exist 4d classes of d-dimensional deterministic discretehyperbolic dynamical systems. We obtain a criterion for topological con-jugacy of two linear hyperbolic cocycles and show that the number ofclasses depends crucially on the ergodic properties of the metric dynami-cal system over which they are defined. Our result is a generalization ofthe deterministic theorem of Robbin.

KEY WORDS: Linear hyperbolic cocycle, random homeomorphism,Oseledets splitting, orientation, dimension, coboundary.

AMS Subject Classification: 58F15, 28D05, 58F19.

1 Introduction

In the theory of dynamical systems the classification problem plays an importantrole. It provides a method of simplifying the objects under investigation andgives an insight into the structure of dynamical systems. To classify dynamicalsystems one needs a notion of equivalence relation. Among different possibilitiesof introducing an equivalence relation the notion of topological conjugacy is oneof the most useful and interesting. Roughly speaking, two dynamical systemsare conjugate if there exists a homeomorphism mapping their trajectories intoeach other. The problem of classifying linear deterministic dynamical systemsand, closely related with it, the problem of structural stability have been inves-tigated by many authors, for a bibliography we refer to the works of Robbin(1972), Irwin (1980) and the references therein. In this paper we shall dealwith the problem of topological classification of linear hyperbolic cocycles (inother words, discrete-time linear hyperbolic random dynamical systems). We

∗Institut fur Dynamische Systeme, Universitat Bremen, Postfach 330 440, 28334 Bremen,Germany, and Hanoi Institute of Mathematics, P.O.Box 631 Bo Ho, 10000 Hanoi, Vietnam.

1

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Topological classification of linear hyperbolic cocycles 2

are mainly interested in the problem of classification and, therefore, in this papera theorem on structural stability plays only the role of the key tool for studyingthe classification problem. We would like to emphasize that the problem ofstructural stability is one of the important problems in the theory of dynamicalsystems. Structural stability is one of the most important local properties ofdynamical systems; on the other hand, the classification problem is concernedwith their global characteristics.

The work of Robbin (1972) on deterministic dynamical systems gives us alook over random dynamical systems both from the “local” and “global” pointof view. It gives us also very important tools and initial ideas for dealing withrandom dynamical systems. For late use and for the convenience to comparethe classification theorems in deterministic and random cases we formulate herethe theorem of Robbin on classification of discrete-time deterministic linearhyperbolic dynamical systems. Two linear automorphisms f and g of the d-dimensional Euclidean space Rd are called (deterministically) topologically con-jugate if there exists a homeomorphism ϕ on Rd such that g = ϕ−1 f ϕ. Alinear automorphism f of Rd is called hyperbolic if its eigenvalues are containedin the complement of the unit circle in the complex plane. The set of all linearhyperbolic automorphisms of Rd (which will be denoted by Hyp(Rd)) is openand dense in the space of all linear automorphisms of Rd with respect to theoperator norm of linear automorphisms. For a hyperbolic automorphism f wedenote by W±(f) the eigenspaces of f corresponding to eigenvalues with modu-lus less (or greater) than 1. We set the number w±(f) equal to dim W±(f) if therestriction of f to W±(f) preserves the orientation and equal to −dim W±(f)otherwise. The following theorem of Robbin (1972) on topological classificationof deterministic linear hyperbolic dynamical systems is well-known.

Theorem 1.1. For f, g ∈ Hyp(Rd) the following statements are equivalent:1) f and g are topologically conjugate,2) w+(f) = w+(g) and w−(f) = w−(g),3) f and g belong to the same component of Hyp(Rd).

The central objects of our research are linear cocycles generated by linearrandom maps over a discrete-time (metric) dynamical system. We give hereseveral necessary definitions and assumptions which are used throughout thepaper.

Let (Ω,F , P) be a probability space, and θ an automorphism of (Ω,F , P)preserving the probability measure P. Throughout this paper, we assume thatθ is ergodic. The non-ergodic case can be reduced to the ergodic by using ergodicdecomposition of dynamical systems (see Cornfeld at al., 1982); moreover, someresults of the paper, e.g. Theorems 2.1, 2.8 and 2.9, hold without assumption ofthe ergodicity of θ as seen from their proofs.

Consider a linear random map A(·) : Ω → Gl(d, R), i.e. A is a measurablemapping from the probability space (Ω,F , P) to the topological space Gl(d, R)of linear nonsingular operators of Rd equipped with its Borel σ-algebra. Itgenerates a linear cocycle (see Arnold and Crauel, 1991) over the dynamical

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system (θ, Ω) via

ΦA(n, ω) :=

A(θn−1ω) . . . A(ω), n > 0,id, n = 0,A−1(θnω) . . . A−1(θ−1ω), n < 0.

Conversely, if we are given a linear cocycle over (θ, Ω), then its time-one mapis a linear random map. Therefore, the correspondence between A and ΦA isone-to-one and we are free to choose one from them to work with as any of themis given. Throughout this paper we shall deal only with linear random mapsA(·) satisfying the integrability conditions

log+ ‖A(·)±1‖ ∈ L1(P), (1)

where ‖ · ‖ denotes the operator norm of linear operators of Rd, so that theMultiplicative Ergodic Theorem of Oseledets (1968) (see also Arnold, 1994),which we shall abbreviate as MET, applies to the cocycle ΦA. According tothe MET, ΦA has Lyapunov exponents λ1, . . . , λp with multiplicities d1, . . . , dp

which are independent of ω due to the ergodicity of θ. Furthermore, the phasespace Rd is decomposed into the direct sum of subspaces Ei(ω) of dimensionsdi corresponding to the Lyapunov exponents λi, i = 1, . . . , p, i.e.

limn→±∞

n−1 log ‖ΦA(n, ω)x‖ = λi(ω) ⇐⇒ x ∈ Ei(ω)\0,

where ‖ · ‖ denotes the standard Euclidean norm of Rd. The subspaces Ei(ω)are measurable and invariant with respect to A, i.e. A(ω)Ei(ω) = Ei(θω).This decomposition is called Oseledets splitting and Ei(ω) are called Oseledetssubspaces of ΦA. In particular, Rd = Es

A(ω)⊕ EcA(ω)⊕ Eu

A(ω), where

EsA(ω) = x ∈ Rd | lim

n→±∞n−1 log ‖ΦA(n, ω)x‖ < 0,

EcA(ω) = x ∈ Rd | lim

n→±∞n−1 log ‖ΦA(n, ω)x‖ = 0,

EuA(ω) = x ∈ Rd | lim

n→±∞n−1 log ‖ΦA(n, ω)x‖ > 0

are stable, center and unstable subspaces of ΦA.We note that the statements of the MET holds on an invariant set of full

P-measure. Since we deal with discrete-time case we can always neglect sets ofnull measure, and when needed we shall assume, without loss of generality, thatthe assertions of the MET hold on the whole Ω.

Definition 1.2. A linear cocycle ΦA is called hyperbolic if its Lyapunovexponents are different from 0; in other words, if its central subspace is trivial.

Note that dim EsA(ω) + dimEc

A(ω) + dimEuA(ω) = d, so for hyperbolic ΦA

we have dim EsA(ω) + dim Eu

A(ω) = d. Furthermore, since dim Ei(ω) = di, i =1, . . . , p, are independent of ω’s, dim Es

A(ω) and dim EuA(ω) are also independent

of ω’s.

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The next important notion in this paper is the notion of random homeomor-phism h(·) of Rd which is, by definition, a measurable map from the probabilityspace (Ω,F , P) into the topological space Homeo(Rd) with the compact-opentopology.

Definition 1.3. Two linear cocycles ΦA and ΦB are called conjugate ifthere exists a random homeomorphism h of Rd such that for almost all ω ∈ Ωthe following relations hold

• h(ω)0 = 0,

• B(ω) = h(θω)−1 A(ω) h(ω).

We see that h maps trajectories of ΦB into trajectories of ΦA. Clearly, therelation of topological conjugateness is a equivalence relation, i.e. it is reflexive,symmetric and transitive; in particular, if ΦA is conjugate to ΦB and ΦB isconjugate to ΦC , then ΦA is conjugate to ΦC .

In our work, the presence of the dynamics on the probability space makesthe problems complicated and calls for a new approach to tackle them. The def-inition of topological conjugacy is changed corresponding to the presence of thedynamics. Perhaps, the only case where the methods of the theory of determin-istic dynamical systems are applicable to is that of periodic random dynamicalsystems. Studying the problem of classification of linear hyperbolic cocycles wediscovered an interesting generalization of the condition in Theorem 1.1 con-cerning orientation property of hyperbolic linear mappings. Our orientationcondition is expressed in terms of some sets in Ω to be coboundaries, i.e. to bepresented in the form of the symmetric difference of a measurable set and itsimage with respect to the automorphism θ. For two linear hyperbolic cocyclesΦA and ΦB we denote by Cs

AB (resp. CuAB) the set of all ω ∈ Ω such that the

restrictions of the linear maps A(ω) and B(ω) to the stable (resp. unstable)subspaces Es

A(ω) and EsB(ω) (resp. Eu

A(ω) and EuB(ω)) have different degrees,

i.e. one of them is orientation preserving and the other is orientation reversing(for detailed formulas and definitions, see Section 2 below).

Our main result is the following theorem on the classification of linear hy-perbolic cocycles.

Theorem 7.1. Two linear hyperbolic cocycles ΦA and ΦB are conjugateif and only if the following conditions hold

• dim EsA(ω) = dim Es

B(ω) (hence dim EuA(ω) = dim Eu

B(ω)),• their sets Cs,u

AB are coboundaries.

This theorem is a generalization of the deterministic Theorem 1.1. In thedegenerate case, when Ω consists of only one element, it is equivalent to thatone.

This paper is organized as follows. In Section 2 we prove theorems on nec-essary conditions for topological conjugacy of linear cocycles. In Section 3 we


investigate the problem of structural stability of linear cocycles; we shall provea theorem on structural stability of a linear cocycle generated by a contracting(or expanding) bounded (with its inverse) linear random map. In the remainingpart of the paper we use the result on structural stability of Section 3 and thetheorems on necessary conditions for topological conjugacy in Section 2 to solvethe problem of topological classification of linear hyperbolic cocycles. To dothis we reduce the problem in several steps to simpler problems. So, in Section4 we study the problem of diagonalization of linear cocycles; in Section 5 weclassify one-dimensional hyperbolic cocycles and in Section 6, assuming thatΩ is non-atomic, we prove a criterion for topological conjugacy of two linearhyperbolic diagonal cocycles. We present our main result on classification oflinear hyperbolic cocycles in Section 7, where we also treat the case of atomicΩ which was excluded in Section 6, and give some discussion and comments onthe results.

2 Necessary conditions for topological conjugacy

In this section we prove theorems on necessary conditions for topological con-jugacy of linear cocycles. Theorem 2.1 is a dimension condition and Theorems2.8 and 2.9 are orientation conditions.

Theorem 2.1. If two linear hyperbolic cocycles ΦA and ΦB are conjugate,then for almost all ω ∈ Ω we have

dim EsA(ω) = dim Es


B(ω)).

Proof. Denote by h the random homeomorphism providing the topologicalconjugacy between ΦA and ΦB , i.e. B(ω) = h−1(θω) A(ω) h(ω). Then foralmost all ω ∈ Ω and all n ∈ Z we have

ΦB(n, ω) = h−1(θnω) ΦA(n, ω) h(ω). (2)

For K ∈ N putΩK := ω | h(ω)B1 ⊂ BK,

where Br denotes the closed ball of radius r and centered at 0. Then ΩK =ω | supq∈Qd∩B1

|h(ω)q| ≤ K, so ΩK are measurable. Furthermore, sinceh(ω)0 = 0 the sets ΩK are increasing and tend to Ω (mod 0), as K tends to∞. For fixed K put

Ω∞K := lim sup

n→∞θ−nΩK = ω | θnω ∈ ΩK for infinitely many n ∈ N.

