A semi-invertible Oseledets Theorem with applications to transfer operator cocycles

Oseledets' celebrated Multiplicative Ergodic Theorem (MET) is concerned with the exponential growth rates of vectors under the action of a linear cocycle on R^d. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of R^d into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated that a splitting over R^d is guaranteed even without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.


Introduction
Oseledets-type ergodic theorems deal with dynamical systems σ : Ω → Ω where for each ω ∈ Ω there is an operator (or in the original Oseledets case a matrix) L ω acting on a linear space X. One then studies the properties of the operator L (n) ω = L σ n−1 ω • · · · • L ω , giving an ω-dependent decomposition of X into subspaces with a hierarchy of expansion properties.
Prior to the previous work of the current authors, [10], all of the Oseledets-type theorems in the literature split into two cases according to the hypotheses: the invertible and non-invertible cases.
Invertible case: In this case the base dynamical system σ is assumed to be invertible and the operators L ω are assumed to be invertible (or in some cases just injective). Integrability conditions may be imposed on L −1 ω . The conclusion here is that the space X admits an invariant splitting E 1 (ω) ⊕ E 2 (ω) ⊕ · · · , finite or countable, possibly with a 'remainder' in the infinite-dimensional case. Non-zero vectors in E i (ω) expand exactly at rate λ i . Non-invertible case: In the non-invertible case no assumptions are made about invertibility of the base nor about injectivity of the operators. The weaker conclusion here is that X admits an invariant filtration V 1 (ω) ⊃ V 2 (ω) ⊃ · · · such that vectors in V i (ω) \ V i+1 (ω) expand at rate λ i .
The conclusion in the invertible case may be seen to be much stronger as one obtains an invariant family of complements to V i+1 (ω) in V i (ω). These are in general finite-dimensional so that one 'sees the vectors responsible for λ i expansion'. This is of considerable importance in applications.
Our principal contribution here is to focus on the semi-invertible case. Here assumptions are made about the invertibility of the base transformation, but there are no assumptions about invertibility or injectivity of the operators L ω . In spite of this we are able to show that one can obtain an invariant splitting rather than the weaker invariant filtration, for the setting investigated by Thieullen [27] where the random compositions have some quasi-compactness properties. In [10] we obtained an analagous result for the original Oseledets setting of matrices acting on R d .
1.1. Set-up. Let (Ω, F , P) be a probability space and (X, · ) a Banach space. A random dynamical system is a tuple R = (Ω, F , P, σ, X, L), where σ is an invertible measure-preserving transformation of (Ω, F , P), called the base transformation, and L : Ω → L(X, X) is a family of bounded linear maps of X, called the generator. We will later impose suitable measurability conditions on L.
• We say that R is quasi-compact if for almost every ω there is an α < λ(ω) such that V α (ω) is finite co-dimensional. Of particular interest is the infimal α with this property. We call this quantity α(ω). • For each isolated Lyapunov exponent r ∈ Λ(ω), let ǫ r > 0 be small enough that Λ(ω) ∩ (r − ǫ r , r) = ∅. If the codimension d of V r−ǫr (ω) in V r (ω) is finite, then we say r is a Lyapunov exponent of multiplicity d. • The Lyapunov exponents greater than α(ω) are said to be exceptional.
In the setting where P is ergodic and the generator L satisfies suitable measurability conditions, λ(ω) = λ * , α(ω) = α * and the exceptional Lyapunov spectrum will be independent of ω P-a.e.
If X is a finite dimensional space, then α(ω) = −∞ for each ω and so all Lyapunov exponents are exceptional. Since the sets V α (ω) are subspaces for each α ∈ R, the number of Lyapunov exponents counted with multiplicity is bounded by the dimension of X, and so each is isolated and of finite multiplicity.
We are interested in Banach space analogues of the multiplicative ergodic theorem. In order to make sense of this it will be necessary to put a topology on suitable collections of subspaces of Banach spaces. The Grassmannian G(X) of a Banach space X is defined to be the set of complemented closed subspaces E of X (that is, those for which there is a second closed subspace F with the property that X = E ⊕ F ). Since every finite dimensional subspace of X is closed and complemented, the collection of d-dimensional subspaces of X forms a subset of G(X), which we denote by G d (X). Also, finite codimensional subspaces are necessarily complemented, so the collection of closed c-codimensional subspaces of X forms a subset of G(X), which we denote by respectively G c (X). More details on the Grassmannian are given in Section 2 along with proofs of some basic theorems concerning Grassmannians that we shall need later.
Definition 2. Consider a random dynamical system R = (Ω, F , P, σ, X, L) with ergodic base, and suppose R is quasi-compact with exceptional spectrum , such that for all ω in a full measure σ-invariant subset Ω ′ ⊂ Ω and for each i = 1, . . . , p: Oseledets subspaces, such that for all ω in a full measure σ-invariant subset Ω ′ ⊂ Ω and for each i = 1, . . . , p: We say a Lyapunov filtration is measurable if the maps V i : Ω → G ci (X) are measurable with respect to the Borel σ-algebra on G(X) for each 1 ≤ i ≤ p, where the topology will be defined in the next section. We say an Oseledets splitting is measurable if, in addition, the maps E i : Ω → G di (X) are measurable.
1.2. Multiplicative Ergodic Theorems. The first result on the existence of Lyapunov filtrations and Oseledets splittings in the finite dimensional setting is the Multiplicative Ergodic Theorem of Oseledets. Throughout, we define log + (x) = max{0, log x}.
Theorem 3 (Oseledets [22]). Let R = (Ω, F , P, σ, R d , L) be a random dynamical system with ergodic base, and suppose that the generator L is measurable and log + L ω dP < +∞. Then R admits a measurable Lyapunov filtration. Moreover, if the base is invertible, L ω is invertible a. e. and log + L ±1 ω dP < +∞, then R admits a measurable Oseledets splitting.
