Constructing the Oseledets decomposition with subspace growth estimates

The semi-invertible version of Oseledets' multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work represents a significantly simplified means of obtaining the result, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman's subadditive ergodic theorem.


Introduction
The multiplicative ergodic theorem is a fundamental tool in the study of linear dynamical systems.Given a Banach space X write B(X) for the space of bounded linear operators on X.Let Ω be a Lebesgue probability space.Given an ergodic system σ : Ω → Ω and function L : Ω → B(X), one may compose copies of L along orbits and investigate long term behaviour of any x ∈ X.Such an L is referred to as a cocycle, and is said to be forward-integrable if log + L ∈ L 1 (Ω).Iteratively set L (0) ω = 1X : X → X, and for n < 0 extends the cocycle to all n ∈ Z.If L −1 is forward-integrable then L is said to be backwardintegrable.One may seek ways of describing X in terms of long term behaviour of vectors under L (n) ω as n → ∞.Given any normed space V write SV = {x ∈ V : x = 1} and BV = {x ∈ V : x < 1}.Oseledets [12] proved the following in 1965: Let Ω be a Lebesgue probability space and σ : Ω → Ω be an invertible measure preserving transformation.Let L : Ω → GL d (R) be forward and backward integrable, where GL d denotes the invertible d−dimensional matrices.Then there are measurable numbers λi(ω), i ∈ {1, • • • , rω} and a direct sum decomposition of invariant measurable subspaces R d = i Ei(ω) such that for x ∈ S E i (ω) , lim n→±∞ 1 n log L (n)  ω x = ±λi(ω).
This limit converges uniformly in x.
The case where the underlying vector space is a separable Banach space was first presented by Lian and Lu [9].Their monograph obtains the decomposition result assuming almost-everywhere injectivity of the cocycle in a separable Banach space.
The injectivity condition isn't necessary to obtain a decomposition: Froyland, Lloyd and Quas [10] demonstrated that as long as σ is invertible the space may still be written as a sum of fast and slow spaces.That paper dealt with the finite dimensional case, but in [3] González-Tokman and Quas presented a generalisation to a separable Banach space.However, there is some ambiguity to the statement and consequentially the proof -it is claimed that a measurable composition of X exists without a precise discussion of what this means, which is crucial given that in the infinite dimensional setting no statement may be made of the slow spaces, as discussed in Horan's thesis [5].González-Tokman and Quas' subsequent shorter proof in the case where the dual X * = B(X, R) is separable used intuitive notions of volume growth that this work partially builds upon, but relied on less than fully rigorous references to cocycles whose domain varies depending on ω.Another example of this kind of emphasis on volume growth may be seen in the noninvertible result of Blumenthal in [1] where a flag is obtained with no separability assumptions but with a much stronger uniform measurability condition for the cocycle.
Given measurable spaces A and B write M(A → B) for the space of measurable functions from A to B. Let F be the Borel sigma algebra induced by the strong operator topology on X.The space SM of strongly measurable functions consists of measurable maps with respect to this choice of sigma algebra: Write G k X = {V ≤ X : dim V = k} for the Grassmannian of subspaces of dimension k.Given T ∈ B(X) define the slowest growth of vectors in a given subspace under T : The ρ k are called Bernstein numbers in the work of Pietsch [13], which gives an overview of similar kinds of statistics in Banach spaces.Given a strongly measurable forward-integrable cocycle L ona Lebesgue probability space there are decreasing sequences (µi) i∈N and λi of invariant functions While there are countably many µi there may only be finitely many λis, referred to henceforth as the Lyapunov exponents.The main result, an extension of the semi-inveritble result to separable Banach spaces, may now be stated: Theorem 3. Let (Ω, σ, X, L) be a strongly measurable forward-integrable random linear dynamical system with (Ω, σ) an ergodic invertible map on a Lebesgue probability space and having Lyapunov exponents (µi) ∞ i=1 and (λi) L i=1 , where 1 ≤ L ≤ ∞.Then for each 0 ≤ l < L there is a direct sum decomposition into equivariant spaces The projection Π : X → j<i Ej parallel to Vi is strongly measurable and tempered, that is to say, limn→∞ 1 n log Πσnω = 0 almost surely.There is a nontrivial decomposition, L ≥ 2, exactly when ν = limn→∞ µn < λ1.