Then θ−1Ω∞K = Ω∞

K , and the Poincare recurrence theorem (see Halmos, 1956,p. 10) implies that P(Ω∞

K ∩ ΩK) = P(ΩK), hence P(Ω∞K ) tends to 1 as K tends

to ∞.


Choose ω from the invariant set of full P-measure on which the statementsof the MET hold for both ΦA and ΦB . Suppose x ∈ Es

B(ω) and h(ω)x isdecomposed in h(ω)x = y1 + y2 with y1 ∈ Es

A(ω) and y2 ∈ EuA(ω) such that

y2 6= 0. Then, due to the MET, there exists n0(ω) ∈ N such that for every n ≥n0(ω) we have ΦB(n, ω)x ∈ B1, ΦA(n, ω)y1 ∈ B1, ΦB(n, ω)y2 ∈ Bc

K+1, whence,ΦA(n, ω)h(ω)x ∈ Bc

K . Therefore, by (2), θnω /∈ ΩK for every n ≥ n0(ω). Thusω /∈ Ω∞

K . Since K is arbitrary and P(Ω∞K ) tends to 1 we get that the set of all

ω ∈ Ω such that h(ω)EsB(ω) 6⊂ Es

A(ω) is a null set.Using analogous arguments for the unstable subspaces and changing the role

of A and B to get converse inclusions we have for almost all ω ∈ Ω

h(ω)EsB(ω) = Es

A(ω),h(ω)Eu

B(ω) = EuA(ω). (3)

Theorem 2.1 follows from (3) and the fact that dimension is a topological in-variant (see Pears, 1975).

Next we study necessary conditions for topological conjugacy of linear cocy-cles concerning orientation property of linear random maps. For this we shallneed the notion of a coboundary from algebraic theory of dynamical systemsand some of its properties.

Definition 2.2. A measurable set E ⊂ Ω is called a coboundary if thereexists C ∈ F such that E = C4θC (mod 0), where C4θC denotes the sym-metric difference of C and θC.

This definition is a special case of a much more general notion of a coboundaryas a taking values in a Polish group (1-)cocycle on a measurable space with anonsingular equivalence relation which is cohomologous to the trivial cocycle(see Schmidt, 1990 and Moore and Schmidt, 1986).

Define the return time of a set E ∈ F by (see Cornfeld et al., 1982, p. 15)

kE(ω) = minn ≥ 1| θnω ∈ E for ω ∈ Ω.

If θnω /∈ E for all n ∈ N we set kE(ω) = +∞. Then kE(·) : Ω → R ∪ ∞ ismeasurable. By virtue of the Poincare recurrence theorem, for any measurableset E ⊂ Ω the function kE(·) is finite almost surely on E.

Lemma 2.3. A measurable set G is a coboundary if and only if it can bedecomposed into two disjoint measurable sets G1, G−1 such that almost surely

ω ∈ Gj =⇒ kGj(ω) > kG−j

(ω) (j = ±1), (4)

andP(G1) = P(G−1) =

12

P(G). (5)

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Proof. If P(G) = 0, then G is a coboundary and (4)–(5) trivially hold.Assume that P(G) > 0.

Suppose that (4) and (5) hold. Put Fn := ω ∈ G1| kG−1(ω) = n (n ∈ N).Then, by (4), we have almost surely

Fn ∩ Fm = ∅ for all n 6= m,∞⋃

n=1

Fn = G1,

θnFn ⊆ G−1 for all n ∈ N.

Furthermore, almost surely,⋃n−1

i=1 θiFn ⊆ Ω\G for all n ∈ N. Therefore,θnFn ∩ θmFm = ∅ (mod 0), for all n 6= m. Hence,

P(G1) =∞∑

n=1

P(Fn) =∞∑

n=1

P(θnFn) ≤ P(G−1),

which, by virtue of (5), implies

G−1 =∞⋃

n=1

θnFn (mod 0).

This yields G = C4θC (mod 0), where C :=⋃∞

n=1

⋃n−1i=0 θiFn. Therefore, G

is a coboundary.Conversely, suppose that G is a coboundary, so that G can be represented

in the form G = C4θC (mod 0), for some C ∈ F . Set G1 := C\θC andG−1 := θC\C. Then, clearly,

P(G1) = P(G−1) =12

P(G) > 0,

G1 ∩G−1 = ∅,G1 ∪G−1 = G (mod 0).

It remains to prove that the following statement is true almost surely

ω ∈ Gj =⇒ kGj(ω) > kG−j

(ω) (j = ±1).

We recall that for almost all ω ∈ Gj the value kGj (ω) is finite. Let ω ∈ G1 bearbitrary with finite value kG1(ω). If kG−1(ω) = 1, then obviously kG1(ω) >kG−1(ω). If kG−1(ω) > 1, then by induction one can easily prove that θiω ∈C∩θC for all i ∈ 1, . . . , kG−1(ω)−1. Hence, kG1(ω) > kG−1(ω)−1. Therefore,kG1(ω) > kG−1(ω) because kG1(ω) 6= kG−1(ω) for the disjoint sets G1, G−1.Analogously, for almost all ω ∈ G−1 we have kG−1(ω) > kG1(ω).

From Lemma 2.3 we can deduce the following relation between coboundariesand induced automorphisms. Let G ∈ F and P(G) > 0. Then the inducedautomorphism θG is defined as (see Cornfeld et al., 1982, pp. 20–21)

θG(ω) := θkG(ω)ω for ω ∈ G,


and the probability space (G,FG, PG) is generated by G via

FG := A ∈ F| A ⊆ G,

PG(A) :=P(A)P(G)

for all A ∈ FG.

Corollary 2.4. If G is a coboundary and P(G) > 0, then almost surely

θGG1 = G−1 and θGG−1 = G1,

where G1, G−1 are the sets of the decomposition of G from Lemma 2.3.

Proof. For almost all ω ∈ Gj , j = ±1, by Lemma 2.3 we have kG(ω) =kG−j

(ω), hence θG(ω) = θkG(ω)ω = θkG−j(ω)ω ∈ G−j . Corollary 2.4 now followsfrom this and the fact that the sets G1, G−1 have the same measure.

Remark 2.5. If G is a coboundary and P(G) > 0, then by Lemma 2.3 G1

and G−1 are invariant with respect to (θG)2. Hence (θG)2 is not ergodic. Thiscondition appears to be sufficient for a measurable set to be a coboundary asshown in Proposition 2.6 below.

We present here two further properties of coboundaries from Knill (1991),which will be needed later.

Proposition 2.6. A set E ∈ F with P(E) > 0 is a coboundary if and onlyif (θE)2 is not ergodic.

Proposition 2.7. Let Z ⊂ Y ⊂ Ω be measurable sets and P(Z) > 0. ThenZ is a coboundary for θ if and only if it is a coboundary for θY .

Now we introduce the notion of a random orientation on Rd and its sub-spaces, which depend measurably on ω ∈ Ω, in particular, Oseledets subspacesof linear cocycles (Rd is considered to depend trivially, hence measurably, onω). Let Rl(ω) be an l-dimensional random subspace of Rd which depends mea-surably on ω ∈ Ω. Choose and fix an (ordered) random basis f1(ω), . . . , fl(ω)of Rl(ω), i.e. fi(·) : Ω → Rd, i = 1, . . . , l, are measurable. A random orientationon Rl(ω) is, by definition, a measurable choice of orientation on Rl(ω), i.e. thereexist a measurable function s(·) : Ω → 1,−1 such that f1(ω), . . . , fl(ω) is apositively oriented basis of Rl(ω) if s(ω) = 1, and negatively oriented otherwise.It is easily seen that the above definition is independent of the choice of therandom basis f1(ω), . . . , fl(ω). Furthermore, given a random orientation wecan always choose a positively oriented random basis, i.e. a random basis whichis positively oriented at every point of Rl(ω). Note that although Rd dependstrivially on ω’s, there are many different random orientations on Rd.

Choose and fix a random orientation and a positively oriented random basisof Rd. Let A and B be two linear random maps of Rd. Denote by deg A(ω)


and deg B(ω) the degrees of the maps A(ω), B(ω) with respect to the chosenrandom orientation, respectively. It is known that these degrees take values inthe set 1,−1 and are equal to the signs of the determinants of the matricesrepresenting A(ω), B(ω) in the chosen positively oriented random basis of Rd.In this paper we shall deal with the degrees of linear random maps and of ran-dom homeomorphisms. We recall some simple properties the degree of bijectivecontinuous maps on oriented Euclidean spaces (see Dold, 1972, Chapters 4, 8):

deg (h1 h2) = deg h2 · deg h1, (6)deg id = 1,

deg h = ±1,

and, therefore, deg h−1 = deg h. The maps h1 and h2 are such that (6) makessense, i.e. the domain of h1 must coincide with the image of h2.

Now let ΦA and ΦB be two linear hyperbolic cocycles. Put

CAB := ω ∈ Ω| deg A(ω) · deg B(ω) = −1.

Then CAB is the set of all ω ∈ Ω at which the degrees of the linear mapsA(ω) and B(ω) are different. A condition on the linear cocycles ΦA and ΦB

affecting their set CAB might be interpreted in a natural way as an orientationcondition on the pair ΦA and ΦB . Since ΦA is hyperbolic, by the MET we haveRd = Es

A(ω)⊕EuA(ω). Denote the restrictions of A(ω) to Es

A(ω) and EuA(ω) by

As(ω) and Au(ω), respectively. For simplicity of notation, in case one of thespaces Es

A(ω) or EuA(ω) is trivial we set the corresponding restricted linear map

equal to id. The meaning of Es,uB (ω) and Bs,u is clear. Choose and fix random

orientations on EsA(ω), Eu

A(ω), EsB(ω) and Eu

B(ω). We introduce the notation

CsAB := ω ∈ Ω| deg As(ω) · deg Bs(ω) = −1,

CuAB := ω ∈ Ω| deg Au(ω) · deg Bu(ω) = −1.

The following two theorems on an algebraic property of the sets CAB , Cs,uAB

of a pair of conjugate linear cocycles ΦA and ΦB give us an algebraic invariantproperty of topological conjugacy. We note that the sets CAB , Cs,u

AB depend onthe choice of random orientations on Rd and Es

A(ω), EuA(ω), Es

B(ω) and EuB(ω),

but whether they are coboundaries is independent of the choice of random ori-entations as Theorem 2.11 below shows, so the statements of the following twotheorems do not depend on the choice of orientations.

Theorem 2.8. If two linear cocycles ΦA and ΦB are conjugate, then theirset CAB is a coboundary.

Proof. Denote by h the random homeomorphism providing the topologicalconjugacy between ΦA and ΦB , i.e. B(ω) = h−1(θω) A(ω) h(ω) for almostall ω ∈ Ω. Then, by (6) we have

deg h(θω) = (deg A(ω)deg B(ω))deg h(ω). (7)


SetH := ω ∈ Ω| deg h(ω) = −1.

We prove our theorem by showing that

CAB = H4θ−1H. (8)

Suppose ω ∈ CAB , then we have deg A(ω)deg B(ω) = −1. If ω ∈ H,then deg h(ω) = −1. By (7) this implies deg h(θω) = 1. Hence, θω ∈ Ω\H.Consequently, ω ∈ H\θ−1H ⊂ H4θ−1H. If ω ∈ Ω\H, then deg h(ω) = 1,which, by (7), implies deg h(θω) = −1. Hence, θω ∈ H. Consequently,ω ∈ θ−1H\H ⊂ H4θ−1H.