This situation may be summarized by saying that Oseledets splittings can be found when the base is invertible and the linear actions in the cocycle are invertible with bounded inverses, whereas in the non-invertible linear action cases the theorem only guarantees a Lyapunov filtration. This situation persisted in all subsequent versions [24,19,27] and extensions of the Oseledets theorem, to our knowledge, until the result stated below by the current authors which obtained a Oseledets splitting in the semi-invertible case where the base is invertible but the operators themselves are not assumed to be invertible (or they are invertible but there is no bound on the logarithmic norms of their inverses).
Remark 5. It is natural to ask whether one can obtain an invariant splitting in the absence of invertibility of either the base or the operators. In section 3.3 we show that if the base is non-invertible then even in the case where the operators are invertible one cannot in general obtain an invariant splitting.
The result of Oseledets has been extended by many authors. Of particular relevance to us are the result of Ruelle [24] dealing with the case where X is a Hilbert space and the result of Mañé [19] on random dynamical systems of compact operators in Banach spaces. This was subsequently extended to the quasi-compact case by Thieullen [27]. Thieullen's result will be stated precisely in Section 3. A key requirement for Thieullen's extension is that the dependence of the operator L ω on ω is required to be P-continuous (the definition follows in Section 3) and it is upon this that we build. It should be pointed out that this is a significant limitation as many natural random dynamical systems fail to satisfy this condition (e.g. if T ω is a family of Lasota-Yorke maps, it is almost never the case that their Perron-Frobenius operators depend in a P-continuous way on ω). A parallel approach was taken in the recent thesis of Lian [17] where the measurability condition is relaxed to the weaker 'strongly measurable' condition (meaning that for each fixed x ∈ X, the map ω → L ω x is measurable). The cost (which is again heavy from the point of view of applications) is that in order to obtain suitable measurability Lian imposes the condition that the Banach space X be separable.
Our main theorem is related to Thieullen's theorem in exactly the way that our theorem from [10] is related to Oseledets' Theorem: it provides an Oseledets splitting for the category of a quasi-compact linear action in the semi-invertible case where the base is invertible without any invertibility assumptions on the operators. We include the statement here, but defer some relevant definitions to section 3.
Main Theorem (Theorem 17). Let Ω be a Borel subset of a separable complete metric space, F the Borel sigma-algebra and P a Borel probabilty. Let X be a Banach space and consider a random dynamical system R = (Ω, F , P, σ, X, L) with base transformation σ : Ω → Ω an ergodic homeomorphism, and suppose that the generator L : Ω → L(X, X) is P-continuous and satisfies log + L ω dP < +∞.
If κ(ω) < λ(ω) (where κ is the "index of compactness" of L) for almost every ω, then R is quasi-compact and admits a unique P-continuous Oseledets splitting.

1.3.
Overview. An outline of the paper is as follows. In Section 2 we prove some basic results concerning Grassmanians. In Section 3 we describe the result of Thieullen [27] and introduce the key notions of P-continuity and index of compactness from that work. We then prove our main result. Section 4 describes two applications of our main result: Perron-Frobenius cocycles generated by random "Rychlik" maps (generalisations of Lasota-Yorke maps), and transfer operator cocycles generated by subshifts of finite type with random weight functions.

The Grassmannian of a Banach Space
Let X be a Banach space and suppose that E, F ⊂ X are subspaces forming a direct (algebraic) sum: that is, E + F = X and E ∩ F = {0}. This decomposition specifies a linear map Pr F E (e + f ) = f with range F and kernel E, called the projection onto F along E. Conversely, any projection P : X → X (that is, a linear map satisfying P 2 = P ) determines a decomposition X = ker(P ) + ran(P ), where ker(P ) ∩ ran(P ) = {0}.
Unlike in finite dimensions, not all projections in infinite-dimensional Banach spaces are continuous. A necessary (and sufficient) condition for a projection to be continuous is that it has a closed range. Since every continuous linear map has a closed kernel, it follows that every continuous projection determines a topological direct sum: a direct sum decomposition into complementary closed subspaces. We denote the topological direct sum of subspaces E, F ⊂ X by E ⊕ F . Conversely, it follows from the Closed Graph Theorem that if E ⊂ X is a closed subspace with a closed complementary subspace F ⊂ X, then Pr F E is continuous.
As mentioned before the Grassmannian of X, denoted G(X), is the collection of complemented closed subspaces of X. The set G(X) admits a Banach manifold structure as follows. Given E 0 ∈ G(X), fix any F 0 ∈ G(X) for which E 0 ⊕ F 0 = X. We can use F 0 to define a neighbourhood of E 0 : we set U F0 = {E ∈ G(X) : We now prove some basic properties of the Grassmannian G(X).
Lemma 6. Let X be a Banach space and let Ω be a topological space. Suppose that for each ω ∈ Ω there are closed subspaces V (ω) and W (ω) whose topological direct sum is X. Suppose further that V (ω) and W (ω) depend continuously on ω.
Let R(ω) = Pr V (ω) W (ω) be the projection of X onto V (ω) along W (ω). Then the mapping ω → R(ω) is continuous (with respect to the operator norm on L(X, X)).
Given any ǫ > 0, since V (ω) and W (ω) are continuous there is a neighbourhood N 2 of ω 0 contained in N 1 such that for ω ∈ N 2 one has . Write the right side as x 1 + x 2 . Now we split x 1 and x 2 into parts lying in V (ω 0 ) and W (ω 0 ) as x 1 = x 11 + x 12 and Rearranging we have 22 and similarly x 22 < C x + ǫ x 11 . Summing and rearranging we obtain Since ǫ may be chosen arbitrarily small, this establishes continuity of R at ω 0 . Lemma 7. Let R : Ω → L(X, X) and E : Ω → G(X) be continuous. Then ω → R(ω)| E(ω) is continuous.
Lemma 8. Suppose that the map V : Ω → G(X) is continuous and that there are elements E 0 and F 0 of the Grassmannian such that In particular we see that E 0 is a topological complementary subspace to V (ω) ⊕ F 0 for ω ∈ N 1 . Let ω 1 ∈ N 1 be fixed. We need to establish that for ω sufficiently close to ω 1 , Pr E0 V (ω)⊕F0 | V (ω1)⊕F0 is small. We demonstrate this by writing the operator as the composition of three parts: two of them bounded and the third one small.