The choice of construction method for the Lyapunov exponents is a fundamental aspect to each proof: in [14] µi are written simultaneously using finite dimensional singular value decomposition, while by contrast in [9] the first Lyapunov exponent is defined according to the asymptotic growth rate of L (n) ω followed by the Lyapunov exponents being first described as the asymptotic growth rate for nonzero elements of the fast spaces in the statement of the thorem.While singular value decomposition is no longer available in these contexts, some construction must be written down that may be viewed as reminiscent of a proof of the singular value decomposition of a transformation.The work presented here explicitly relies on a notion of singular values for arbitrary elements of B(X).Given a T ∈ B(X), sufficient conditions on these singular values for contracting fast growing regions of GX under the action V → T V are derived.A modified version of Kingman's subadditive theorem is established in reminiscent used to guarantee the asymptotic growth rates of the singular values of The results from past papers stated thus far have not required ergodicity.Throughout the rest of this paper, ergodicity will be assumed, simplifying the classification of invariant functions, although all results as is typical may be formulated with this assumption dropped.In this Banach space setting, the main theorem yields in a trichotomous classification of forward-integrable cocycles on separable Banach spaces: one of • L fails to be quasicompact, with µi = ν for all i ∈ N -no fast spaces may be detected • L is quasicompact, with finitely many finite dimensional fast spaces growing at rates λ1 > • L is quasicompact, with a countable sequence of finite dimensional fast spaces growing at rates λ1 > λ2 > • • • → ν.
These three possibilities just correspond to situations where there are no, finitely many or countably many fast spaces.The number ν is an alternative choice to the typical index of compactness κ = limn→∞ 1 n L (n) ω c that proves simpler to work with in this proof.In the appendix it is verified that these quantities are equal, so that the final result represents an extension of the result of Lian and Lu with the injectivity assumption dropped, avoiding measurability or domain concerns present in [3] and [4].In particular, the advantages of the current work are in its brevity, the preciseness of the conclusion and the use of an intuitive geometric perspective.
The author is indebted to Anthony Quas for his enthusiastic encouragement and guidance throughout.

The Grassmannian
When discussing subspaces of X, we may also consider the space of subspaces of X they lie in: Definition 4. Given vector spaces U, V ≤ X with U ⊕ V = X, write Π U |V for the projection defined by Π U V (u + v) = u for any u ∈ U and v ∈ V .The Grassmannian is defined as the set GX may be metrised by any of a few equivalent choices of distance between spaces, such as the Hausdorff distance between unit spheres.G k X ⊂ GX has already been defined.Write G k X = {V ≤ X : codim V = k} ⊆ GX.Lemma 5. Suppose that X is a Banach space.Then Proof.Completeness is established in section 2.1 of chapter IV of [6].The pushforward result is established in corollary B.13 of [3].
Lemma 6.If we take B = B(X) then we have the following characterisation of the space of strongly measurable functions: Proof.Lemma A.4 of [3] checks the equivalence of these conditions.
The following hold: We make use of the following straightforward construction: Lemma 8. let Ω be a measurable space.Suppose that {xn} n∈N ⊆ X for some measurable space X.
Then if we can write down a measurable set Sn ⊆ Ω for each n ∈ N such that n Sn = Ω then there is an associated measurable N : Ω → N given by Nω = inf{n : ω ∈ Sn}.Further, the map ω → xN ω is measurable.
This principle is applied to check measurability in the context of spaces which are separable.
3 Measurable statistics for linear operators Lemma 9. Let X be a normed space.Then g : Here use the following metric inducing the product topology: First note that g is continuous at (0, V ) for any V since g(0, V ) = 0 and if d(V, W ) + T − 0 < then g(T, W ) < .Otherwise, the projection onto the sphere Since V and W are finite dimensional their spheres are compact and we may choose an x ∈ SV with Sx = g(S, V ).Choose a y ∈ SW with x − y < .Then By the same argument swapping (S, V ) and (T, W ) then, g(S, V ) < g(T, W ) + 2 and continuity (in fact, uniform continuity) on S B(X) × G k X is clear.Thus g is continuous on B(X) × G k X as required.
Proof.The ρ k may now be written as the supremum of the functions {T → g(T, V ) : V ∈ G k X} which may be written as the supremum of a countable family since G k X is separable.
Lemma 11.Grassmannian contraction estimates: Let T ∈ B(X) and Θ ∈ (ρ k+1 T, ρ k T ).Suppose that V, W ∈ G k X are choices of fast growing spaces: In particular, if Θ > 2ρ k+1 T then Then since a isn't in the kernel then . Since the choice of a was arbitrary, In other words, U s i (V ) is the first member of the dense collection to fall in the nonempty open set Then Us i is Cauchy and so the pointwise limit of measurable functions Again we get that xt i converges pointwise to some measurable b with the desired properties.