Now suppose that ω ∈ H4θ−1H = (H\θ−1H)∪ (θ−1H\H), then either ω ∈H and θω ∈ Ω\H or θω ∈ H and ω ∈ Ω\H. Therefore, deg h(ω)deg h(θω) = −1.By (7), this implies deg A(ω)deg B(ω) = −1, hence ω ∈ CAB . Thus, (8) isproved and so is the theorem.

Notice that for a hyperbolic linear cocycle ΦA, one can choose random orien-tations on Es

A(ω) and EuA(ω) such that deg A(ω) = deg As(ω) deg Au(ω). There-

fore, the following theorem is a generalization of Theorem 2.8.

Theorem 2.9. If two linear hyperbolic cocycles ΦA and ΦB are conjugate,then their sets Cs

AB and CuAB are coboundaries.

Proof. Denote by h(ω) the homeomorphism providing the topologicalconjugacy between ΦA and ΦB , i.e.

B(ω) = h−1(θω) A(ω) h(ω).

In the proof of Theorem 2.1 we obtained relation (3) which states that for almostall ω ∈ Ω

h(ω)EsB(ω) = Es

A(ω),h(ω)Eu

B(ω) = EuA(ω).

Therefore, with an abuse of language in denoting by the same characters h(ω)and h−1(ω) the restrictions of h(ω) and h−1(ω) to the subspaces Es

B(ω), EuB(ω)

and EsA(ω), Eu

A(ω), we have

Bs(ω) = h−1(θω) As(ω) h(ω),Bu(ω) = h−1(θω) Au(ω) h(ω).

Consequently, by the arguments analogous to those of the proof of Theorem 2.8Cs

AB and CuAB are coboundaries.

Remark 2.10. Being more familiar with algebra one can easily see thatthe deg maps our space of linear cocycles into the space of linear cocycles with


values in the Abelian group Z2 = +1,−1. Furthermore, by (7), it mapsconjugate cocycles to cohomologous cocycles, hence CAB is a coboundary.

In Theorems 2.8 and 2.9 we have chosen and fixed random orientations onRd, Es

A(ω), EuA(ω), Es

B(ω) and EuB(ω). The following theorem shows that the

statements of Theorems 2.8 and 2.9 are independent of the choice of thoserandom orientations.

Theorem 2.11. 1. If the set of all ω ∈ Ω at which A(ω) and B(ω) havedifferent degrees with respect to a random orientation on Rd is a coboundary,then the set of all ω ∈ Ω at which A(ω) and B(ω) have different degrees withrespect to another (arbitrary) random orientation on Rd is also a coboundary.2. If the sets of all ω ∈ Ω at which As,u(ω) and Bs,u(ω) have different degreeswith respect to a choice of random orientations on Es

A(ω), EuA(ω), Es

B(ω) andEu

B(ω) are coboundaries, then the sets all ω ∈ Ω at which As,u(ω) and Bs,u(ω)have different degrees with respect to another (arbitrary) choice of random ori-entations on Es

A(ω), EuA(ω), Es

B(ω) and EuB(ω) are also coboundaries.

Proof. 1. Take and fix a random basis f1(ω), . . . , fd(ω) which is positivelyoriented with respect to the chosen random orientation on Rd. Suppose we aregiven another random orientation. Introduce a function s(·) : Ω −→ 1,−1,such that s(ω) = 1 if the new orientation coincides with the old orientationand s(ω) = −1 otherwise. Then s(·) is measurable and the random basiss(ω)f1(ω), f2(ω), . . . , fd(ω) is positively oriented in the new orientation. De-note by s1(ω) and s2(ω) the degrees of the map A(ω) with respect to the oldand the new orientations, respectively. Then we have,

s2(ω) = s(θω)s1(ω)s(ω).

Therefore, by the arguments of the proof of Theorem 2.8, the set of elementsω ∈ Ω at which A(ω) has different degrees with respect to the above two orien-tations is a coboundary. The same argument may be applied to the map B(·).Consequently, by virtue of the group property of coboundaries with respect tothe operation of taking the symmetric difference of sets, if the set of elementsω ∈ Ω at which A(ω) and B(ω) have different degrees with respect to the oldorientation is a coboundary, then the set of elements ω ∈ Ω at which A(ω)and B(ω) have different degrees with respect to the new orientation is also acoboundary. Therefore, part 1 of the theorem is proved.

2. The proof of part 2 of the theorem is analogous to that of part 1.

Remark 2.12. We note that the set CAB is, in fact, independent of thechoice of the random orientation on Rd, but the sets Cs,u

AB depend on the choiceof the random orientations on Es

A(ω), EuA(ω), Es

B(ω) and EuB(ω).


3 Structural stability of linear cocycles

In this section we investigate structural stability of linear cocycles. For this weshall need a concept of random norms. We present here necessary informationon random norms and refer to Arnold’s book (1994) for more details.

Definition 3.1. A random norm on Rd is a measurable function on Ω×Rd

which is a norm ‖ · ‖ω for each fixed ω ∈ Ω (in particular, the norm can begenerated by a random scalar product 〈·, ·〉ω). In this case we say that Rd isequipped with the random norm ‖ · ‖ω.

We remark that linear random maps and linear cocycles are defined as linearoperators of Rd, hence they are independent of the choice of a norm on Rd.However, in the formulation of the classical MET (see Oseledets, 1968) thestandard Euclidean norm of Rd is used, and for the application of the METone needs a norm on Rd. For a random norm generated by a random scalarproduct the MET (reformulated accordingly) is applicable as well (see Arnold,1994, §3.7). (In this paper we shall consider only random norms which aregenerated by random scalar products.) The change of the (possibly random)norm on Rd affects only the quantitative properties, e.g. Lyapunov spectrum,which are defined via norms of linear maps and vectors but does not affect thelinear random maps and linear cocycles themselves. Furthermore, for a certainclass of random norms the Lyapunov spectrum of linear cocycles is invariant(see Lemma 3.7 below).

Definition 3.2. A linear cocycle ΦA on Rd equipped with a random norm‖ · ‖ω,1 is called Lyapunov cohomologous to a linear cocycle ΦB on Rd equippedwith a random norm ‖ · ‖ω,2 if there exists a measurable map L : Ω → Gl(d, R)such that almost surely B(ω) = L−1(θω) A(ω) L(ω) and

limn→±∞

1n

log supx ∈ Rd

‖x‖ω,2 = 1

‖L(θnω)x‖θnω,1 =

= limn→±∞

1n

log supx ∈ Rd

‖x‖ω,1 = 1

‖L−1(θnω)x‖θnω,2 = 0.

In this case L is called a Lyapunov cohomology.

Remark 3.3. If two linear cocycles are Lyapunov cohomologous, then theyhave the same Lyapunov spectrum (see Arnold, 1994). A Lyapunov cohomologyis a linear topological conjugacy.

Definition 3.4. Given ε > 0. A random variable R : Ω → (0,∞) is calledε-slowly varying (with respect to the dynamical system (θ, Ω)) if almost surely

e−ε|n|R(ω) ≤ R(θnω) ≤ eε|n|R(ω) for all n ∈ Z.


Now we formulate a theorem of Arnold which includes the definition of Lya-punov scalar product, Lyapunov norm and their properties (see Arnold, 1994,Theorem 3.9.6). Recall that Rd =

⊕pi=1 Ei(ω) is the Oseledets splitting of ΦA.

Proposition 3.5. Choose and fix a constant a > 0. Introduce for ω ∈ Ω,where Ω is the set of full measure in Ω on which the assertions of the MET holdfor ΦA, and any x = ⊕p

i=1xi and y = ⊕pi=1yi with xi, yi ∈ Ei(ω)

〈x, y〉a,ω :=p∑

i=1

〈xi, yi〉a,ω,

where for u, v ∈ Ei(ω)

〈u, v〉a,ω :=∑n∈Z

〈ΦA(n, ω)u, ΦA(n, ω)v〉e2(λin+a|n|) ,

where 〈·, ·〉 denotes the standard Euclidean scalar product of Rd. Put 〈x, y〉a,ω :=〈x, y〉 for ω /∈ Ω. Then(i) 〈·, ·〉a,ω is a random scalar product on Rd which depends measurably on ωand with respect to which the Ei(ω)’s are orthogonal.(ii) For each ε > 0 there exists an ε-slowly varying function Bε : Ω → [1,∞)such that

1Bε(ω)

‖ · ‖ ≤ ‖ · ‖a,ω ≤ Bε(ω)‖ · ‖,

where

‖x‖2a,ω = 〈x, x〉a,ω =p∑

i=1

‖xi‖2a,ω,

‖xi‖2a,ω =∑n∈Z

‖ΦA(n, ω)xi‖2

e2(λin+a|n|) ,

is the random norm corresponding to 〈·, ·〉a,ω.(iii) For all i = 1, . . . , p, x ∈ Ei(ω), n ∈ Z

eλin−a|n|‖x‖a,ω ≤ ‖ΦA(n, ω)x‖a,θnω ≤ eλin+a|n|‖x‖a,ω.

Definition 3.6. The above scalar product 〈·, ·〉a,ω and random norm‖ · ‖a,ω and their relatives are called Lyapunov scalar product and Lyapunovnorm (of the linear cocycle ΦA, and with parameter a). The selection the-orem for multi-valued functions (see Deimling, 1985, Theorem 24) and theGram-Schmidt orthogonalization procedure assure the existence of a (measur-able) random orthonormal (with respect to the Lyapunov scalar product 〈·, ·〉a,ω

of ΦA) basis f1(ω), . . . , fd(ω) such that f1(ω), . . . , fd1(ω) belong to E1(ω),fd1+1(ω), . . . , fd1+d2(ω) belong to E2(ω), ..., fd−dp+1(ω), . . . , fd(ω) belong to


Ep(ω). We call such a random basis Lyapunov random basis of ΦA correspond-ing to 〈·, ·〉a,ω.

The following property of Lyapunov norms allows us to use them for solvingour classification problem (see Arnold, 1994, Corollary 3.9.9).

Lemma 3.7. A linear cocycle with respect to a Lyapunov norm is Lyapunovcohomologous to itself with respect to the standard Euclidean norm. (In otherwords, the identity map of Rd is a Lyapunov cohomology with respect to any pairof a Lyapunov norm and the standard Euclidean norm.) Hence, the Lyapunovspectrum of linear cocycles is invariant in the class of Lyapunov norms on Rd.

We shall use the result of Wanner (1992) on a Hartman-Grobman theoremfor discrete random dynamical systems and the idea of Robbin (1972) to provea theorem on structural stability of linear hyperbolic cocycles generated bycontracting (or expanding) bounded (together with its inverse) linear randommaps.

Definition 3.8. A linear cocycle ΦA on Rd equipped with a random norm‖ · ‖ω is called structurally stable (with respect to ‖ · ‖ω) if there exists a positivenumber ε such that for any linear random map B satisfying ‖B(ω)−A(ω)‖ω,θω ≤ε for all ω ∈ Ω the linear cocycle ΦB is conjugate to ΦA.

Remark 3.9. The above definition of structural stability depends cruciallyon the random norm ‖ · ‖ω.

Definition 3.10. We say that the linear cocycle ΦA on Rd equipped with arandom norm ‖ · ‖ω exhibits an exponential dichotomy (with respect to ‖ · ‖ω)if there exist positive numbers K > 0, α > 0 and a family of projections Pω onRd depending measurably on ω ∈ Ω such that


i) ‖ΦA(n, ω)PωΦ−1A (m,ω)‖θmω,θnω ≤ K exp(−α(n−m))

for all n ≥ m,ω ∈ Ω,

ii) ‖ΦA(n, ω)(id− Pω)Φ−1A (m,ω)‖θmω,θnω ≤ K exp(−α(n−m))

for all n ≤ m,ω ∈ Ω.