Proof. Fix ω 0 ∈ Ω. Let F 0 be a topological complement of V (ω 0 ) ⊕ W (ω 0 ) (this exists as all finite-dimensional subspaces have a topological complement). In order to demonstrate continuity we need to show that is of small norm with a similar result for the restriction to W (ω 0 ). We write We have from Lemmas 6 and 8 that It remains to show that Pr F0 W (ω) remains bounded on a neighbourhood of ω 0 . To see this we note from Lemma 6 that Pr F0⊕V (ω0) W (ω) is continuous on a neighbourhood of ω 0 and Pr F0 V (ω0) is a bounded operator since F 0 ⊕ V (ω 0 ) is a topological direct sum. Composing these two operators gives the required result.
Lemma 10. Let X be a Banach space, K a compact metrizable space and let E : K → G d (X) be a continuous map. Let P be a finite measure on K. Then there exists an open and dense measurable subset U of K with full P-measure and maps e 1 , . . . , e d : K → X with e i |U continuous, i = 1, . . . , d such that for each ω ∈ U , e 1 (ω), . . . , e d (ω) is a basis for E(ω).
Furthermore, the basis can be chosen so that for each ω ∈ U and all a ∈ R d , Proof.
. By a theorem of F. John (see [5,Chapter 4 Theorem 15] for example), there exists a basis v 1 , .
Notice that these vectors depend continuously on ω by Lemma 6. Replacing U ω0 by a smaller neighbourhood of ω 0 if necessary, we may assume that for all ω ∈ U ω0 and a ∈ R d , It follows that the vectors are linearly independent and hence form a basis for E(ω).
We have that {U ω : ω ∈ K} is an open cover of K. Let ρ be a metric on K compatible with the topology and let δ > 0 be the Lebesgue number of the cover: that is, for every 0 < r < δ and ω ∈ K, there exists ω ′ ∈ Ω such that B r (ω), the open ball of radius r centred at ω, is contained in U ω ′ . Fix 0 < r 0 < δ and consider the open cover {B r0 (ω) : ω ∈ K} of K. By compactness, we have a finite subcover {B r0 (ω i ) : i = 1, . . . , k}. For each i = 1, . . . , k, the collection {∂B r (ω i ) : r 0 < r < δ} is an uncountable family of pairwise disjoint sets (contained in the sphere of radius r about ω i ), and so there exists r ∈ (r 0 , δ) such that P(∂B r (ω i )) = 0 for each i.
We have that {B i := B r (ω i ) : i = 1, . . . , k} is a cover of K by open sets whose boundaries have zero P-measure. These sets have the additional property that for for ω ∈ D j , for each i = 1, . . . , d and j = 1, . . . , k gives maps with the required properties.

Oseledets splitting
Thieullen [27] in his work on multiplicative ergodic theorems for operators introduced a framework on which this paper will be based. A key notion introduced in that paper is P-continuity.
Definition 11. For a topological space Ω, equipped with a Borel probability P, a mapping f from Ω to a topological space Y is said to be P-continuous if Ω can be expressed as a countable union of Borel sets such that the restriction of f to each is continuous.
Remark 12. As noted in [27], if Ω is homeomorphic to a Borel subset of a separable complete metric space, then a function f : Ω → Y is P-continuous if and only if there exists a sequence (K n ) n≥0 of pairwise disjoint compact subsets of X such that µ( n≥0 K n ) = 1 and the restriction f | Kn is continuous for each n ≥ 0.
We shall call a Lyapunov filtration or Oseledets splitting P-continuous if all of the exponents and all maps into the Grassmannian are P-continuous (with respect to the topology defined in Section 2 in the case of maps into the Grassmannian).
Remark 13. If P is a Radon measure on Ω (that is, locally finite and tight) and Y is a metric space, then a map f : Ω → Y is P-continuous if and only if it is measurable (see [9]). In particular, this is the case in the 'Polish noise' setting (see, for example, [15,1]), where Ω is a separable topological space with a complete metric, F is the Borel sigma-algebra and P is any Borel probability.
Consider a random dynamical system R = (Ω, F , P, σ, X, L). If σ is invertible with a measurable inverse, we say R has an invertible base. If Ω is a Borel subset of a complete separable metric space, F is the Borel sigma-algebra and σ is continuous (or a homeomorphism), we say R has a continuous (or homeomorphic) base.
Suppose R is a random dynamical system with a homeomorphic base. Provided ω → L ω is P-continuous we see that ω → L (n) ω is P-continuous and hence Fmeasurable. We shall assume throughout that log is a subadditive sequence of functions it follows from the subadditive ergodic theorem that for almost every ω, 1 n log L (n) ω is convergent and hence the quantity λ(ω) defined in (3) may be re-expressed as The boundedness of log + L ω dP(ω) ensures that λ(ω) is finite P-almost everywhere.
ω v ≤ N e nr , ∀n ∈ N}. The set A N is closed, and by the choice of r, we have N ∈N A N = X for each ω ∈ Ω. Thus by the Baire Category Theorem, there exists an A N containing an interior point u. Let δ > 0 be small enough that B δ (u) ⊂ A N . For any v ∈ B δ (0) and n > 0, we have L Since r is an arbitrary quantity greater than Λ(ω), the result follows.
We concentrate on the setting in which σ is ergodic. The function λ(ω) is then constant along orbits, and thus essentially constant. We denote by λ * ∈ R the constant satisfying λ(ω) = λ * for almost every ω ∈ Ω.
A second key concept introduced by Thieullen is that of the index of compactness of a random composition of operators. For a bounded operator A, A ic is defined to be the infimal r such that A(B X ) may be covered by a finite number of r-balls, where B X is the unit ball in X. We have AA ′ ic ≤ A ic A ′ ic for any bounded linear operators on X. One can check that is a continuous function of the operator. In particular for each n, ω → L n ω ic is Pcontinuous and hence F -measurable. By sub-additivity we have (1/n) log L (n) ω ic is convergent.
ic is called the index of compactness of the random composition of operators.
Since κ(ω) is σ-invariant it is equal almost everywhere to a constant which we call κ * .