Applying this result recursively we obtain the following: Lemma 14.Let X be a separable Banach space.Then for every k > 0 there is a measurable map Proof.
Proof.Make use of the inequality We may then conclude that gT (U ) ≥ e − ρ l (T •Π)

Π Π
with > 0 arbitrary, so that ρ l (T • Π) ≤ 4ρ l+k (T • Π) Π Π as required.σ a ω , which may be thought of as the evolution rule for X from time a to time b.Under this notation it is easy to see that for any a < c < b ∈ Z, L c→b • L a→b = L a→b .The inequality L a→b ≤ La→c L c→b holds.Taking log of each side, one obtains the following triangle inequality-like bound on the growth of points in X from time a to b:

A balanced subadditive ergodic theorem
Then F is referred to as a subadditive family of measurable functions.
The following view is useful: Definition 18.A subadditive family {fn} n∈N 0 generates a stationary subadditive process {f a→b : a < b, a, b ∈ Z} and vice versa via the relation A subadditive process is a collection {f a→b } a<b∈Z ⊆ M(Ω → R) such that for all a < c < b ∈ Z, and the stationarity condition is As such both notations may be interchanged as appropriate.A similar formalism is outlined in [8].
The Kingman theorem concerns subadditive families of functions, here a slight refinement when the underlying transformation is invertible will be required.
In the case where σ is invertible, one easily obtains the following useful corollary: Corollary 20.Let (Ω, σ, P) be an invertible ergodic system on a Lebesgue probability space and let {fn} n∈N ⊂ L 1 Ω be subadditive.Then pointwise almost everywhere.
Proof.Simply set gn = fn • σ −n .Then gn is subadditive with respect to σ −1 : so that certainly there exists some L such that 1  n fn • σ −n → L. It remains to check that this limit and 1  n fn(ω) → L coincide.Let > 0. Then there exists an N such that and was arbitrary though, so that L = L .
Kingman's original result may be modified in a few directions.One may very rapidly obtain the following, which may be found in [2]: Lemma 21.Let {fn} n∈N ⊆ L 1 Ω be a subadditive family of functions.Then Proof.The lower bound is immediately clear, since On the other hand, let > 0, then since 1 n fn → C there exists some N such that for all n ≥ N , 1 n fn ≤ C + .Given some fixed n ≥ N and some j < N , set j = n N and break up the interval

Applying subadditivity in this manner
The above is valid for all j ∈ [0, N ), so averaging and dividing by nN we obtain N n n(N (C + )) for sufficiently large n.Since was arbitrary, the result is proven.
A similar result for f−n→n will be crucial to the main result.The proof here uses the following lemma, which is a simplified version of result 3.9 in [15]: Lemma 22 (Backward Vitali).Let (Ω, P, σ) be an invertible ergodic system on a Lebesgue probability space.Suppose that there is a sequence of integer valued functions j k ∈ M(Ω → N0) such that j k (ω) → ∞ as k → ∞.Then there is a measurable A ⊆ Ω and a measurable j ∈ M(A → N0) such that writing Iω = {σ i ω : i ∈ {−jω, • • • , jω}}, the Iω are disjoint and Theorem 23 (Kingman's theorem for balanced intervals).let (Ω, σ, P) be an ergodic system with invertible base, and let (fn) n∈N be subadditive sequence of measurable functions on Ω.Then Proof.Kingman's theorem immediately yields an upper bound: since ) where Θ(C ¯, ) → C ās → 0. The set of j such that f−j→j is small is infinite, so denote the ordered elements Since f is integrable, it is uniformly integrable and so there exists a δ > 0 such that if P(A) < δ then A f < .Apply Backward Vitali to the jis and the −jis with parameter := min{ , δ} to obtain an A ⊆ Ω and a j : A → N according to theorem 22 such that writing B = Ω \ ω∈A Iω, we have P(B) < δ and so While every j(ω) = ji(ω), for some i so that for every ω ∈ A, Write Λ± = {σ ±j i (ω) (ω) : ω ∈ A}.In addition we may define measurable maps T± ∈ M(Ω → N) by Ts is the earliest nonnegative time such that σ Ts(ω) ω is the start of a period at growth rate guaranteed close to C ¯, Te by contrast seeks the first time in the past which was the end of such an interval.Each are almost surely finite since P(Λ±) = P(A) > 0.
