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OSELEDETS

1. S

Let (X, µ,A) be a probability space, and letT :X→ Xbe an ergodic measure-preserving transformation.

Given a measurable mapA: X→GL(d,R), we can define a skew- product F : X× Rd → X ×Rd by (x,v) 7→ (T(x),A(x)· v). Write Fn(x)=(Tn(x),A(n)(x)·v). The mapFis called alinear cocycle.

Fix some normk·konRd, and use the same symbol to indicate the induced operator norm.

Theorem 1 (Multiplicative Ergodic Theorem of Oseledets [O]). Let T be an invertible bimeasurable ergodic transformation of the probability space(X, µ). Let A: X→GL(d,R)be measurable such thatlog+kAkand log+kA1kare integrable. Then there exist numbers

(1) λ1 >· · ·> λk

and forµ-almost every x∈X, there exist a splitting (2) Rd =E1x⊕ · · · ⊕Ekx

such that for every1≤i≤k we have A(x)·Eix =EiT(x) and

(3) lim

n→±∞

1

nlogkA(n)(x)·vik=λi for all non-zero vi ∈Eix.

The subspaces Eix are unique µ-almost everywhere, and they depend mea- surably on x.

The numbers (1) are calledLyapunov exponents, and the splitting (2) is called theLyapunov splitting.

Invariance of the bundlesEix implies that their dimensions di are constant a.e. Measurability means the following: there exist measur- able mapsu1, . . . ,ud :X→Rdsuch that for each j=1, . . . ,k, the space Eixis spanned by the vectorsui(x) withd1+· · ·+di1 <i≤d1+· · ·+di. Exercise i. If T is not assumed to be ergodic then a similar result still holds. The Lyapunov exponents (and their number) become measurableT-invariant a.e. defined functions. State and prove this assuming Theorem 1 and using ergodic decomposition.

1

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Exercise ii. Use the last exercise, state and prove a version of Os- eledets’ Theorem for diffemorphisms of a compact manifold.

In the next section, we explain some additional conclusions that could be added to Oseledets’ Theorem, and also state some results that will be need later. The proofs will take the rest of the note. Some exercises are given along the way. Those that are need for the proof of Theorem 1 are marked with the symbolX.

2. A 

2.1. One-sided version. There is a version of Theorem 1 for non- (necessarily)-invertible transformations:

Theorem 2. Let T be an ergodic transformation of the probability space (X, µ). Let A : X → GL(d,R) be measurable such that log+kAk and log+kA1kare integrable. Then there exist numbers

λ1 >· · ·> λk

and forµ-almost every x∈X, there exist a flag

(4) Rd =V1x ⊃Vx2⊃ · · · ⊃ Vkx⊃Vxk+1 ={0}

depending measurably on the point, and invariant by the cocycle, such that

(5) lim

n→∞

1

nlogkA(n)(x)·vik=λi for all vi ∈ VixrVxi+1. We call (4) theLyapunov flagof the cocycle.

Exercise iii. In the caseT is invertible, the flag is given in terms of the Lyapunov splitting byVix=E1x⊕ · · · ⊕Eix.

2.2. Subadditivity. We will use the following result (see e.g. [L] for a proof):

Theorem 3(Kingman’s Subadditive Ergodic Theorem). Assume T is ergodic. If(φn)n=1,2,... is a sequence of functions such thatφ+1 is integrable and φm+n ≤ φmn◦Tm for all m, n ≥ 1. Then n1φn converges a.e. to some c∈R∪ {−∞}. Moreover, c=infn1

n

R φn.

An immediate corollary of Theorem 3 in the setting of cocycles is the existence of the limit limn→∞ 1

nlogkA(n)(x)k. (This is called the Furstenberg-Kesten Theorem.) It can be shown that this limit is the upper Lyapunov exponentλ1given by Oseledets’ Theorem:

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Addendum 4. We have:

λ1= lim

n+

1

nlogkA(n)(x)k= lim

n→−∞

1

nlogm(A(n)(x)), λk = lim

n+

1

nlogm(A(n)(x))= lim

n→−∞

1

nlogkA(n)(x)k.