Moreover, if the base is invertible and L ω is injective a. e., then R admits a P-continuous Oseledets splitting.
Our main result in this article is the extension of Thieullen's theorem to show that one obtains an Oseledets splitting in Thieullen's setting without making the assumption of invertibility of the L ω .

Theorem 17.
Let Ω be a Borel subset of a separable complete metric space, F the Borel sigma-algebra and P a Borel probabilty. Let X be a Banach space and consider a random dynamical system R = (Ω, F , P, σ, X, L) with base transformation σ : Ω → Ω an ergodic homeomorphism, and suppose that the generator L : Ω → L(X, X) is P-continuous and satisfies log + L ω dP < +∞.
If κ * < λ * for almost every ω, then R is quasi-compact and admits a unique Pcontinuous Oseledets splitting.
The proof of this theorem (which makes extensive use of Theorem 16) is given in the next two subsections, in which existence and uniqueness of the Oseledets splitting, respectively, are proved.
3.1. Existence of an Oseledets splitting. Consider a random dynamical system R = (Ω, F , P, σ, X, L) with an ergodic homeomorphic base. Suppose L is P-continuous and log the Lyapunov filtration. Following Thieullen, we construct an extension Banach spaceX, and a new generatorL : Ω → L(X,X) whose cocycle retains all the dynamical information of the original system but has the advantage thatL ω is injective.
The extended random dynamical systemR = (Ω, F , P, σ,X,L) is defined as follows:X for a positive sequence (α n ) ∞ n=0 decaying to zero. We endowX with the norm ṽ X = sup n v n X whereṽ = (v n ) ∞ n=0 . EveryL ω is injective onX. In Thieullen's article sufficient conditions on the speed of decay of the sequence (α n ) are given to ensure that the indices of compactness ofR and R are equal (κ * = κ * ) and that λ * = λ * . In fact we check in Subsection 3.4 that this holds for any sequence (α n ) of positive numbers tending to 0.
For each 1 ≤ i ≤ p, we define E i (ω) = πẼ i (ω). As the linear image of a finite dimensional space, E i (ω) is a closed subspace. We now demonstrate that (E i : Ω → G(X)) p i=1 is the splitting we seek.

3.2.
Uniqueness of the Oseledets splitting. Consider R = (Ω, F , P, σ, X, L), a quasi-compact random dynamical system, and assume that σ is ergodic and the Oseledets subspaces constructed above.
The following lemma gives us exponential uniformity in a finite-dimensional subspace all of whose Lyapunov exponents are equal. A result of this type first appeared in the Euclidean case in a paper of Barreira and Silva [4] (see also [10] for an independent proof). The proof here follows by choosing a suitable basis.
Proof. By Lemma 10, since E : where · 2 represents the Euclidean norm on The linear map A is invertible a. e. and satisfies 1/(4 Since A(ω) is a bijection, it follows that lim n→∞ (1/n) log τ (n) (ω)a 2 = λ i for each a ∈ R d \{0}. Applying the theorem of Barreira Reusing the above inequalities the proof of the Lemma is complete.
A sequence (v n ) n∈Z is called a full orbit at ω ∈ Ω if L(σ n ω)v n = v n+1 for all n ∈ Z. For full orbits, we may consider growth rates as n → −∞.
Lemma 20. Let (v n ) n∈Z ⊂ X be a full orbit for ω ∈ Ω ′ and suppose v n ∈ V i (σ n ω) for all n ∈ Z. Then lim inf If we have v n ∈ E i (σ n ω) for all n ∈ Z, then we have the stronger statement Proof. We have lim n→∞ (1/n) log L (n) ω |V i (ω) = λ i , and by [10,Lemma 8.2], it follows that lim n→∞ (1/n) log L (n) (σ −n ω)|V i (σ −n ω) = λ i . Thus for any full orbit {v n } n∈Z satisfying 0 = v n ∈ V i (σ n ω) for all n ∈ Z, we have Thus lim inf For the second statement we shall assume that v n ∈ E i (σ n ω) for all n ∈ Z. The mapping L ω |E i (ω) is a bijection, so we denote by S(ω) : E i (σω) → E i (ω) the inverse map. We let S (n) (ω) := S(σ −n ω) · · · S(σ −1 ω) = [L (n) σ −n ω | Ei(σ −n ω) ] −1 denote the cocycle for the map σ −1 generated by S. As log S (n) (ω) is a subadditive sequence of functions over σ −1 , using [10, Lemma 8.2] again we have where the last equality follows from Lemma 19. Suppose now that 0 = v n ∈ E i (σ n ω) for all n ∈ Z. Then we have lim sup Claim 21. The P-continuous Oseledets splitting is unique on a full measure subset of Ω.
for almost every ω ∈ Ω. Assume for a contradiction that there is a measurable subset J ⊂ Ω, P(J) > 0, such that Let (U n ) n≥0 be a sequence of measurable subsets of Ω, P( n≥0 U n ) = 1, such that the maps V i+1 | Un , E i | Un and F i | Un are continuous. By Lemma 9, the map E i ⊕ F i is continuous on U n for each n ≥ 0. By Lemma 6, the map R(ω) := Pr Vi+1(ω) Ei(ω)⊕Fi(ω) is continuous on U n for each n ≥ 0. Thus, by Lemma 7, the mapping g(ω) = R(ω)| E ′ i (ω) is P-continuous, and in particular, is F -measurable. We first prove that lim n→∞ g(σ n ω) = 0 for almost all ω. Let ω ∈ Ω ′ be given. For any fixed u ∈ E ′ i (ω) \ {0}, we have R(ω)u ∈ V i+1 (ω) so that for any ǫ > 0 there exists a C < ∞ with L (n) ω R(ω)u ≤ Ce n(λi+1+ǫ) for all n > 0. On the other hand since u ∈ V i (ω) \ V i+1 (ω), there is a C ′ > 0 such that L (n) ω u ≥ C ′ e n(λi−ǫ) for all n. Fix ǫ < 1 4 (λ i − λ i−1 ). We have for each fixed u there is a constant C u such that We now use a Baire category argument. Define D N by Since these sets are closed and their union is all of E ′ i (ω), one of them must contain a ball B δ (u) ∩ E ′ i (ω). By scale-invariance it contains a ball B 1 (u/δ) ∩ E ′ i (ω). Set u 0 = u/δ and let x ∈ E ′ i (ω) satisfy x = 1. Then we have for each n ω , subtracting the above two inequalities and using the triangle inequality we obtain The numerator is bounded above by an expression of the form Ce n(λi+ǫ) . Similarly, by Lemma 19 the denominator is bounded below by an expression of the form C ′ e n(λi−ǫ) . It follows that g(σ n ω) ≤ (N C/C ′ )e −n(λi−λi+1−4ǫ) . By our choice of ǫ we see that g(σ n ω) → 0 as claimed. Now let ω ∈ J and let (v n ) be a full orbit over ω with v 0 ∈ E ′ i (ω)\E i (ω). Such an orbit exists since L ω maps E ′ i (ω) bijectively to E ′ i (σ(ω)). Let u n = v n − R(σ n ω)v n and w n = R(σ n ω)v n . Since E ′ i (ω) ⊂ E i (ω)⊕V i+1 (ω) we see that u n ∈ E i (σ n ω) (i.e. u n has no component in F i (σ n ω)). We also have w n ∈ V i+1 (σ n ω). Since w 0 = 0 we have w n = 0 for all n < 0.