The idea of the next step is that [0, n) may be measurably broken up into intervals of slow growth and with small gaps inbetween.Set b0 = 0 and recursively iterate along the orbit for i > 0: For any m ∈ N0 we may set tm(ω) = max{t : bt(ω) ≤ m}.Since j and T± are measurable and the tm(ω) → ∞ as m → ∞, it is possible to pick an N such that for all n ≥ N , P(j ≥ N ) < 1  4 , P(tn = 0) < 1   4   and P(T± ≥ N ) < 1 4 .Let n ≥ N .The orbit of ω is considered over the following intervals of times: Time ai → bi is guaranteed to have low growth: ¯ω + ), and in addition there is a good chance that the gaps between the ais and the bis won't be too large: 4 .Since T− seeks the endpoints of the periods of guaranteed growth, it holds that either T−(σ n ω) = n − b tn(ω) (ω) or tn(ω) = 0. Therefore, so that it is immediate from above that P(Λ) < .Given some measurable g write Spg = p−1 i=0 g •σ i .We then repeatedly apply subadditivity to f0→n: In the case where tn(ω) = 0 the first two terms are empty sums.Outside Λ it is guaranteed that It remains to check that each of these terms are small enough to yield the result when we divide through by n and integrate: Putting these together we obtain ¯as required.Linearity of the cocycles yields subadditivity of these families, which allow us to apply Kingman.
Definition 24.Given a random linear dynamical system R = (Ω, σ, X, L), write Write mi for the multiplicity of λi, ie, mi is defined as the largest m ∈ N such that Lemma 25.The quantities λi, µi and ν are almost everywhere constants.Further, λ2 exists if and only if λ1 > ν.
Proof.The existence of the limits are guaranteed by applying Kingman to particular choices of subadditive sequences: The second statement is a trivial consequence of the definitions ν = limn→∞ µn and λ2 = µ inf{t:µ t <µ 1 } .

Decomposing a cocycle
The tools obtained thus far are now used to decompose quasicompact cocycles.
Lemma 26.Let R = (Ω, σ, X, L) be a strongly measurable random linear dynamical system with ergodic base on a separable Banach space.Suppose that L satisfies the quasicompactness condition ν < λ1.Then there is a unique measurable choice of fast space E : Ω → Gm 1 X for which the following hold almost surely: Proof.Since X is separable, by lemma 5 Gm 1 X is separable: choose some dense {Ei} i∈N ⊆ Gm 1 X.Let ∈ (0, 1 5 (λ1 − λ2)).The sets σ −n ω ))} are measurable and cover Ω, since for fixed ω, density of the Eis and continuity of g L (2n) σ −n ω means that we may find an i with g(L (2n) σ −n ω ) as we please.Then lemma 8 above provides measurable functions and the pushforward is also then measurable by lemma 5.
First, we establish that this sequence is Cauchy, and therefore convergent, to a family of spaces E ∈ M(Ω → Gm 1 X).
For almost every ω ∈ Ω the fastest m1 dimensional growth rate ).Thus for each ω in this full measure set we may choose Mω such that for n ≥ Mω we can usefully estimate growth under L in a few cases: Applying the inequalities in lemma 7 then, and In addition, L σ −(n+1) ω Ẽ(n+1) (ω) is also guaranteed to have fast growth under L 4 ) .E (n) (ω) consists then of the image of vectors that were fast from time −n to 0, and will grow fast from time 0 to n.Then by lemma 11 with Θ = e n(λ 1 −4 ) we have 5 ) .
Thus E (n) (ω) is Cauchy and convergent since Gm 1 X is complete, say to E(ω).
To prove equivariance, observe that for n ≥ max{Mω, Mσω}, we find Then, once again, by closeness of images of fast spaces, Thus we may conclude that as well as being equivariant, E(ω) is fast for all sufficiently large n; since the choice of x was arbitrary the growth is uniform: On the other hand, for n sufficiently large we also have ω , E(ω)) → λ1.Finally, we check uniqueness: Suppose that E(ω) and E (ω) are both equivariant and fast, so that for every ω there is some N such that for n ≥ N , Define ϕ ∈ M(Ω → [0, 1]) by ϕ(ω) = d(E(ω), E (ω)).Applying lemma 11, for almost every ω we have ϕ tends to zero along all orbits.Therefore the sets {ϕ(ω) > } all have measure zero, whence ϕ vanishes almost everywhere and E = E .
The following lemma provides the top fast space for a general quasicompact cocycle: Lemma 27.Let R = (Ω, σ, X, L) be a quasicompact, semi-invertible random linear dynamical system.Then there exists a forward-equivariant decomposition X = E(ω) ⊕ V (ω) , where V : Ω → G k X and the corresponding projection is a strongly measurable Π : Ω → B(X).V is a slow growing space: ω | V (ω) → λ2 almost surely.Finally, Πω is tempered.