For the proof of our main theorem on structural stability of linear cocycleswe need the following result of Wanner (1992).

Proposition 3.11. Let ϕ(n, ω, x) := ΦA(n, ω)x+Ψ(n, ω, x) be a nonlinearcocycle on Rd equipped with a random norm ‖ · ‖ω. Suppose that the linear partΦA of ϕ exhibits an exponential dichotomy with respect to the given randomnorm with Pω = 0 or id and K = 1 and that the nonlinear part Ψ(n, ω, x)satisfies the following inequalities for some fixed positive numbers L, M:

‖Ψ(1, ω, x)−Ψ(1, ω, x′)‖θω ≤ L‖x− x′‖ω for all x, x′ ∈ Rd, ω ∈ Ω,

‖Ψ(1, ω, x)‖θω ≤ M for all x ∈ Rd, ω ∈ Ω.

Suppose further that the map ϕ(n, ω, ·) is invertible and continuous together withits inverse. Set

L∗ :=

1− e−α if Pω = id,eα − 1 if Pω = 0.

Then the condition L < L∗ implies that the nonlinear cocycle ϕ is conjugate toits linear part ΦA by a random homeomorphism h with h(ω)0 = 0, i.e.

ϕ(k, ω, ξ) = h(θkω)−1 ΦA(k, ω) h(ω)ξ for all k ∈ Z, ξ ∈ Rd.

Remark 3.12. One can find a more general version of the Hartman-Grobman theorem for a random dynamical system exhibiting an exponentialdichotomy in Wanner (1994).

Now we formulate and prove our theorem on structural stability of linearcocycles.

Theorem 3.13. Assume that ΦA exhibits an exponential dichotomy withrespect to a random norm ‖ · ‖ω on Rd with projections Pω equal to 0 or idand constant K equal to 1. Suppose further that there exists a positive numberM > 0 such that max‖A(ω)‖ω,θω, ‖A−1(ω)‖θω,ω ≤ M for all ω ∈ Ω. Thenthere exists a positive number ε > 0 depending only on the constant α (appearingin the definition of the exponential dichotomy of ΦA) and M such that for anylinear random map B satisfying ‖B(ω)−A(ω)‖ω,θω ≤ ε for all ω ∈ Ω the linearcocycle ΦB is conjugate to ΦA.

In other words, if A is contracting or expanding and is bounded together withits inverse then ΦA is structurally stable with respect to the given random norm.

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Proof. I. First, we prove our theorem for the case when A is a contractionwith coefficient e−α < 1, α > 0, i.e. ΦA exhibits an exponential dichotomy withPω = id, K = 1 and α > 0.

1. Let B be a random linear map. We introduce a function c(·, ·) : Ω×Rd →R by the following formula

c(ω, x) :=

1 if x ∈ Rd, ‖x‖ω ≤ 1,2− ‖x‖ω if x ∈ Rd, 1 < ‖x‖ω < 2,0 if x ∈ Rd, ‖x‖ω ≥ 2.

Then c(ω, x) is measurable on Ω, Lipschitz continuous on Rd with Lipschitzconstant 1 and the support of c(ω, ·) is the ball of radius 2 in Rd.

Introduce a nonlinear random map g(·)· : Ω× Rd → Rd by

g(ω)x := c(ω, x)B(ω)x+(1−c(ω, x))A(ω)x = A(ω)x+c(ω, x)(B(ω)x−A(ω)x).

Put ε1 := 12M > 0. It is easily seen that if ‖B(ω)−A(ω)‖ω,θω ≤ ε1, then g(ω) is

invertible, because the map id−u is invertible for any contraction u. Moreover,g(ω) and its inverse depend continuously on x. The random difference equationxk+1 = g(θkω)xk generates a nonlinear cocycle

ϕ(k, ω, ξ) :=

g(θk−1ω) . . . g(ω)ξ for k > 0,id for k = 0,g−1(θkω) . . . g−1(θ−1ω)ξ for k < 0.

We consider this nonlinear cocycle as a perturbation of ΦA by the followingdefinition of the perturbation (nonlinear part)

Ψ(k, ω, ξ) := ϕ(k, ω, ξ)− ΦA(k, ω)ξ.

Then

Ψ(1, ω, x) = ϕ(1, ω, x)−ΦA(1, ω)x = g(ω)x−A(ω)x = c(ω, x)(B(ω)x−A(ω)x).

Consequently,

‖Ψ(1, ω, x)‖θω ≤ 2‖B(ω)−A(ω)‖ω,θω for all x ∈ Rd,

and

‖Ψ(1, ω, x)−Ψ(1, ω, x′)‖θω

= ‖c(ω, x)(B(ω)−A(ω))x− c(ω, x′)(B(ω)−A(ω))x′‖θω

≤ 3‖B(ω)−A(ω)‖ω,θω‖x− x′‖ω.

Put ε2 := min 12M , 1

4 (1 − e−α) > 0. Then, by Proposition 3.11, in case‖B(ω)−A(ω)‖ω,θω ≤ ε2 for all ω ∈ Ω we can find a random homeomorphism hwith h(ω)0 = 0 providing a topological conjugacy between ϕ and its linear partΦA, i.e.

ϕ(k, ω, ξ) = h(θkω)−1 ΦA(k, ω) h(ω)ξ for all k ∈ Z, ξ ∈ Rd.


Therefore, from the definition of g we have

g(ω)x = h−1(θω) A(ω) h(ω)x for all x ∈ Rd,

B(ω)x = h−1(θω) A(ω) h(ω)x for all ‖x‖ω ≤ 1, x ∈ Rd. (9)

2. In this step we construct a random homeomorphism h which will, and weare going to prove that now, furnish a topological conjugacy between ΦB andΦA for B close to A.

Put ε3 := 12 (1− e−α) > 0 and set

j(k) := (e−α + ε3)k, k ∈ Z.

Then the function j(n), n ∈ Z, is positive and decreasing to zero as n tends to∞.From the assumptions of the theorem it follows that if ‖B(ω)−A(ω)‖ω,θω ≤ ε3

for all ω ∈ Ω then ϕ(k, ω, x) satisfies the following inequality

‖ϕ(k, ω, ξ)‖θkω ≤ j(k)‖ξ‖ω for all ξ ∈ Rd, k ∈ N. (10)

Now we fix a point ω ∈ Ω and construct the inverse h−1(ω) of h(ω) in thefollowing way:

Let η ∈ Rd. By the definition of h we have

h−1(θkω) ΦA(k, ω)η = h−1(θkω) ΦA(k, ω) h(ω) h−1(ω)η= ϕ(k, ω, h−1(ω)η),

which, by (10), implies

‖h−1(θkω) ΦA(k, ω)η‖θkω = ‖ϕ(k, ω, h−1(ω)η)‖θkω

≤ j(k)‖h−1(ω)η‖ωk→∞−→ 0. (11)

Therefore, there exists a number m(η, ω) ∈ N such that

‖h−1(θkω) ΦA(k, ω)η‖θkω ≤ 1 for all k ≥ m(η, ω).

For k ≥ m(η, ω) we set

h−1(ω)η := Φ−1B (k, ω) h−1(θkω) ΦA(k, ω)η. (12)

We show that the definition of h−1(ω) is independent of k ≥ m(η, ω). For that,we have

Φ−1B (k + 1, ω) h−1(θk+1ω) ΦA(k + 1, ω)η = Φ−1

B (k, ω) B−1(θkω) h−1(θk+1ω) A(θkω) h(θkω) h−1(θkω) ΦA(k, ω)η.

Consequently, by virtue of the inequality ‖h−1(θkω) ΦA(k, ω)η‖θkω ≤ 1 and(9), we have

Φ−1B (k + 1, ω) h−1(θk+1ω) ΦA(k + 1, ω)η

= Φ−1B (k, ω) B−1(θkω) B(θkω) h−1(θkω) ΦA(k, ω)η

= Φ−1B (k, ω) h−1(θkω) ΦA(k, ω)η.


This proves the independence of the definition of h−1(ω) of the choice of k ≥m(η, ω). Hence h−1(ω) is well-defined.

3. Now we prove that the above constructed map h−1(ω) is a homeomor-phism on Rd depending measurably on ω ∈ Ω.

a) Let η1 and η2 be different vectors of Rd. Take k > maxm(η1, ω),m(η2, ω).Then

h−1(ω)η1 = Φ−1B (k, ω) h−1(θkω) ΦA(k, ω)η1,

h−1(ω)η2 = Φ−1B (k, ω) h−1(θkω) ΦA(k, ω)η2.

Therefore h−1(ω)η1 6= h−1(ω)η2, because the maps in the right-hand sides arebijective. Consequently, h−1(ω) is injective.

b) Clearly, from the definition of h−1(ω) we have h−1(ω)0 = 0.c) From (11) and the definition of h−1(ω) it follows that m(η, ω) can be

chosen common for all η from a compact subset of Rd, hence we can choosem(η, ω) = mr(ω) for all ‖η‖ω ≤ r, r ∈ R+. This implies that the map h−1(ω)ηdepends continuously on η ∈ Rd, because all the maps in the formula definingh−1(ω) are continuous.

d) From the contracting property of A with coefficient e−α < 1 it follows thatany linear random map B satisfying ‖B(ω)−A(ω)‖ω,θω ≤ ε3 (= 1

2 (1−e−α) > 0)for all ω ∈ Ω is contracting. Therefore, for any ξ ∈ Rd there exists an integer ksuch that

‖ΦB(k, ω)ξ‖θkω < 1. (13)

Put ξ := ΦB(k, ω)ξ and η := Φ−1A (k, ω) h(θkω)ξ. Then, by (13) and the

definition of h−1(ω) we have ξ = Φ−1B (k, ω) h−1(θkω) ΦA(k, ω)η = h−1(ω)η.

Therefore, h−1(ω) is surjective.Combining the above results we obtain that the map h−1(ω) is bijective

continuous, and h−1(ω)0 = 0. This implies that its inverse h(ω) is well-definedand h(ω)0 = 0. Using the arguments of part c) we can easily prove that the maph(ω) is continuous, too, because the maps on the right-hand side of the formuladefining h−1(ω) are continuously invertible and we can assume that k is fixedfor a compact set of Rd. Moreover, by virtue of (11) and (12) it is easily seenthat h(ω)ξ and h−1(ω)η depend measurably on ω ∈ Ω on any compact subsetof Rd, hence h is a random homeomorphism.

4. We show that the random homeomorphism h constructed in Steps 2 and3 furnishes a topological conjugacy between ΦA and ΦB .

Fix ω ∈ Ω. Let η ∈ Rd be an arbitrary vector. By (12), fork > maxm(η, ω),m(A(ω)η, θω) we have

h−1(ω)η = Φ−1B (k + 1, ω) h−1(θk+1ω) ΦA(k + 1, ω)η,

and

B−1(ω) h−1(θω) A(ω)η= B−1(ω) Φ−1

B (k, θω) h−1(θk+1ω) ΦA(k, θω) A(ω)η

= Φ−1B (k + 1, ω) h−1(θk+1ω) ΦA(k + 1, ω)η = h−1(ω)η.


Consequently, B(ω) = h−1(θω) A(ω) h(ω). Hence, h furnishes a topologicalconjugacy between ΦA and ΦB .

Choosing ε = minε1, ε2, ε3 = ε2 = min 12M , 1

4 (1 − e−α) > 0 we haveTheorem 3.13 proved for the case when A is a contraction, i.e. ΦA exhibits anexponential dichotomy with Pω = id and K = 1.