We now have R(σ −n ω)(w −n + u −n ) = w −n . Lemma 20 tells us that u −n ≤ Ce −n(λi−ǫ) and that w −n ≥ C ′ e −n(λi+1+ǫ) . We deduce that If we consider the set A = {ω ∈ Ω : g(ω) < 1/2} we have for almost every ω, σ n ω ∈ A for all large positive n whereas σ n ω ∈ A for all large negative n. This contradicts the Poincaré recurrence theorem, and hence the promised uniqueness is established.

3.3.
Necessity of invertibility of the base. The Main Theorem provides an invariant splitting in the absence of invertibility of the operators as long as the base is invertible. It is natural to ask whether one can obtain an invariant splitting in the absence of invertibility of the base. The following example establishes that in general this is not possible.
Example 22. Let Σ = {0, 1} Z be equipped with the shift-transformation σ and the ( 1 2 , 1 2 )-Bernoulli measure and let A 0 and A 1 be two non-commuting invertible 2 × 2 matrices which we consider as operators on R 2 . Let L ω : R 2 → R 2 be given by L ω = A ω0 . We assume further that the two Lyapunov exponents of the random dynamical system differ. As is standard we define for n > 0, L We call this random dynamical system R.
We define an inverse system R as follows: Σ = Σ where the base map is σ = σ −1 . We define the operators on this inverse system by L ω = A −1 ω−1 . Oseledets' theorem guarantees that the splitting E 1 (ω) ⊕ E 2 (ω) also works for R.
Suppose now for a contradiction that there are Oseledets splittings for R + and Then one can check that E + 1 (π + (ω)) ⊕ E + 2 (π + (ω)) is an invariant splitting for R which gives the correct rates of expansion as n → ∞. Theorem 17 guarantees that there is only one such splitting and hence we see that (8) E i (ω) = E + i (π + (ω)) for almost every ω.
Similarly E − 1 (π − (ω)) ⊕ E − 2 (π − (ω)) is an invariant splitting for R which gives the correct rates of expansion as the power n of the inverse random dynamical system approaches ∞. Since the splitting for R was the same as that for R we deduce that (9) E i (ω) = E − i (π − (ω)) for almost every ω.
Since the intersection F − ∩ F + is the trivial sigma-algebra it follows that E i is constant almost everywhere, equal to E * i say. From this it follows that A 0 (E * i ) = A 1 (E * i ) = E * i so that A 0 and A 1 have common eigenspaces and hence are simultaneously diagonalizable. Since they do not commute by assumption this is a contradiction.

3.4.
Reduction to the invertible case in Thieullen's Theorem. As mentioned above Thieullen deduces the non-invertible version of his theorem from the invertible case by constructing an invertible extension of the given system. More specifically if the original system has maps L ω acting on a Banach space X the new system has mapsL ω acting on a Banach spaceX wherẽ Thieullen then defines γ n = k≤n log α k and states conditions on the (α n ) and (γ n ) which suffice to ensure that the exceptional spectrum of the extension agrees with the exceptional spectrum of the original system.
We claim that Condition (1) implies the other two conditions. That (1) implies (2) is immediate. We now indicate a brief proof that (2) implies (3).
We remark that in Thieullen's proofs it is sufficient to take a sequence (α k ) for which (2) is satisfied. Clearly the most natural way to do this is to take any sequence satisfying (1).

Applications
The motivation for the development of Theorem 4 is the desire to extend transfer operator approaches for the global analysis of dynamical systems from deterministic autonomous dynamical systems to random or non-autonomous dynamical systems.
A common setting for deterministic systems is: M ⊂ R m is a smooth manifold and T : M → M a C 1 map with some additional regularity properties. The (deterministic) dynamical system T : M → M has an associated Perron-Frobenius operator L T : X → X defined by L T f (x) = y∈T −1 x f (y)/| det DT (y)|, where X is a Banach space of complex-valued functions on M . The Perron-Frobenius operator evolves density functions on M forward in time, just as the map T evolves single points x ∈ M forward in time.
More generally, the "weight" 1/| det DT (y)| may be replaced with a sufficiently regular generalised weight g(y) to form a transfer operator. Perron-Frobenius operators and transfer operators have proven to be indispensable tools for studying the long term behaviour of dynamical systems. An ergodic absolutely continuous invariant probability measure (ACIP) describes the long term distribution of forward trajectories {T k x} ∞ k=0 in M for Lebesgue almost-all initial points in x ∈ M . An early use of Perron-Frobenius operators was to prove the existence of ACIMs for piecewise C 2 expanding maps [16]. A study of the peripheral spectrum of L T yielded information on the number of ergodic ACIPs [13,25]. The particular weight function 1/| det DT (y)| is attuned to ACIPs. Other "equilibrium states" can be read off from the leading eigenfunction of the transfer operator by varying the weight function g (in statistical mechanics terms, g describes the local energy of states in M ).