Proof.By lemma 14 we may choose a measurable family of bases Write vi(ω) = bi(E(ω)) which is itself then measurable.Let {(qij) k i=1 } j∈N be dense in R k .Let Tj ∈ M(Ω × X → X) be defined by Π (n) ∈ M(X → X) may then be defined piecewise by x otherwise, and To see that ι is finite on X \ E(ω), first note that the set {Tjx : j ∈ N} is dense in the set S = E(ω) + x.The set is open and scale invariant -for every θ = 0 we have θU = U , so is open and nonempty in S, whence U ∩ {Tjx}j = ∅, so that an ιn(ω, x) may be found in finite time.To see that ιn is measurable, note that Immediately from the definition, P ω x ∈ E(ω).By convergence of the 1 n log ρ k s, for all n greater than or equal to some Nω, In addition there is an easy bound independent of x ∈ SX on Π (n) ω : which rearranged yields Consider differences between successive approximate slow components: whence gaps between subsequent points decay exponentially, so that (Π ω (x)) n∈N forms a Cauchy and thus convergent sequence.Set Note that ΠωSX is then bounded, since the final line being independent of choice of x.Not only then is Π (n) ω x convergent, but we have the estimate Πω is linear: to see this let b, c ∈ X and t ∈ R, and set Then applying the estimate for Π Thus d = 0 and Πω ∈ B(X).By construction ΠωE(ω) = 0. On the other hand, PωX = E(ω) since for any x ∈ X and n ∈ N we have x − Π and is thus a projection.Set V (ω) = ΠωX so that X = V (ω) ⊕ E(ω).For all n ≥ Nω, and any x ∈ Vω(x), because of the exponential rate of convergence there is a sequence of approximants which means by dimension counting that there is always some xn ∈ S V (ω)∩Yn with 1 n L ω xn → λ2 and so Further, the map ω → Πω is strongly measurable, since for each x ∈ X the map ω → Πωx is the limit of a sequence of measurable functions.
To see that V (ω) is equivariant it is sufficient to show that Pσω • Lωx = 0 for all x ∈ V (ω).If this were not the case, then Lωx would have a nonzero component in E(σω).From this is would follow that λω(x) = λ1 which would contradict the fact that L (n) ω | V (ω) < e n(λ 2 +8 ) for sufficiently large n.
As for temperedness of the projections: Since Πω is bounded pointwise we may choose an A ⊆ Ω of positive measure on which Πω is at most some M > 0. Then define a new cocycle L by Then L ω is forward-integrable since Since A has positive measure, there is almost surely an n such that L (n) ω E(ω) = 0, and so We may then conclude that for each x ∈ X, λ ω (x) ≤ λ2 and λ 1 ≤ λ2.Applying lemma 21 to the subadditive families gn = log L (n) ω and g n = log L (n) ω there is an N1 such that for n ≥ N1, L n→2n < e n(λ 2 + ) and Ln→2n > e n(λ 1 − ) .
Clearly L n→2n = Ln→2n since the latter has a greater norm.Therefore there must be some j such that L n→2n = L n→j • Π σ j ω • L j→2n = Ln→2n • Πσnω.
was arbitrary and the norms of Π and P differ by at most 1 so 1 n log Pσnω , 1 n log Πσnω → 0 as required.

Proof of main result
Finally, we may conclude with the main result, a well behaved decomposition of the space acted on by a random linear dynamical system: Proof of thorem 3. The decomposition is obtained inductively.At each stage it is shown that if λi+1 exists, there exists an equivariant decomposition with E ≤i ∈ SM(ω → GM i X) and bounded projections Πi+1ω : X → Vi+1(ω) and Piω : X → E ≤i (ω).
The existence of the top fast space has already been established -here denote this E<2(ω) ⊕ V2(ω) with measurable projections Π2ω and P2ω.
Suppose that the statement is true up to i = l − 1.If λ l = ν then we are done, so suppose otherwise -that there exists λ l+1 ≥ ν.The projection Π lω is pointwise bounded, so that there exists some M > 0 such that A = { Π lω < M } has positive measure.
As before L is forward integrable.Write λ , µ i , λ i , ν , E l ⊕ V , M i , P and Π for the asymptotic growth rates, Lyapunov exponents, decomposition, fast space multiplicities, fast projection and slow projection with L .There almost surely exists an N ∈ N such that for each n ≥ N , the followng hold: • L Consider a decomposition of the interval [a, b) = [a, c) [c, b) ⊂ Z.In discrete time, where these intervals represent different intervals in which dynamics may occur.Given a cocycle L : Ω → B(X), for fixed ω and each b ≥ a ∈ Z we may define the map L a→b (ω) := L (b−a)