Recall that theco-norm of a linear mapLis m(L) = infkvk=1kLvk; it equalskL1k1ifLis invertible.

2.3. Angles. The angles between the spaces of the Lyapunov split- ting are sub-exponential, in the sense made precise by the following:

Addendum 5. Let Rd = E1x ⊕ · · · ⊕Ekx be the Lyapunov splitting, and assume k ≥ 2. Let J1tJ2 be a partition of the set of indices {1, . . . ,k}into two disjoint non-empty sets. Let Fix = L

jJiExj for i = 1, 2. Then, for µ-a.e. x we have

nlim→±∞

1

nlog sin]

F1Tn(x),F2Tn(x)

=0.

We will deduce Addendum 5 directly from Oseledets Theorem.

For that, we need the following result (that will also be usuful in the proof of the theorem itself):

Lemma 6. Letφ:X→ Rbe a measurable function such thatφ◦T−φis integrable in the extended sense1. Then

nlim→±∞

1

|n|φ(Tnx)=0 for a.e. x∈X.

(n<0is allowed if T is invertible.) Proof. Let ψ = φ◦T−φand let c = R

ψ ∈ [−∞,+∞]. By Birkhoff’s Theorem,

φ◦Tn

n = φ

n + 1 n

n1

X

j=0

ψ◦Tj →ca.e.

Ifcwere non-zero then we would have|φ(Tnx)| → ∞for a.e.x. But, by the Poincar´e’s Recurrence Theorem, the set of pointsxwhich satisfy the latter condition has zero measure. Thereforec=0, as we wanted to show. IfTis invertible thenφ◦T1−φis quasi-integrable as well,

so the previous reasoning applies.

1ψ is said to be integrable in the extended sense if ψ+ or ψ are integrable.

Another exercise: Verify that the BirkhoffTheorem still applies.

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Exercise iv. Let L : Rd → Rd be a invertible linear map and letv, w be non-zero vectors. Then

1

kLk kL1k ≤ sin](Lv,Lw)

sin](v,w) ≤ kLk kL1k.

Proof of Addendum 5. The function φ : X → R defined by φ(x) = log sin](F1x,F2x) is measurable. By Exercise iv, we have|φ(Tx)−φ(x)| ≤ logkA(x)k+logkA(x)1k, which is integrable, by the hypothesis of the Oseledets Theorem. So Lemma 6 givesφ(Tn(x))/n→0 for a.e.x.

2.4. Conjugacy. For a fixed T : X → X, we say a measurable map C:X →GL(d,R) istemperedif limn→∞ 1

nlogkC±1(Tnx)k=0.

We say that two cocycles A, B : X → GL(d,R) are conjugate (or equivalent) if there is a temperedC:X →GL(d,R) such that

B(x)=C(Tx)1A(x)C(x) That is, (T,B)=(C,id)1◦(T,A)◦(C,id).

Exercise v. If two cocycles are conjugate then they have the same Lyapunov exponents.

2.5. Determinants. Another addendum to Oseledets Theorem (that we will obtain along its proof) is this:

Addendum 7. For a.e. x, the convergence in (3) is uniform over unit vectors in Eix.

Using this, do the following:

Exercise vi. Let∅, J⊂ {1, . . . ,k}andFx =L

jJExj. Prove that

nlim→±∞

1

nlog|detA(n)(x)|Fx|=X

jJ

λjdimEj.

3. LO

The proof of Theorem 1 will take the rest of this note.

It is comparatively very easy to prove Theorem 2 if limits are replaced by lim sup’s. Let

λ(x,¯ v)=lim sup

n→∞

1

nlogkA(n)(x)·vk, v,0.

Lemma 8. There exist numbersλ1 > · · · > λk such that for a.e. x, λ(x,¯ ·) takes only these values. For a.e. x there is a flag into linear spaces

Rd=Vx1 ⊃V2x ⊃ · · · ⊃Vxk ⊃Vk+1x ={0}

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such that

v∈VixrVxi+1 ⇒ λ(x,¯ v)=λi.