The spectrum of the Perron-Frobenius operator provides information on the exponential rate at which observables become temporally decorrelated. The essential spectral radius of Perron-Frobenius operators [14] establishes a threshold beyond which spectral values are necessarily isolated. Furthermore, this radius is typically connected with the average rate at which nearby trajectories separate. Thus, these isolated spectral values are of particular interest in applications because they predict decorrelation rates slower than one expects to be produced by local separation of trajectories. The eigenfunctions associated with these isolated eigenvalues have been used to detect slowly mixing structures in a variety of physical systems, see, for example, [26,8,12,7].
From a physical applications point of view, it is natural to study random or time-dependent (non-autonomous) dynamical systems using a transfer operator methodology. Theorem 4 considered this question in the setting of a finite number of piecewise linear, expanding interval maps, sharing a joint Markov partition, where the Perron-Frobenius operators acted on the space of functions of bounded variation. In the present work, in our first application, we remove the assumptions of finiteness, piecewise linearity and Markovness, and allow random compositions that are expanding-on-average. Our second application is to subshifts of finite type with random continuously-parametrized weight functions. 4.1. Application I: Interval maps. We now show that Theorem 17 can be applied in the context of random compositions of expanding-on-average mappings acting through their Perron-Frobenius operators on the space BV of functions of bounded variation. In this context a major drawback of the Thieullen approach becomes clear: if T 1 and T 2 are any two distinct expanding mappings then their Perron-Frobenius operators L T1 and L T2 are far apart in the operator norm on BV. In fact the set of Perron-Frobenius operators acting on BV is discrete. As a consequence, in order for ω → L Tω to be a continuous map on a compact space, the maps range of ω → T ω is forced to be finite. If we want ω → L Tω to be P-continuous then it can have at most countable range.
Let I = [0, 1] ⊂ R denote the closed unit interval, B denote the Borel σ-algebra and m denote Lebesgue measure.
The class of Rychlik maps is closed under composition. Recall that the variation of a function f : I → R is the quantity (10) var(f ) := sup A function on the interval is said to be of bounded variation if var(f ) < ∞.
The Perron-Frobenius operator for a Rychlik map T is defined, for a function f ∈ L 1 (I) by The Perron-Frobenius operator is a Markov operator : that is, if f ∈ L 1 (I), then We consider the action of L T on the Banach space for all x. We shall assume versions are chosen so as to satisfy this condition, unless stated otherwise. We shall need a lemma that is a combination of Lemmas 4, 5 and 6 from Rychlik [25].
Lemma 24 (Rychlik [25]). Let T be a Rychlik map of the unit interval and let L T be its Perron-Frobenius operator. Suppose ess inf x |T ′ (x)| > 1. Let a = 3/ ess inf |T ′ |. Then there is a partition P of the unit interval into finitely many subintervals and a constant D such that for all f ∈ BV We define a random composition of Rychlik maps as follows. Let {T i } i∈I , be a finite or countably infinite set of Rychlik maps. Let I denote the one-point compactification of I (with the discrete topology) and let S = I Z . Let σ : S → S be the shift map and let P be an ergodic shift-invariant probability measure supported on Ω = I Z . For ω ∈ Ω let L ω = L Tω 0 be the Perron-Frobenius operator of the map T ω0 acting on the space BV. We make the further assumption that log + L ω dP(ω) < ∞ (or equivalently i∈I P({i}) log L Ti < ∞). If these conditions are satisfied we refer to the 6-tuple R = (Ω, F , P, σ, BV, L) as a Rychlik random dynamical system.
One can then verify that the system R satisfies the assumptions of Theorem 17. We denote the n-fold composition T σ n−1 ω • · · · T σω • T ω by T (n) ω . It is well known that the composition, L (n) ω , of the Perron-Frobenius operators of T ω , T σω , . . . , T σ n−1 ω is equal to the Perron-Frobenius operator of T (n) ω . A random composition may also be considered as a single transformation on the space Ω × I which we endow with the sigma-algebra F ⊗ B: the skew product Θ : Ω × I → Ω × I is given by Θ(ω, x) = (σω, T ω x).
We shall need a well-known inequality relating the index of compactness to the essential spectral radius. For a version of the converse inequality the reader is referred to work of Morris [21]. Let A : X → X be a linear operator on a Banach space. We write A fr for inf{ A − F : F has finite rank}. Recall from earlier A ic is defined to be inf{r : A(B X ) may be covered by a finite number of r-balls}.

Lemma 25. For a linear operator A between Banach spaces
Proof. Let A = F + R where F has finite rank and R = r. Let ǫ > 0. Since F (B X ) is compact it may be covered by a finite number of ǫ-balls for any Since it is possible to find decompositions with r arbitrarily close to A fr the lemma follows.
Keller [14] used Lemma 24 together with a supplementary argument to identify the essential spectral radius of the Perron-Frobenius operator of an expanding Rychlik map acting on the space of functions of bounded variation. We show that Keller's argument applies equally in our context of random dynamical systems.
Theorem 26. Let R = (Ω, F , P, σ, BV, L) be a Rychlik random dynamical system. Then there exists a χ such that for P-almost every ω, Further if χ < 1 then L (n) ω 1/n ic → χ. Definition 27. We say that the Rychlik random dynamical system appearing in the theorem is expanding-on-average if χ < 1.
Proof. We note that both L (n) ω ic and a n (ω) = 1/ ess inf x T (n) ω ′ (x) are submultiplicative. It follows from the subadditive ergodic theorem that both of the limits appearing in the statement of the theorem exist for P-almost every ω. In the case where χ < 1 we claim the following inequalities: (13) a n (ω) ≤ L (n) ω ic ≤ L (n) ω fr ≤ 3a n (ω) provided a n (ω) < 1.