II. For the case ΦA exhibits an exponential dichotomy with Pω = 0 andK = 1 our theorem follows from the use of the above arguments in the oppositedirection of time k → −∞.

4 Diagonalization of linear cocycles

In this section we study the problem of diagonalizing linear cocycles. It is knownthat for a linear cocycle with simple Lyapunov spectrum one can choose a ran-dom basis of vectors from the one-dimensional Oseledets’ spaces such that thegiven cocycle has diagonal form with respect to the chosen basis. Therefore, onecan try to diagonalize a cocycle by proving that it has a simple Lyapunov spec-trum. Many authors have investigated the class of linear cocycles with simpleLyapunov spectrum. We mention the work of Gol’dsheid and Margulis (1989)giving a criterion for the simplicity of the Lyapunov spectrum of products of in-dependent identically distributed matrices, the work of Virtser (1979) on theMarkov case. Knill (1992) has proved that the set of two-dimensional lin-ear cocycles over an aperiodic ergodic dynamical system on a Lebesgue spacewhich have simple Lyapunov spectrum is dense in the space of all boundedlinear two-dimensional cocycles equipped with L∞-topology. Thus, in any L∞-neighborhood of a two-dimensional bounded linear cocycle one can find a linearcocycle with simple Lyapunov spectrum, and which, therefore, is diagonalizable.

To classify linear hyperbolic cocycles we can approach to the problem ofdiagonalization of linear cocycles from another point of view. Namely, we shalldirectly construct a diagonal cocycle conjugate to the given hyperbolic linearcocycle. The constructed diagonal cocycle is, in general, L∞-far away from theoriginal cocycle.

Choose and fix a nonrandom basis f1, . . . , fd of Rd which is orthonormalwith respect to the standard Euclidean scalar product of Rd. A linear randommap A is called diagonal if it is represented by a diagonal matrix with respectto the chosen basis. A linear cocycle generated by a diagonal linear randommap is called diagonal. We emphasize that linear random maps and linearcocycles are defined as linear operators on Rd, hence they do not depend on thechoice of a basis on Rd. Furthermore, as a (possibly random) basis of Rd waschosen it makes sense to speak of the matrix representations of linear randommaps and linear cocycles. We note that, although the definition of diagonalcocycles depends on the choice of the basis f1, . . . , fd, for our problem oftopological classification the choice of (possibly random) basis is not essentialas the following elementary lemma shows.

Let g1(ω), . . . , gd(ω) and g′1(ω), . . . , g′d(ω) be two random bases of Rd.Denote by L the basis change from g′1(ω), . . . , g′d(ω) to g1(ω), . . . , gd(ω), i.e.

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L : Ω → Gl(d, R) with L(ω)g′i(ω) = gi(ω) for all i = 1, . . . , d, ω ∈ Ω. It is easilyseen that L is a linear random homeomorphism.

Lemma 4.1. 1. Let A be a linear random map. ThenB(ω) := L−1(θω)A(ω)L(ω) is a linear random map, and the matrix representa-tion of B with respect to the random basis g′1(ω), . . . , g′d(ω) coincides with thematrix representation of A with respect to the random basis g1(ω), . . . , gd(ω).Conversely, if B is the linear random map which has its matrix representationwith respect to the random basis g′1(ω), . . . , g′d(ω) coinciding with the matrixrepresentation of A with respect to the random basis g1(ω), . . . , gd(ω), thenΦB is conjugate to ΦA by the linear random homeomorphism L.2. If g1(ω), . . . , gd(ω) and g′1(ω), . . . , g′d(ω) are two Lyapunov random basesof two (arbitrary) linear cocycles, in particular they can be nonrandom basesorthonormal with respect to the standard Euclidean scalar product of Rd, thenL is a Lyapunov cohomology.

Proof. The first part of the lemma is elementary and the second part isa direct corollary of Lemma 3.7 and the fact that Lyapunov random bases, bytheir definition, are orthonormal with respect to the corresponding Lyapunovscalar product.

Next we show that the relation of topological conjugateness respects theoperation of taking direct sums. Let A and B be two linear random maps.Assume that we have the decomposition

Rd = E ⊕ F, (14)

where E is a r-dimensional linear subspace of Rd and F is a (d− r)-dimensionallinear subspace of Rd, such that E = A(ω)E = B(ω)E and F = A(ω)F =B(ω)F for all ω from an invariant set Ω of full P-measure. Denote by A1, A2

and B1, B2 the restrictions of A and B to E,F , respectively. Then A1, B1 andA2, B2 are r- and (d− r)-dimensional linear random maps, respectively. In thiscase we say A and B are direct sums of A1, A2 and B1, B2, respectively.

Lemma 4.2. If ΦA1 is conjugate to ΦB1 and ΦA2 is conjugate to ΦB2 ,then ΦA is conjugate to ΦB.

Proof. Denote by hi the random homeomorphisms providing topologicalconjugacies between ΦAi

and ΦBi, i = 1, 2. Let ω ∈ Ω be arbitrary. By (14)

every x ∈ Rd is decomposed uniquely in x = x1 + x2 with x1 ∈ E, x2 ∈ F .Define a mapping from Rd into itself:

h(ω)x := h1(ω)x1 + h2(ω)x2.

It is easily seen that h(·) is a random homeomorphism of Rd furnishing a topo-logical cojugacy between ΦA and ΦB .


Theorem 4.3. Every linear hyperbolic cocycle is conjugate to a linearhyperbolic diagonal cocycle.

Proof. 1. Let ΦA be a linear hyperbolic cocycle. Since the assertionsof the MET for ΦA hold on an invariat set of full P-measure we can assume,without loss of generality, that that set coincides with Ω. Denote by a(ω) thematrix representation of A(ω) with respect to the chosen basis f1, . . . , fd.Then a(ω) is a measurable map from the probability space (Ω,F , P) into thespace of nonsingular (d×d)-matrices equipped with the metric of operator normof matrices.

2. Let α be a positive number. The exact value of α will be specified later.Denote by λ1, . . . , λp the Lyapunov exponents of ΦA and by d1, . . . , dp theirmultiplicities. Take a Lyapunov random basis g1(ω), . . . , gd(ω) correspondingto the Lyapunov scalar product 〈·, ·〉α,ω of ΦA. Denote by a1(ω) the matrixrepresentation of A(ω) with respect to the basis g1(ω), . . . , gd(ω). By virtueof Proposition 3.5 a1(ω) has quasi-diagonal form:

a1(ω) = diaga11(ω), . . . , a1p(ω),

where a1k(ω) are (dk × dk)-matrices, k = 1, . . . , p, and

eλk−α‖x‖ ≤ ‖a1k(ω)x‖ ≤ eλk+α‖x‖, for all x ∈ Rdk . (15)

Recall that ‖ · ‖ denotes the standard Euclidean norm of Rd. Denote by A1(ω)the linear random map which has matrix representation a1(ω) with respect tothe basis f1, . . . , fd. Then, by Lemma 4.1, ΦA is linearly conjugate to ΦA1

3. Consider one of the matrices a11(ω), . . . , a1p(ω), say a11(ω). For simplicityof notation we denote it by a2(ω), the corresponding to it Lyapunov exponentλ1 by λ and the multiplicity d1 by m. Consider the corresponding subspaceRm, where Rm is the linear subspace of Rd spanned by the vectors f1, . . . , fm.Denote by A2 the restriction of A1 to Rm. Then A2 is a m-dimensional linearrandom map which has a2(ω) as its matrix representation with respect to thebasis f1, . . . , fm of Rm. Furthermore, a2(ω) satisfies

eλ−α‖x‖ ≤ ‖a2(ω)x‖ ≤ eλ+α‖x‖, for all x ∈ Rm, ω ∈ Ω.

According to the theorem on polar decomposition (see Gantmacher, 1977,p. 286) every nonsingular matrix C can be decomposed into a product C = SQ,where S =

√CCT = g(CCT ) is a positive definite symmetric matrix, g(·) is

a real polynomial, Q is an orthogonal matrix. (Here CT denotes the trans-posed matrix of C.) Therefore, matrix S is defined uniquely by C and dependscontinuously on C, and hence the same is valid for Q = S−1C (the continu-ity is understood as the continuity in the topology generated by the operatornorm of matrices). The symmetric matrix S is similar to a diagonal matrixS = Q−1

1 DQ1, where D = diags1, . . . , sm with s1 ≤ . . . ≤ sm being the eigen-values of S, which coincide with the positive square roots of the eigenvalues ofCCT (they are called the singular values of C). Therefore, by the theorem on


the continuity of eigenvalues (see Kato, 1976, pp. 107–109) D depends contin-uously on S. Consequently, any nonsingular matrix C can be decomposed intothe product C = Q−1

1 DQ1Q, where Q,Q1 ∈ O(m, R) with O(m, R) denotingthe group of orthogonal (m×m)-matrices, D = diags1, . . . , sm. The numberss1, . . . , sm are the positive square roots of the eigenvalues of CCT . The matricesQ,D depend continuously on C.

Applying the above arguments to the matrix a2(ω) we obtain the decompo-sition

a2(ω) = Q−11 (ω)D(ω)Q1(ω)Q(ω),

where Q(ω), Q1(ω) ∈ O(m, R), D(ω) = diags1(ω), . . . , sm(ω) with s1(ω) ≤. . . ≤ sm(ω). The matrices Q(ω), D(ω) depend continuously on a2(ω), thereforethey depend measurably on ω ∈ Ω. By virtue of (15) and the definition of thenumbers s1(ω), . . . , sm(ω) we have

eλ−α ≤ ‖(a2(ω))−1‖−1 = s1(ω) ≤ sm(ω) = ‖a2(ω)‖ ≤ eλ+α. (16)

4. Puta3(ω) := D1(ω)Q(ω),

where D1(ω) := eλI with I denoting the unit (m × m)-matrix, and denoteby A3(ω) the linear random map on Rm which has a3(ω) as its matrix rep-resentation with respect to the basis f1, . . . , fm. Then ΦA3 exhibits an ex-ponential dichotomy with respect to the standard Euclidean norm of Rm withPω = 0 or id (depending on the sign of λ) and K = 1. Furthermore, ‖A3(ω)‖ =eλ, ‖A−1

3 (ω)‖ = e−λ. Therefore, by virtue of Theorem 3.13 there exists a posi-tive number ε1 > 0 depending only on λ such that for any linear random mapB (on Rm) satisfying ‖B(ω)−A3(ω)‖ < ε1 for all ω ∈ Ω we have ΦB conjugateto ΦA3 . Choosing suitably small α we obtain that ΦA2 is conjugate to ΦA3 ,because due to (16)

‖A2(ω)−A3(ω)‖ = ‖a2(ω)− a3(ω)‖ == ‖Q−1

1 (ω)(D(ω)−D1(ω))Q1(ω)Q(ω)‖ ≤ eλ(eα − 1).

Thus, ΦA2 and ΦA3 are conjugate if α < log(1 + ε1e−λ).

5. By the theorem on the canonical form of an orthogonal matrix (seeGantmacher, 1977, p. 285) any orthogonal (m×m)-matrix R with det R = 1 isorthogonally similar to a canonical orthogonal matrix

R = R1diag(

cos ϕ1 sinϕ1

− sinϕ1 cos ϕ1

),

(cos ϕ2 sinϕ2

− sinϕ2 cos ϕ2

), . . . ,

. . .