Moreover, these spaces depend measurably on the point, and are invariant by the cocycle.

Proof. First, the integrability of log+kA±1kimplies that ¯λ(x,v) is finite for a.e.xand allv,0. It is easy to see that:

λ(x,¯ av)=λ(x,¯ v), λ(x,¯ v+w)≤max

λ(x,¯ v),λ(x,¯ w)

with equality if ¯λ(x,v),λ(x,¯ w).

It follows that for anyt∈R, Vx(t)=n

v∈Rd; ¯λ(x,v)≤torv=0o

is a linear subspace. Also, vectors with different ¯λ are linearly independent, so for all x, ¯λ(x,·) takes at most d distinct values λ1(x)>· · ·> λk(x)(x). LetVix=Vxi(x)).

Exercise vii(X). Show that everything is measurable and conclude the proof of the Lemma.

4. I: S    -

LetPbe a compact metric space. LetS: X×P→ X×Pbe a skew- product map overT(this means thatπ◦S=Twhereπ:X×P→X is the projection) such thatS(x,·) : P→Pis continuous for a.e.x.

Givenφ: X×P→R, we write ψ(n) =

(ψ+ψ◦S+· · ·+ψ◦Sn1 ifn>0;

−ψ◦S1−ψ◦S2− · · · −ψ◦Sn ifn<0 andSis invertible.

LetF be the set of measurable functionsφ : X×P →Rsuch that φ(x,·) is continuous for a.e.xandx∈ X7→ supuP|φ(x,u)|belongs to L1(µ).

Exercise viii. Prove that the normkφk= R

supu|φ(x,u)|dµ(x) makes F a separable Banach space.

Lemma 9. Assumeψ ∈ F and c∈Rare such that forµ-a.e. x ∈X, lim sup

n→∞

1

(n)(x,u)≥c for all u∈ P.

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Then the assertion remains true with the lim sup replaced with a lim inf that is uniform over P. More precisely, forµ-a.e. x ∈ X and every ε > 0 there is n0 such that

1

(n)(x,u)>c−ε for all u∈P and n>n0. Furthermore, if S is invertible then for a.e. x, lim infn→−∞1

nψ(n)(x,u) ≥ c uniformly in u.

Corollary 10. Assumeψ ∈ F and c∈Rare such that forµ-a.e. x ∈X, lim sup

n→∞

1

(n)(x,u)=c for all u∈ P.

Then the assertion remains true with lim sup replaced with uniform lim (orlimn→±∞if S is invertible).

The proof of Lemma 9 is a Krylov–Bogoliubov argument. For that, we need:

Exercise ix (X). Let M be the set of probability measures on X×P that project onµ(that is, πν =µ). There is a topology on Mwhich makes it compact, and such that a sequenceνnconverges toνif and only if

Z

φdνn→ Z

φdν for everyφ∈ F. Proof of Lemma 9. Let

In(x)=inf

uPψ(n)(x,u).

ThenInis measurable,I1 is integrable, andIn+m ≥In◦Tm+Im. So the Subadditive Ergodic Theorem 3 applies to−In, and thusn1Inconverges a.e. to a constantb.

For a.e.x, letun(x) be such thatψ(n)(x,un(x))=In(x).

Exercise x(X). It is possible to chooseunmeasurable.

Takeunmeasurable and define the following probability measures onX×P:

ν0n = Z

δ(x,un(x))dµ(x) and νn=

n1

X

j=0

Sjν0n.

All this measures project to µ. By Exercise ix, there is a converging subsequenceνni →ν.

Exercise xi(X). νis aS-invariant measure.2

2Hint: for anyη∈ F,|R

ηd(Fνnνn)| ≤(2/n)kηk, wherek·kis the obvious norm on the Banach spaceF.

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Moreover, R

ψdν = limR

ψdνn = lim1nR

Indµ = b. By Birkhoff’s Theorem, there is a set of fullν-measure formed by points (x,u) such that the sequence n1ψ(n)(x,u) converges tob. By the assumption, the lim sup’s of those sequences are ≥ c ν-a.e. Therefore b ≥ c. Since b=limn→∞n1infuψ(n)(x,u), we obtain the desired uniform lim inf.