The middle inequality is Lemma 25. To see the upper bound, let P be the partition of the interval into subintervals guaranteed by Lemma 24. Let E P be the conditional expectation operator defined by We then have L The second term has finite rank and Lemma 24 guarantees that var(L (n) has integral 0 and therefore that the L 1 norm is bounded above by the variation. This yields L (n) For the lower bound fix an ǫ > 0 and suppose that 1/|T (n) ω ′ (x)| > (1 − ǫ)a n (ω) for x in an interval J. Suppose further that J lies in a single branch of T ω f I ′ > 2(1 − ǫ)a n (ω). It follows that no (1 − ǫ)a n (ω) ball contains more than two L (n) ω f I 's with distinct endpoints and so in particular L (n) ω B BV does not have a finite cover by (1 − ǫ)a n (ω) balls. We see that L (n) ω ic ≥ (1 − ǫ)a n (ω). Since ǫ is arbitrary, we see that (13) follows.
Taking nth roots and taking the limit, the theorem follows.
Corollary 28. Let R = (Ω, F , P, σ, BV, L) be a Rychlik random dynamical system. Assume that R is expanding-on-average. Then R is quasi-compact, with κ * = lim The random dynamical system therefore admits a P-continuous Oseledets splitting.
The Oseledets splitting provides information on the invariant measures and rates of mixing of the random system. A natural generalisation of the notion of 'invariant measure' to the random setting is the concept of 'sample measure'. A family {µ ω } ω∈Ω of sample measures (see [2]), is a family of probability measures µ ω on I satisfying (1) for all U ∈ F , the map ω → µ ω (U ) is F -measurable.
Sample measures for random compositions of expanding interval maps have previously been studied by Pelikan [23], Morita [20], and in a more general setting by Buzzi [6]. He considers random compositions of Lasota-Yorke maps that have neither too many branches nor too large distortion, and proves that the associated skew product transformation possesses a finite number of mutually singular ergodic ACIPs µ, each giving a family {µ ω } ω∈Ω of sample measures with densities of bounded variation. Returning to the present setting of a random composition of Rychlik maps, any such family {f ω } ω∈Ω of sample measures with densities of bounded variation satisfies dµ ω /dm ∈ E 1 (ω) for P-almost every ω. It follows that the number of such mutually singular ergodic ACIPs (whose sample measure densities are necessarily linearly independent for P-a. e. ω) is bounded by d 1 , the dimension of the Oseledets subspace E 1 (ω).
Furthermore, the exceptional Lyapunov spectral values strictly less than 0, and their corresponding Oseledets subspaces, provide information on exponential decay rates that are slower than the decay produced by local separation of trajectories. The authors discuss and provide examples of such spectral values and Oseledets subspaces in [10]. Corollary 28 provides conditions under which Oseledets subspaces exist in much greater generality than in [10], removing the assumptions of piecewise linearity and Markovness, and allowing the system to be expanding on average. In non-rigorous numerical experiments, Oseledets subspaces have been shown to effectively capture so-called "coherent sets" in aperiodic fluid flow [11]. The present work represents a first step toward making such calculations rigorous by extending the study of Perron-Frobenius operator cocycles to Banach spaces that are more representative of fluid flow.

4.2.
Application II: Transfer Operators with Random Weights. Let Σ be a one-sided 1-step shift of finite type on N symbols. We assume that for each symbol j in the alphabet there is at least one i for which ij is a legal transition (if not we restrict our attention to the subset of Σ obtained by deleting all symbols that have no preimage). For x, y ∈ Σ we let ∆(x, y) be min{n : x n = y n } (or ∞ if x = y). The θ-metric on Σ is d θ (x, y) = θ ∆(x,y) (so that the standard metric is d 1/2 ).
We will write S for the usual left shift map on Σ. If x ∈ Σ and v is a word of some length k ≥ 1 in the alphabet such that v k−1 x 0 is a legal transition then we will write vx for the point in S −k x obtained by concatenating v and x.
Let C θ denote the set of θ-Lipschitz functions: those functions f for which there is a C such that |f (x) − f (y)| ≤ Cd θ (x, y) for all x and y. We define |f | θ to be the smallest C for which such an inequality holds. As usual we endow C θ with the topology generated by the norm f θ = max(|f | θ , f ∞ ). Let W θ be the collection of those functions g in C θ such that min x g(x) > 0.
Denote P g f (x) = y∈S −1 x f (y)g(y) and consider P g as an operator on (C θ , · θ ). For the purposes of the following lemma we consider arbitrary g ∈ C θ but we shall later restrict to g ∈ W θ .
Lemma 29. The map P : C θ → L(C θ , C θ ) is continuous with respect to the operator norm on L(C θ , C θ ).
We now bound |P g f | θ . Let x = y ∈ Σ. We need to estimate |P g f (x) − P g f (y)|/d θ (x, y). If x 0 = y 0 then the denominator is 1 and the numerator is at Combined with the estimate in the case x 0 = y 0 , this shows |P g f | θ ≤ 2N g θ · f θ and so P g ≤ 2N g θ .
Baladi's book [3] contains a number of detailed calculations of the spectral radii and essential spectral radii of Perron-Frobenius operators acting on the Lipschitz spaces. We now develop some of these arguments in the case of random compositions.
Suppose that G : Ω → W θ ; ω → g ω is a continuous mapping. Since Ω will be assumed to be compact there will be a constant γ such that g ω (x) ≥ γ for all x ∈ Σ and ω ∈ Ω. Similarly there will be a constant such that g ω θ ≤ C for all ω ∈ Ω. We assume as usual that σ : Ω → Ω is ergodic. We write P (n) ω for the composition of Perron-Frobenius operators P g σ n−1 ω • · · · • P gω .
A linear map on C θ is said to be positive if it maps non-negative functions to non-negative functions. In particular if g ∈ W θ then P g is positive.
Lemma 30. Let R = (Ω, F , P, σ, C θ , P ) be a continuous ergodic random dynamical system of Perron-Frobenius operators with random weights on a shift of finite type Σ. Suppose that Ω is compact and P : Ω → W θ is continuous. Let R n (ω) = P (n) ω 1 ∞ . Then R n (ω) 1/n converges P-almost everywhere to a constant R * .