(cos ϕq sinϕq

− sinϕq cos ϕq

), 1, . . . , 1

R−1

1 ,

where R1 ∈ O(m, R) and the numbers cos ϕ1 ± i sinϕ1, . . . , cos ϕq ± i sinϕq,1, . . . , 1 are the eigenvalues of R, and hence they depend continuously on R. Atheorem on the structure of the transforming matrix (see Gantmacher, 1977, p.148, supplement to Theorem 7) assures us the choice of the transforming matrix


R1 in continuous dependence of R (R1 = P (R), where P (·) is a polynomial).Therefore, we can assume that the transforming matrix R1 depends continuouslyon R.

PutD2(ω) := diagdet Q(ω), 1, . . . , 1.

Then D2 is diagonal, orthogonal and measurable. Set

Q(ω) := D2(ω)Q(ω).

Then Q(ω) is orthogonal and det Q(ω) = 1 for all ω ∈ Ω. Applying the canonicaldecomposition to the matrices Q(ω), ω ∈ Ω, we have

Q(ω) = Q2(ω)diag(

cos ϕ1(ω) sinϕ1(ω)− sinϕ1(ω) cos ϕ1(ω)

),

(cos ϕ2(ω) sinϕ2(ω)− sinϕ2(ω) cos ϕ2(ω)

),

. . . ,

(cos ϕq(ω) sinϕq(ω)− sinϕq(ω) cos ϕq(ω)

), 1, . . . , 1

Q−1

2 (ω),

where the functions ϕ1(ω), . . . ϕq(ω) are measurable functions from Ω into thehalf-open interval [0, 2π) and the random orthogonal matrix Q2(·) is a measur-able map from Ω into O(m, R). We introduce a family of orthogonal matrices

Q(β, ω) := Q2(ω)diag(

cos βϕ1(ω) sinβϕ1(ω)− sinβϕ1(ω) cos βϕ1(ω)

),(

cos βϕ2(ω) sinβϕ2(ω)− sinβϕ2(ω) cos βϕ2(ω)

), . . .

. . . ,

(cos βϕq(ω) sinβϕq(ω)− sinβϕq(ω) cos βϕq(ω)

), 1, . . . , 1

Q−1

2 (ω),

where β ∈ [0, 1] is a parameter. For each β ∈ [0, 1] the matrix Q(β, ω) dependsmeasurably on ω ∈ Ω, and for each ω ∈ Ω it depends continuously on β ∈ [0, 1].Put

a4(β, ω) := D1(ω)D2(ω)Q(β, ω),

and denote by A4(β, ω) the linear random map on Rm which has matrix rep-resentation a4(β, ω) with respect to the basis f1, . . . , fm. Then for everyβ ∈ [0, 1] the linear cocycle ΦA4(β,·) on Rm exhibits an exponential dichotomywith respect to the standard Euclidean norm with Pω = 0 or id (depending onthe sign of λ) and constant K = 1, because

‖a4(β, ω)x‖ = eλ‖x‖ for all x ∈ Rm.

Therefore, all these cocycles are structurally stable. It is easily seen that‖a4(β, ω) − a4(β1, ω)‖ ≤ 8πeλ|β − β1| for all ω ∈ Ω and β, β1 ∈ [0, 1]. Usingcontinuous induction we obtain that all the cocycles of this family are conjugateto each other: the set of β ∈ [0, 1] such that ΦA4(β,·) is conjugate to ΦA4(0,·)


is non-void and both open and closed in [0, 1] due to the structural stability ofΦA4(β,·) and the above estimation, hence it coincides with [0, 1]. We have

a4(1, ω) = a3(ω),a4(0, ω) = D1(ω)D2(ω).

So, A4(1, ·) = A3(·) and A4(0, ·) is a diagonal linear random map on Rm. There-fore, ΦA3 is conjugate to the linear diagonal cocycle ΦA4(0,·).

6. Now, it is easily seen that we can choose the positive number α introducedin the beginning of Step 2 as follows

α :=12

minlog(1 + ε1e−λ1), log(1 + ε2e

−λ2), . . . , log(1 + εpe−λp) > 0.

Here, by virtue of Theorem 3.13, each constant εi depends only on the value λi,i = 1, . . . , p; more precisely, the value εi can be chosen equal to min 1

2e|λi|, 14 (1−

eλi) in case λi < 0 and min 12e|λi|, 1

4 (eλi − 1) in case λi > 0, i = 1, . . . , p.Fixing the so chosen value α and applying the arguments of Steps 3–5 separatelyto each block of the quasi-diagonal matrix a1(ω), by virtue of Lemma 4.2, weobtain that ΦA1 is conjugate to a diagonal linear cocycle. Consequently, ΦA isconjugate to a diagonal linear cocycle.

Remark 4.4. It is known that every linear cocycle is Lyapunov cohomol-ogous to a triangular one, possibly after enlarging the probability space (seeOseledets, 1968). However, in general, a linear cocycle is not Lyapunov coho-mologous to a diagonal one. Therefore, Theorem 4.3 gives us a nice property oflinear hyperbolic cocycles which can be achieved, in general, only by nonlinearconjugacy.

5 Classification of one-dimensional linear hyper-bolic cocycles

In this section we classify one-dimensional linear hyperbolic cocycles. Chooseand fix a unit vector f ∈ R (here d = 1). Let A be a one-dimensional linearrandom map. Denote by aS(ω) the matrix representation of A(ω) with respectto the basis f of R, and by λA the Lyapunov exponent of ΦA. Then aS(ω) isa non-vanishing real number. Consider the sign of aS(ω):

sign aS(ω) := aS(ω)‖aS(ω)‖−1.

Denote by A the linear random map which has the following matrix represen-tation with respect to the basis f:

aS(ω) := eλAsignaS(ω).

Theorem 5.1. If ΦA is hyperbolic, then ΦA is conjugate to ΦA.


Proof. Let ε > 0 be arbitrary. Choose and fix a Lyapunov norm ‖ · ‖ω ofΦA such that A has its operator norm with respect to the Lyapunov norm ‖.‖ω

ε-close to eλA (this can be achieved by Proposition 3.5):

| ‖A(ω)‖ω,θω − eλA | < ε. (17)

Put v(ω) := ‖f‖−1ω f for ω ∈ Ω. This is a Lyapunov random basis corresponding

to the above Lyapunov norm of ΦA. Denote by A1 the linear random map whichhas aS(ω) as its matrix representation with respect to the random basis v(ω)of R. By Lemma 4.1 ΦA1 is conjugate to ΦA. Furthermore, by Theorem 3.13 andthe definition of ΦA1 there exists ε1 > 0 depending only on λA such that for anylinear random map B satisfying ‖A1(ω)−B(ω)‖ω,θω < ε for all ω ∈ Ω, the linearcocycle ΦB is conjugate to ΦA1 . Denote by a2(ω) the matrix representation ofthe map A(ω) with respect to the random basis v(ω). From the choice ofv(ω) it follows that sign a2(ω) = sign aS(ω) for all ω ∈ Ω. Consequently,

sign a2(ω) = sign aS(ω) = sign aS(ω).

From (17) it follows that| |a2(ω)| − eλA | < ε.

Hence, |a2(ω)− aS(ω)| < ε, which yields

‖A(ω)−A1(ω)‖ω,θω < ε for all ω ∈ Ω.

By the above proved structural stability of ΦA1 and ε-closeness of A and A1 wehave ΦA conjugate to ΦA1 . This, in turn, implies that ΦA is conjugate to ΦA.

Corollary 5.2. Let ΦA and ΦB be two one-dimensional linear hyperboliccocycles. If their Lyapunov exponents have the same sign and the matrix rep-resentations of the maps A and B with respect to the basis f of R have thesame sign for almost all ω ∈ Ω, then ΦA is conjugate to ΦB.

Proof. By Theorem 5.1 ΦA and ΦB are conjugate to ΦA and ΦB , whereA and B have eλAsign aS(ω) and eλB sign bS(ω) as their matrix representationswith respect to the basis f, respectively. From our assumption it follows thatΦA and ΦB are conjugate by the random homeomorphism h defined by

h(ω)xf = x|x|βf for all x ∈ R, ω ∈ Ω,

where β = λA

λB− 1. Hence, ΦA and ΦB are conjugate.

Theorem 2.8 of Section 2 gives a necessary orientation condition for topo-logical conjugacy of linear cocycles. We show that in the one-dimensional caseit is also a sufficient condition. Choose and fixed the orientation on R suchthat the basis f is positively oriented. Recall that CAB denotes the set of allω ∈ Ω such that deg A(ω) deg B(ω) = −1. Here we have deg A(ω) = signaS(ω),


deg B(ω) = signbS(ω). We recall also that the return time function of a setE ∈ F is kE(ω) = minn ≥ 1| θnω ∈ E.

Proposition 5.3. Two one-dimensional linear hyperbolic cocycles ΦA andΦB are conjugate if and only if the following conditions hold:

• signλA = signλB, (18)• their set CAB is a coboundary. (19)

Proof. If ΦA and ΦB are conjugate, then by Theorems 2.1 and 2.8 theconditions (18) and (19) hold.

Suppose that (18) and (19) hold. If P(CAB) = 0 then ΦA and ΦB areconjugate by Corollary 5.2. Assume that P(CAB) > 0. By Lemma 2.3 CAB canbe decomposed into two disjoint sets C1, C−1 such that almost surely

ω ∈ Cj =⇒ kCj (ω) > kC−j (ω), j = ±1,

P(C1) = P(C−1) =12

P(CAB).

We construct a measurable function l(·) : Ω → 1,−1 as follows:

l(ω) :=

1 if ω ∈ C1,−1 if ω ∈ C−1,j if ω ∈ Ω\CAB and kCj (ω) < kC−j (ω) (j = ±1),1 for all remaining ω ∈ Ω.

Seth1(ω) := l(ω) id.

Then h1 is a random linear homeomorphism furnishing a topological conjugacybetween ΦB and ΦB , where

B(ω) := h−11 (θω) B(ω) h1(ω) = l(θω)l(ω)B(ω).

Denoting by bS(ω) the matrix representation of B(ω) with respect to the basisf of R, we have sign bS(ω) = l(θω)l(ω)sign bS(ω). We show that for almostall ω ∈ Ω

sign bS(ω) = sign aS(ω).

For ω ∈ Cj , by virtue of Lemma 2.3, we have kCj (ω) > kC−j (ω). This impliesl(θω) = −j. Hence, l(θω)l(ω) = −1. Therefore, sign bS(ω) = −sign bS(ω) =sign aS(ω). For ω ∈ Ω\CAB we have l(θω) = l(ω), which implies sign bS(ω) =sign bS(ω) = sign aS(ω).

By the definition of B we have λB = λB , hence signλB = signλA. Therefore,by Corollary 5.2, ΦB is conjugate to ΦA. This implies that ΦB and ΦA areconjugate.


6 Classification of linear hyperbolic diagonal co-cycles

In this section we classify linear hyperbolic diagonal cocycles. For our purposewe need further properties of coboundaries.

Lemma 6.1. 1) For every E ∈ F with P(E) < 1 there exists a coboundaryF such that E ⊂ F ;2) Assume that (Ω,F , P) is non-atomic. Then for every E ∈ F with P(E) > 0there exists a coboundary G such that G ⊂ E.

Proof. 1. If E is a coboundary, then we choose F = E. Suppose E is nota coboundary, then 0 < P(E) < 1. Set

Cn :=

(n−1⋂i=0

θ−iE

)\(θE ∪ θ−nE), n = 1, 2, . . .

Then Cn is the set of all points ω ∈ E such that θ−1ω /∈ E and kΩ\E(ω) = n.It is easily seen that all θiCn (i = −1, 0, 1, . . . , n− 1; n = 1, 2, . . .) are disjointand

θ−1Cn ⊂ Ω\E, n ∈ N,

E =∞⋃

n=1

n−1⋃i=0

θiCn, (mod 0).