Notice that for any S-invariant probability measure ν0, we have R ψdν0 ≥b(by Birkhoffagain).

Now assume thatS is invertible. Repeating the above reasoning forS1, we obtain aS-invariant probabilityν0such that

nlim→−∞inf

uP

1

(n)(x,u)= Z

ψdν0. So lim infn→−∞n1ψ(n)(x,u) ≥ R

ψdν0 ≥ b ≥ c uniformly with respect

tou.

5. T    

Consider the exponents λ1 > · · · > λk and the flag V1 ⊃ · · · ⊃ Vk given by Lemma 8. We will focus the attention onλk andVk. Write

λmin(A)=λk, Ex =Vxk

We callEthebundle of least stretchforA. It has positive dimension.

By definition, lim supn→∞n1logkA(n)vk=λmin(A) for a.e.xand all non-zerov∈Ex. Let us improve this information:

Lemma 11. We have:

nlim→±∞

1

nlogkA(n)(x)·vk=λmin(A) for a.e. x and all v∈Exr{0}. (n → −∞allowed if T is invertible.) Moreover the limit is uniform over unit vectors.

The conclusion of the lemma can be rephrased as:

nlim→±∞

1

nlogkA(n)|E (x)k lim

n→±∞

1

nlogm(A(n)|E (x))=λmin(A).

Proof of Lemma 11. Let m = dimE. By conjugating with some mea- surableC:X→O(d,R), we can assume thatEx =Rm ⊂Rd.

Let Pbe the projective space of Rm. Consider the induced skew- productF : X×P → X×PoverT. Letψ(x,v)¯ = logkA(x)kvk·vk, where ¯v denotes the element ofPcorresponding tov. Then

1

(n)(x,v)¯ = 1 n

n1

X

j=0

logkA(j+1)(x)·vk kA(j)(x)·vk = 1

nlogkA(n)(x)·vk kvk

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has a lim sup as n→ ∞ equal toλmin , for a.e. xand all ¯v ∈P. Thus Corollary 10 implies that the lim sup can be replaced with uniform

lim.

By Lemma 8, we have lim sup

n→∞

1

nlogkA(n)(x)·vk ≥λmin(A) for a.e.xand allv∈Rdr{0}. Using Lemma 9 and a similar argument as in the proof of Lemma 11, we conclude that this lim sup can by replaced by a lim inf that is uniform with respect tou. This can be rephrased as:

lim inf

n+

1

nlogm(A(n)(x))≥λmin(A). On the other hand,

lim sup

n+

1

nlogm(A(n)(x))≤lim sup

n+

1

nlogm(A(n)|E (x))≤λmin(A). So we obtain

nlim+

1

nlogm(A(n)(x))=λmin(A).

This proves one of the assertions in Addendum 4. The others are proven similarly.

6. I: H       

Let M(d) indicate the set ofd×dreal matrices.

Lemma 12. Let B : X → M(d) be measurable with log+kBkintegrable.

Assumeγ ∈Ris such that lim sup

n+

1

nkB(n)(x)k ≤γ a.e.

Then for any ε > 0there exists a measurable function bε : X → R+ such that

kB(n)(Tix)k ≤bε(x)e(γ+ε)n+ε|i| for a.e. x and all n≥0, i∈Z.

(i<0is allowed if T is invertible.) Proof. The formula

c(x)=sup

n0

e(γ+ε)nkB(n)(x)k

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defines a measurable a.e. finite function. By definition, kB(n)(x)k ≤ c(x)e(γ+ε)n. Now we need to bound the expressionc(Tix). We have

c(Tx)=sup

n0

e(γ+ε)nkB(n)(Tx)k

≥ kB(x)k1sup

n0

e(γ+ε)nkB(n+1)(x)k

=eγ+εkB(x)k1sup

n1

e(γ+ε)nkB(n)(x)k

Either the rightmost sup equalsc(x) or elsec(x)=1 and thenc(Tx)≥1.