Proof. Since the operators P g are positive (g being positive), we have P It follows that log R n (ω) is a subadditive sequence of functions so that by the subadditive ergodic theorem, for P-almost all ω, R n (ω) 1/n converges to a quantity R(ω). Since this quantity is σ-invariant, there is a constant R * such that R(ω) = R * for P-almost every ω ∈ Ω.
Lemma 31 (Bounded Distortion). Let R be as in the previous lemma. Let g (n) ω (x) denote the product g ω (x)g σ(ω) (Sx) . . . g σ n−1 ω (S n−1 x). There exists a D > 0 such that for all ω ∈ Ω, if x 0 = y 0 and v is a word of an arbitrary length k such that v k−1 x 0 is a legal transition then ≤ Dd θ (x, y).
Proof. As mentioned above there is a γ > 0 such that g ω (x) ≥ γ for all ω ∈ Ω and all x ∈ Σ. Similarly there is a Γ such that g ω (x) ≤ Γ for all ω and x and also a C such that g ω θ ≤ C for all ω ∈ Ω. We make use of the fact that there exists a constant K such that if γ < a, b < Γ then | log(b/a)| ≤ K|b − a|.
The next lemma appears as an exercise in the deterministic case in Baladi's book [3].
Lemma 32. Let R be as above. Then there exists a constant K such that for Proof. We need to estimate sup x =y |P y). If x and y differ in the zeroth coordinate, the denominator is 1 and we bound the numerator above by R n (ω) f ∞ giving a bound of the given form (with K=1).
If x and y agree in the zeroth coordinate then we estimate as follows. We let W n be the set of words v of length n such that v n−1 x 0 is legal.
We therefore see that |P (n) ω f | θ ≤ R n (ω) (θ n |f | θ + D f ∞ ) as required. Let n > 0 and let [w 1 ], . . . , [w k ] be an enumeration of the n-cylinders. For each 1 ≤ j ≤ k, let x j be a point of [w j ]. Given these choices, define a finite rank operator Π n : C θ → C θ by Proof. Let K be as in Lemma 32. Let f ∈ C θ . We have |P (n) ω f | θ ≤ (K + 1)R n (ω) f θ . Also P (n) ω f ≤ P (n) ω ( f ∞ 1) ≤ R n (ω) f ∞ . Combining these we see that P (n) ω f θ ≤ (K + 1)R n (ω) f θ . On the other hand we have P (n) ω 1 θ ≥ R n (ω) while 1 θ = 1 so the bounds on P (n) ω are established. For the upper bound on P (n) ω ic we use Lemma 25 to compare with P (n) ω fr and we let Π n be as above and give bounds on P (n) ω • (I − Π n ) . Let f ∈ C θ . We have P (n) ω •(I −Π n )f ∞ ≤ R n (ω) (I −Π n )f ∞ ≤ θ n R n (ω)|f | θ where the last inequality made use of Lemma 33. Using Lemmas 32 and 33 we see |P (n) ω ((I − Π n )f )| θ ≤ R n (ω)(θ n max(2θ, 1)|f | θ + Kθ n f ∞ ). Combining these two inequalities leads to an upper bound of the desired form for P ω 1(x) > R n (ω)/2. We show that the index of compactness is large by exhibiting an infinite collection of points in the unit sphere of C θ whose images under P (n) ω are uniformly separated. Let u ∈ U and let C k be the k-cylinder about u. Since U is open there exists a k 0 such that C k0 ⊂ U . Since Σ is an irreducible shift of finite type there exists an infinite sequence k 0 < k 1 < k 2 < . . . such that C ki is a proper subset of C ki−1 for all i ≥ 1. We let f i = θ ki+n−1 1 C k i • S n . To check that f i θ = 1 we note that if x and y agree on at least the first k i + n symbols then f i (x) = f i (y). Since the numerator of |f i (x) − f i (y)|/θ ∆(x,y) takes only the values 0 and θ ki+n−1 the maximum in this expression is obtained by taking x and y that agree for as many symbols as possible, but for which f i (x) = f i (y). By the assumption on Σ and choice of k i there are points agreeing for k i + n − 1 symbols but disagreeing on the k i + n − 1st symbol for which f i (x) = f i (y) so that f i θ = 1 as required.
Example 35. Let σ : Ω → Ω be any homeomorphic dynamical system defined on a compact space Ω preserving an ergodic probability measure P. Let Σ = {0, 1} Z + . Fix 0 < θ < 1 and let {h ω : ω ∈ Ω} be a continuously-parameterized family of antisymmetric monotonic elements of C θ (Σ), where a function is antisymmetic if it satisfies h(x) = −h(x) for x ∈ Σ, wherex i = 1 − x i . A function will be called monotonic if it satisfies h(x) ≤ h(y) whenever x y, where x y means x i ≤ y i for each i.
One can verify that the P ω map antisymmetric functions to antisymmetric functions and monotonic functions to monotonic functions. Following Liverani [18] we define a cone K a = {f : f (x) > 0, ∀x; f (x)/f (y) ≤ e ad θ (x,y) , ∀x, y}. For suitably large a, there is a ′ < a such that P ω (K a ) ⊂ K a ′ . Since 1 is a fixed point the theory of cones guarantees that if f is a positive function in C θ , then P (n) ω f converges at an exponential rate to a constant uniformly in ω. In particular an antisymmetric function f can be written as the difference of two positive functions: f 1 − f 2 . Since P (n) ω f 1 converges exponentially fast to a constant C 1 (ω) and P (n) ω f 2 converges exponentially to a constant C 2 (ω), the fact that P (n) ω f remains antisymmetric implies that C 1 (ω) = C 2 (ω). It follows that P (n) ω f converges at an exponential rate to 0 uniformly over ω ∈ Ω.
Choosing f (x) = 1 [1] − 1 [0] , f is both monotone and antisymmetric. It follows that P (n) ω f decays exponentially. We are able to give a lower bound on the decay rate that guarantees the presence of non-trivial exceptional spectrum. Specifically, using the fact that g ω (0x) + g ω (1x) = 1, we have If the h ω are chosen in such a way that g ω (1111 . . .) is uniformly close to 1 as ω varies then we will ensure that there is non-trivial exceptional spectrum.