Put

D :=

( ∞⋃n=1

n−1⋃i=0

θiCn

)⋃( ∞⋃m=1

θ−1C2m−1

).

Then D is a coboundary because

D = C4θC,

where

C :=

( ∞⋃m=1

m−1⋃i=0

θ2iC2m

)⋃( ∞⋃m=1

m−1⋃i=0

θ2i−1C2m−1

).

From the construction of D it follows that P(E\D) = 0. We set F := E ∪D, soE ⊆ F and F is a coboundary because F = D (mod 0).

2. If P(E) = 1, then we take a measurable set G1 with 0 < P(G1) < 1(such a set G1 exists because (Ω,F , P) is non-atomic by assumption). Set G :=G14θG1. Then G is a coboundary, and P(G) > 0 by virtue of the ergodicity ofθ. Therefore, G ∩ E ⊂ E is a coboundary with positive P-measure.

Let 0 < P(E) < 1. Then θE , FE and PE are well-defined. Take a set G2 ∈FE with PE(G2) > 0 such that G2 is a coboundary for θE on (E,FE , PE) as wehave done above for the case P(E) = 1. Then, by Proposition 2.7, G2 is also


a coboundary for θ on (Ω,F , P). Moreover, we have P(G2) = PE(G2)P(E) > 0and G2 ⊂ E.

Lemma 6.2. Assume that (Ω,F , P) is non-atomic. If E ∈ F has P(E) > 0,then for any Z ∈ F there exists a measurable set Y ⊂ E such that the set Z4Yis a coboundary.

Proof. Let E ∈ F with P(E) > 0 and Z ∈ F be arbitrary.If P(Z) = 0 then we choose Y := ∅.Assume that P(Z) > 0. Put D := E ∪Z. Clearly, P(D) > 0, so θD, FD and

PD are well-defined.If P(E\Z) > 0, then by Lemma 6.1 there exists a coboundary F ∈ FD (with

respect to θD) such that F ⊃ Z. By Proposition 2.7 F is also a coboundarywith respect to θ. Set Y := F\Z. Then Y ⊂ E and Y4Z = F is a coboundarywith respect to θ.

Suppose that P(E\Z) = 0. By Lemma 6.1 there exists a coboundary (withrespect to θ) G ⊂ E with P(G) > 0. From P(G) > 0 it follows that PD(Z4G) =PD(Z\G) < 1. Therefore, by Lemma 6.1 there exists a coboundary (with respectto θD) F ∈ FD such that F ⊃ Z4G (= Z\G). By Proposition 2.7 F is alsoa coboundary with respect to θ. Set Y := G∩F . Then Y4Z = G4F (= Z\Y )is a coboundary (with respect to θ) and Y ⊂ E.

Now we are going to classify linear hyperbolic diagonal cocycles. Recall thatwe have chosen and fixed a basis f1, . . . , fd of Rd, which is orthonormal withrespect to the standard Euclidean scalar product of Rd, and a linear random mapis called diagonal if its matrix representation with respect to the chosen basisis diagonal. In the following proposition, speaking of matrix representations oflinear operators of Rd we mean their matrix representations with respect to thechosen basis f1, . . . , fd of Rd.

Proposition 6.3. Assume that (Ω,F , P) is non-atomic. Two linear hy-perbolic diagonal cocycles ΦA and ΦB are conjugate if and only if the followingconditions hold:


B(ω) (hence dim EuA(ω) = dimEu

B(ω)), (20)• their sets Cs,u

AB are coboundaries. (21)

Proof. If d = 1 then Proposition 6.3 is equivalent to Proposition 5.3.Assume that d ≥ 2.

The “only if” part is an immediate corollary of Theorems 2.1 and 2.9.Suppose that the conditions (20) and (21) hold. Denote by a(ω) and b(ω) the

matrix representations of the linear random maps A(ω) and B(ω), respectively.By the assumption of the theorem a(ω) and b(ω) have diagonal form

a(ω) = diaga1(ω), a2(ω), . . . , ad(ω),b(ω) = diagb1(ω), b2(ω), . . . , bd(ω).


Now in several steps we show that ΦA is conjugate to ΦB .1. Denote by λi the Lyapunov exponent of the one-dimensional cocycle

generated by the one-dimensional random linear map, which has ai(ω) as itsmatrix representation with respect to the basis fi of the linear one-dimensionalsubspaces of Rd spanned by fi (i = 1, 2, . . . , d). Then the Lyapunov spectrumof ΦA consists of the numbers λ1, . . . , λd, where every Lyapunov exponent isrepeated as many times in this set as its multiplicity. For definiteness we assume

λ1 ≥ λ2 ≥ . . . ≥ λr > 0 > λr+1 ≥ . . . ≥ λd,

where r = dim EuA. Any other order of the λi (i = 1, . . . , d) can be reduced to

this case by a coordinate change which is a linear conjugacy.2. Set

a(ω) := diag3l11(ω), 3l12(ω), . . . , 3l1r(ω), 3−1l1r+1(ω), . . . , 3−1l1d(ω),

where l1i (ω) := sign ai(ω) (i = 1, . . . , d). Denote by A(ω) the random linearmap which has a(ω) as its matrix representation. Applying Proposition 5.3separately to each diagonal element of the matrices a(ω) and a(ω), by virtue ofLemma 4.2, we obtain that ΦA is conjugate to ΦA.

3. Suppose that r ≥ 2, so λ1 ≥ λ2 > 0.a) Set

E1 := ω ∈ Ω| l11(ω) = −1,E2 := ω ∈ Ω| l12(ω) = −1.

Choose and fix two arbitrary disjoint measurable sets E3, E4 with P(E3) >0, P(E4) > 0. By Lemma 6.2 there exist measurable sets Y1 ⊂ E3, Y2 ⊂ E4

such that the sets Y14E1, Y24E2 are coboundaries. Set

l21(ω) := 1− 2 · 1Y1(ω),l22(ω) := 1− 2 · 1Y2(ω),

where 1Y (·) denotes the characteristic function of the set Y in the probabilityspace (Ω,F , P). Then l21(·), l22(·) take values in the set +1,−1 and theyare equal to −1 only on the sets Y1, Y2, respectively. From the constructionof Y1, Y2, by virtue of Proposition 5.3, the one-dimensional linear cocyclesgenerated by the one-dimensional linear maps with matrix representation 3l11(ω)and 3l12(ω) are conjugate to the one-dimensional linear cocycles generated bythe one-dimensional linear maps with matrix representations 3l21(ω) and 3l22(ω),respectively. Therefore, by Lemma 4.2, ΦA is conjugate to the linear cocycleΦA′ generated by the linear random map A′(·) which has the following matrixrepresentation:

a′(ω) := diag3l21(ω), 3l22(ω), . . . , 3l2r(ω), 3−1l2r+1(ω), . . . , 3−1l2d(ω),

where l2i (ω) := l1i (ω) (i = 3, . . . , d).


b) Denote by R2 the linear subspace of Rd spanned by the first two basisvectors f1 and f2. Consider the restrictions of our linear random maps andlinear cocycles to R2.

Introduce a family of linear random maps A1(β, ω) (β ∈ [0, 1]) on R2 bythe following formula of their matrix representations with respect to the basisf1, f2 of R2.

a1(β, ω) := R−1(β, ω) diag3l21(ω), 3l22(ω) R(β, ω),

where

R(β, ω) :=(

cos(1E4(ω)βπ2 ) sin(1E4(ω)βπ

2 )− sin(1E4(ω)βπ

2 ) cos(1E4(ω)βπ2 )

).

According to this construction, for any β ∈ [0, 1], n ∈ Z, ω ∈ Ω, the map3−nΦA1(β,·)(n, ω) is an orthogonal transformation on R2. Hence, for any β ∈[0, 1] the linear cocycle ΦA1(β,·) has an exponential dichotomy with Pω = 0, K =1 and α = log 3. Therefore, by virtue of Theorem 3.13, all the linear cocycleΦn(A1(β, ·), ω), β ∈ [0, 1], are structurally stable. Furthermore, we note that‖A1(β, ω)−A1(β1, ω)‖ ≤ 12π|β−β1| for all ω ∈ Ω, β, β1 ∈ [0, 1]. Consequently,the set of all β ∈ [0, 1] such that ΦA1(β,·) is conjugate to ΦA1(0,·) is non-voidand both open and closed in [0, 1], hence coincides with [0, 1]. Thus all cocyclesΦA1(β,·), β ∈ [0, 1], are conjugate to each other, in particular ΦA1(0,·) is conjugateto ΦA1(1,·). Clearly, ΦA1(0,·) coincides with the restriction of ΦA′ to R2. The two-dimensional linear random map A1(1, ·) is diagonal and its matrix representationwith respect to the basis f1, f2 is

a1(1, ω) = R−1(1, ω) diag3l21(ω), 3l22(ω) R(1, ω) == diag3l22(ω), 3l21(ω)1E4(ω) +

+(1− 1E4(ω))diag3l21(ω), 3l22(ω) == diag3l22(ω)1E4(ω) + 3(1− 1E4(ω))l21(ω),

3l21(ω)1E4(ω) + 3(1− 1E4(ω))l22(ω).

Denote by A2(ω) the linear random map which has a2(ω) as its matrix repre-sentation, where

a2(ω) := diag3l31(ω), 3l32(ω), . . . , 3l3r(ω), 3−1l3r+1(ω), . . . , 3−1l3d(ω),l31(ω) := l22(ω)1E4(ω) + (1− 1E4(ω))l21(ω),l32(ω) := l21(ω)1E4(ω) + (1− 1E4(ω))l22(ω),l3i (ω) := l1i (ω) (i = 3, . . . , d).

By virtue of the construction of E1, E2, E3, E4 and the definition of l21, l22, thefunction l32 is equal to 1 on Ω.

By the above arguments and Lemma 4.2 ΦA′ is conjugate to ΦA2 . Hence,ΦA is conjugate to ΦA2 .

4. Repeating the arguments of Step 3 for the pairs of indices (1, 3), . . . , (1, r)we deduce that ΦA is conjugate to the linear cocycle ΦA3 generated by a linear


random map A3(·) which has the following matrix representation:

a3(ω) := diag3l41(ω), 3l42(ω), . . . , 3l4r(ω), 3−1l4r+1(ω), . . . , 3−1l4d(ω),

where

l4i (ω) ∈ +1,−1, i ∈ 1, . . . , d, ω ∈ Ω,

l4j (ω) = 1 j ∈ 2, . . . , r, ω ∈ Ω.

5. Applying arguments similar to those in Steps 3 and 4 to the indicesr+1, . . . , d (corresponding to the negative Lyapunov exponents of ΦA) we deducethat ΦA is conjugate to the linear cocycle ΦA4 generated by the linear randommap A4(·) which has the following matrix representation:

a4(ω) := diag3l51(ω), 3l52(ω), . . . , 3l5r(ω), 3−1l5r+1(ω), . . . , 3−1l5d(ω),

where

l5i (ω) ∈ +1,−1, for i ∈ 1, . . . , d, ω ∈ Ω,

l5j (ω) = 1 for j ∈ 2, . . . , r, r + 1, . . . , d− 1, ω ∈ Ω.

6. Applying the arguments of Steps 1–5 to the linear cocycle ΦB (recallingthat r = dim Eu

A(ω) = dim EuB(ω) is independent of ω ∈ Ω due to the ergodicity

of θ) we have ΦB conjugate to the linear cocycle ΦB4 generated by the linearrandom map B4(·) which has the following matrix representation:

b4(ω) := diag3m51(ω), 3m5

2(ω), . . . , 3m5r(ω), 3−1m5

r+1(ω), . . . , 3−1m5d(ω),

where

m5i (ω) ∈ +1,−1, for i ∈ 1, . . . , d, ω ∈ Ω,

m5j (ω) = 1 for j ∈ 2, . . . , r, r + 1, . . . , d− 1, ω ∈ Ω.