In any case, we have

c(Tx)≥min

eγ+εkB(x)k1,1 c(x). and so

logc◦T−logc ≥min

γ+ε−log+kBk,0 .

is quasi-integrable. So Lemma 6 applies, giving 1i logc◦Ti → 0 a.e.

asi→ ±∞. We apply the sup trick a second time: Let bε(x)=sup

iZ

eε|i|c(Tix).

Thenbεis the desired function.

7. T 

As before, let E be the bundle of least stretch forA. Assume the norm k·k comes from an inner product. For each x, let Ex be the orthogonal complement ofEx; this is a measurable subbundle. With respect to the splittingRd =E⊕E, we write

(6) A= B 0

C A|E

! . SoB(x) :Ex →ETxgives a new cocycle.

Lemma 13. λmin(B)≥λk1. In particular,λmin(B)> λmin(A).

Proof. Notice that m(B(n)(x)) ≥ m(A(n)(x)).3 So, by Addendum 4, λmin(B)≥λmin(A)=λk.

Let E0 ⊂ E be the subbundle the bundle of least stretch for the cocicleB. The measurable bundleF=E0⊕EisA-invariant.

3Let w be the unit vector in Ex the most contracted by B(n)(x), and let v = [A(n)(x)]1B(n)(x)·w. By Pythagoras, kvk ≥ kwk. Hence m(A(n)(x)) kA(n)(x)vk

kvk

kB(n)(x)wk

kwk =m(B(n)(x)).

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Fixε >0 small. We have A(n)|F =





B(n)|E0 0 Cn A(n)|E





 , where Cn(x)=

n1

X

i=0

A(n|Ei+1)(Ti+1x)C(Tix)B(i)|E0(x). Write σ = λmin(B). Applying Lemma 12, we find measurable func- tionsc,b,asuch that for a.e. xand alln,i≥0,

kC(Tix)k ≤c(x)eεi kB(n)|E0(x)k ≤b(x)e(ε+σ)n kA(n)|E (Tix)k ≤a(x)e(ε+λk)n+εi Thus

kCn(x)k ≤n max

0in1

kA(n|Ei+1)(Ti+1x)k kC(Tix)k kB(i)|E0(x)k

≤nd1(x) max

0in1eε(i+1)+(λk+ε)(ni1)eεie(σ+ε)i

≤nd2(x)e4εn max

0in1e(ni1)λk+σi

≤nd3(x)e4εneσn.

(In the last step we used thatσ≥λk.) So, at a.e. point, λk1 ≤lim sup

n→∞

1

nlogkA(n)|F k ≤lim sup

n→∞

1

nlog max

kA(n)|E k kB(n)|E0k kCnk

=max lim sup

n→∞

1

nlogkA(n)|E k,lim sup

n→∞

1

nlogkB(n)|E0k,lim sup

n→∞

1

nlogkCnk

!

≤max(λk, σ, σ+4ε).

Soλk1 ≤σ+4ε. Asε >0 is arbitrary,λk1≤σ=λmin(B).

Exercise xii. In the notation of the proof of Lemma 13, prove that limn1logkA(n)(x)·vkexists and equalsλmin(B) for allv∈FrE. Also show that λmin(B) = λk1 (that is, equality holds in Lemma 13) and F=Vk1is the penultimate space of the Lyapunov flag. Use induction to prove Theorem 2.

8. C     

Lemma 14. There is a measurable invariant subbundle G ofRd such that G⊕E=Rd.

Remark. Here is the first time we needTto be invertible.

Proof. Let Gbe any (non-necessarily invariant) measurable subbun- dle which is complementary toE, that is, such thatG⊕E=Rd.

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Let L(x) : Ex → Ex be linear maps depending measurably on x.

The set of such maps is indicated by L. The graph of L, given by Gx ={L(x)v+v; v∈Ex}, is a subbundle complementary toE.

The (forward) image of a subbundle Gis the bundle whose fiber overxisA(T1x)·GT1x.