7. By virtue of Theorems 2.1 and 2.9 and the fact that coboundaries forma subgroup of the Abelian group of measurable sets of Ω with respect to theoperation 4 the conditions (20) and (21) hold also for the random linear mapsA4 and B4 (with the corresponding change from A and B to A4 and B4 in theirformulation).

8. By virtue of the result of Step 7, Proposition 5.3 and due to the con-struction of A4 and B4 the one-dimensional linear cocycles generated by thelinear random maps having matrix representations 3l51(ω) and 3m5

1(ω) in theone-dimensional linear subspace of Rd spanned by the vector f1 are conju-gate. The one-dimensional linear cocycles generated by the linear random mapshaving matrix representations 3−1l5d(ω) and 3−1m5

d(ω) in the one-dimensionallinear subspace of Rd spanned by the vector fd are also conjugate. There-fore, by Lemma 4.2, ΦA4 is conjugate to ΦB4 because l5j (ω) = 1 = m5

j (ω) forj ∈ 2, . . . , r, r + 1, . . . , d− 1. Consequently, ΦA and ΦB are conjugate.


7 Main result

In this section we present the main result of the paper — Theorem 7.1, and givesome examples showing the dependence of the number of topological classes oflinear hyperbolic cocycles on the ergodic properties of the underlying dynamicalsystems (θ, Ω,F , P). The main work toward the proof of theorem 7.1 was donein previous sections. It remains only the case of atomic (Ω,F , P) which wasexcluded in Section 6. That case is a direct generalization of the deterministictheorem 1.1 and its proof relies on that one. Furthermore, its treatment is of thenature different than the rest of the paper, so we shall deal with it separatelyin Proposition 7.2. We would like to emphasize once again that although forthe definition of the sets Cs,u

AB we had to choose random orientations on EsA(ω),

EuA(ω), Es

B(ω) and EuB(ω), whether they are coboundaries is independent of the

choice of those random orientations. Therefore, the statement of Theorem 7.1is independent of the choice of random orientations.

Theorem 7.1. Assume that the dynamical system (θ, Ω) is ergodic. Thentwo linear hyperbolic cocycles ΦA and ΦB are conjugate if and only if the fol-lowing conditions hold:



B(ω)), (22)• their sets Cs,u

AB are coboundaries. (23)

Proof. Assume that ΦA and ΦB are conjugate. Then by virtue of Theo-rems 2.1 and 2.9 the conditions (22) and (23) hold.

Suppose that (22) and (23) hold and that (Ω,F , P) is non-atomic. By The-orem 4.3 ΦA is conjugate to a linear hyperbolic diagonal cocycle ΦA1 and ΦB isconjugate to a linear hyperbolic diagonal cocycle ΦB1 . From Theorems 2.1 and2.9 it follows that the conditions (22) and (23) (with corresponding change ofnotation) hold for the pair of the linear hyperbolic diagonal cocycles ΦA1 andΦB1 . Therefore, Proposition 6.3 implies that ΦA1 and ΦB1 are conjugate. Thisyields that ΦA and ΦB are conjugate.

If (22) and (23) hold and (Ω,F , P) is atomic then by Proposition 7.2 belowΦA is conjugate to ΦB .

Proposition 7.2. Assume that (Ω,F , P) is atomic. Then two linear hyper-bolic cocycles ΦA and ΦB are conjugate if and only if the conditions (22) and(23) of Theorem 7.1 hold.

Proof. Since (Ω,F , P) is atomic there exists an atom E ∈ F with P(E) > 0.Then by ergodicity of θ there exists n ∈ N such that the sets θnE =: Ei,i = 0, . . . , n − 1, are disjoint and their union is Ω (mod 0). Therefore, thedynamical system (θ, Ω,F , P) is metrically isomorphic to the cyclic permutationof n-point space with uniform probability, i.e. the dynamical system (θ, Ω, F , P),where Ω := 1, . . . , n, F is the collection of all subsets of Ω, P(i) = 1

n for


i = 1, . . . , n, and θi = i + 1 for i = 1, . . . , n− 1, θn = 1. Therefore, the problemof topological classification of linear cocycles over (θ, Ω,F , P) is the same as theproblem of topological classification of linear cocycles over (θ, Ω, F , P). Hence,for simplicity of presentation we assume that (θ, Ω,F , P) = (θ, Ω, F , P).

If ΦA and ΦB are conjugate then by Theorems 2.1 and 2.9 the conditions(22) and (23) hold.

Suppose that (22) and (23) hold. Put

A := A(n) . . . A(1),B := B(n) . . . B(1).

Then

ΦA(mn + j, 1) = A(j) . . . A(1)Am,

ΦB(mn + j, 1) = B(j) . . . B(1)Bm.

This implies that A and B are hyperbolic operators of Rd and their stable andunstable subspaces, which we denote by Es,u

Aand Es,u

B, coincide with Es,u

A (1)and Es,u

B (1), respectively. Therefore, by (22)

dim EsA

= dim EsB

and dim EuA

= dim EuB

. (24)

Furthermore,

Es,uA (i) = ΦA(i− 1, 1)Es,u

A (1) = ΦA(i− 1, 1)Es,u

A,

Es,uB (i) = ΦB(i− 1, 1)Es,u

B (1) = ΦB(i− 1, 1)Es,u

B.

Recall that

As,u(i) = A(i)|Es,uA

(i), Bs,u(i) = B(i)|Es,u

B(i)

,

As,u = A|Es,u

A

, Bs,u = B|Es,u

B

Note that by Lemma 2.3 a subset of Ω is a coboundary if and only if itcontains even number of elements of Ω. Put F := i ∈ Ω | deg As(i) = −1,G := i ∈ Ω | deg Bs(i) = −1. Then, by (23) F4G is a coboundary, henceits cardinality is even. This implies that deg As = deg Bs. Similarly, deg Au =deg Bu. Taking into account (24), by the theorem of Robbin, 1972 (Theorem 1.1in our Introduction), there exists a homeomorphism h0 ∈ Homeo(Rd) such that

A = h−10 B h0. (25)

Set

h(i) :=

h0 for i = 1,B(i− 1) . . . B(1) h0 A−1(1) . . . A−1(i− 1) for i = 2, . . . , n.

Then h is a random homeomorphism of Rd. We show that h furnishes a topo-logical conjugacy between ΦA and ΦB .


For i = 1, . . . , n− 1 we have

h(i + 1) A(i) = B(i) . . . B(1) h0 A−1(1) . . . A−1(i) A(i) = B(i) h(i).

Furthermore, since θn = 1, by (25) we have

h(1) A(n) = h0 A(n) = B h0 A−1 A(n)= B(n) . . . B(1) h0 A−1(1) . . . A−1(n) A(n) = B(n) h(n).

Therefore, for all i ∈ Ω we have A(i) = h−1(θi) B(i) h(i). Hence, ΦA isconjugate to ΦB .

We see from Theorem 7.1 that the classification of linear hyperbolic cocyclesdepends crucially on the properties of the dynamical system (θ, Ω). We givehere some examples for showing the influence of the dynamical system (θ, Ω) onthe classification of linear hyperbolic cocycles.

Example 7.3. Akcoglu and Chacon (1965) have constructed a non-atomicmeasure space of finite measure (S, Σ, µ) and an automorphism T such that forany given ω ∈ F , the equation

f(Ts) = ω(s) · f(s), for all s ∈ S,

has a solution f ∈ F , where F is the class of complex-valued Σ-measurablefunctions of absolute value 1. This means that any measurable set of S is acoboundary with respect to the automorphism T . Therefore, condition (23) ofTheorem 7.1 holds automatically. This implies that we have only d + 1 topo-logical classes of d-dimensional linear hyperbolic cocycles over the dynamicalsystem (T, S).

The arguments of the proof of Proposition 7.2 show that if (Ω,F , P) isatomic, then we have exactly 4d topological classes of d-dimensional linear hy-perbolic cocycles over the dynamical system (θ, Ω) – the same number as inthe deterministic case. We shall show that if the dynamical system (θ, Ω) isthe well-known irrational rotation of the unit circle, then there exist infinitelymany topological classes of d-dimensional linear hyperbolic cocycles over it forany d ∈ N. First, we need the following lemma, the idea of its proof is due toKirillov (1967).

Lemma 7.4. Let (Ω,F , P) be the half-open interval [0, 1) with its Borel σ-algebra and Lebesgue measure dx, and θ be the rotation x 7→ x+α (mod 1) withan irrational α. Then there exist infinitely many measurable sets Gi ∈ F , i ∈N, such that for any pair of indices (i, j), i, j ∈ N the set Gij := Gi4Gj is nota coboundary.

Proof. It suffices to prove that for any n ∈ N we can find n sets Gi ∈ Fwith the above property.

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By Proposition 2.6, Ω is not a coboundary because θ2 is ergodic. Let k ∈ Nbe arbitrary. Divide the interval Ω into k equal disjoint subintervals. If one ofthose subintervals is a coboundary, then, by homogeneity, all the other ones arecoboundaries, too. This implies that Ω, as their union, is a coboundary, whichis a contradiction. Therefore, none of these subintervals is a coboundary.

Now divide Ω into (2n)! equal disjoint subintervals

Gi =[

i

(2n)!,i + 1(2n)!

), i = 0, 1, . . . , (2n)!− 1, n ∈ N.

We show that for any pair of indices (i, j), 0 ≤ i < j ≤ n the set Gij := Gi4Gj isnot a coboundary. Suppose there exists a pair of indices (i1, j1), 0 ≤ i1 < j1 ≤ nsuch that the set Gi1j1 is a coboundary. By homogeneity we can put i1 = 0,j1 = k ≤ n. Moreover, also by homogeneity, we get that the sets Gm,m+k,m = 0, . . . , (2n)!− k − 1 are coboundaries. Therefore, the sets

G2km :=

k−1⋃l=0

Gm+l,m+k+l =2k−1⋃l=0

Gm+l, m = 0, 1, . . . , (2n)!− 2k, (26)

are coboundaries. This implies that Ω is a coboundary, because it is a finiteunion of disjoint sets of type (26). This contradiction proves our lemma.

Proposition 7.5. Let (θ, Ω,F , P) be as in Lemma 7.4. Then for anyd ∈ N there exist infinitely many d-dimensional linear hyperbolic cocycles overthe dynamical system (θ, Ω) such that they are pairwise not conjugate.

Proof. Take the sets Gi ∈ F , i ∈ N, from Lemma 7.4. Construct randommatrices

ai(ω) := 3(1Gi(ω)P + (1− 1Gi

(ω))I), for ω ∈ Ω, i ∈ N,

where P := diag−1, 1, . . . , 1, 1G(·) denotes the characteristic function of G ∈F , and I denotes the unit (d × d)-matrix. Denote by Ai the linear randommaps which have ai(·) as their matrix representations with respect to a fixednonrandom basis of Rd (i ∈ N). It is easily seen that the linear cocycles ΦAi

are hyperbolic and they are pairwise not conjugate.

Acknowledgments

I am greatly indebted to Professor L. Arnold for suggesting the problem, forhis fruitful discussion, constant help and encouragement during the course ofthis work. I would like to thank Doctors D. N. Hao, T. Wanner and V. M.Gundlach for their careful reading of the manuscript and valuable commentsand suggestion for improving the first draft of this paper. I would also liketo thank the referee for providing valuable comments and a helpful suggestionimproving the proof of Theorem 2.1. This work was supported by the Alexandervon Humboldt Foundation, Germany.


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Topological classification of linear hyperbolic cocycles

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