LetΓ: L → Lbe thegraph transformso that the image of the graph ofLis the graph ofΓL. Recalling (6), we see that

(7) ΓL=D+ ΦL,

whereD∈ LandΦ :L → Lare given by:

D(x)=C(T1x) [B(T1x)]1, (ΦL)(x)=A|E(T1(x))L(T1x) [B(T1x)]1. To prove Lemma 14 we need to prove thatΓhas a fixed point. We will prove that there is a measurable functiona : X → Rand τ > 0 such that

(8) k(ΦnD)(x)k ≤a(x)eτn for a.e.xand alln≥0.

Then the formula L= P

n=0ΦnDwill define a element ofLwhich is fixed by the graph transform (7).

We have (ΦnL)(x)=(A|E)(n)(Tn(x))L(Tnx)B(n)(x), so (9) k(ΦnD)(x)k ≤ k(A|E)(n)(Tnx)k kD(Tnx)k kB(n)(x)k

We will bound each of the three factors using Lemma 12. Fix a small ε >0 be small (to be specified later). For the first factor in (9) we have

k(A|E)(n)(Tnx)k ≤a1(x)en(λk+2ε), for some measurablea1. Secondly,

log+kD(x)k ≤log+kC(T1x)k+logkB(T1x)1k

≤log+kA(T1x)k+logkA(T1x)1k. So log+kDkis integrable and Lemma 12 gives

kD(Tnx)k ≤a2(x)eεn, for some measurablea2.

Now look to the third factor in (9). By Addendum 4 and Lemma 13,

nlim+

1

nlogkB(n)(x)k=−λmin(B)≤ −λk1.

The function log+kB1k ≤log+kA1kis integrable. Thus we can apply Lemma 12 to the cocycleB1 overT1and conclude that

kB(n)(x)k ≤a3(x)e(λk1+ε)n, for some measurablea3. Putting the three bounds together in (9) we obtain

nD(x)k ≤(a1a2a3)(x)e(λk1k+4ε)

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Takingεsmall we obtain the desired (8). This proves Lemma 14.

9. C

Proof of Theorem 1 (and Addendum 7). Let V1 ⊃ · · · ⊃ Vk be given by Lemma 8. LetEk = Vk. By Lemma 11, (3) holds fori = k(uniformly with respect to unit vectors). Ifk = 1 then we are done. Otherwise, apply Lemma 14 to find an invariant subbundle G complementary toEk. Now consider the cocycle restricted toG; its Lyapunov flag is:

G=V1∩G⊃ · · · ⊃ Vk1∩G⊃ {0}

with corresponding exponents λ1 > · · · > λk1. Then define Ek1 =

Vk1∩Gand repeat the argument.

10. C   ()

There are several proofs of the Oseledets’ Theorem. The more conventional proofs use heavier Linear Algebra (exterior powers, singular values etc). The more conventional approach has the ad- vantage of providing more useful information about the Lyapunov exponents and splittings: see the detailed treatment in [A], also [L].

The proof here is a combination of the proofs of [V] (which is on its turn based on [M]), and [W]. Lemmas 8 and 9 come from [W], and Lemma 14 comes from [V]. In [M, V], only the two-sided (i.e., invertible) case was considered, while [W] proves only the one-sided4 version of the Oseledets’ Theorem. Unlike [V, M], we used the Subadditive Ergodic Theorem (in the proof of Lemma 9)5.

R

[A] L. Arnold.Random dynamical systems. Springer Verlag, 2003.

[L] F. Ledrappier. Quelques propri´et´es des exposants caract´eristiques. Springer Lect. Notes Math.1097 (1984), 306–396.

[M] R. Ma ˜n´e.Ergodic theory and differentiable dynamics. Springer Verlag, 1987.

[O] V. I. Oseledets. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems.Trans. Moscow Math. Soc.19 (1968), 197–231.

[V] M. Viana. A proof of Oseledets’ theorem. www.impa.br/viana/out/

oseledets.pdf

[W] P. Walters. A dynamical proof of the multiplicative ergodic theoremTrans.

Amer. Math. Soc.355 (1993), 245–257.

Jairo Bochi May 27, 2008

4It is not clear to me how to deduce the two-sided theorem from its one-sided version without using exterior powers.

5Maybe this could be avoided, if desired.

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