A volume-based approach to the multiplicative ergodic theorem on Banach spaces

A volume growth-based proof of the Multiplicative Ergodic Theorem for Banach spaces is presented, following the approach of Ruelle for cocycles acting on a Hilbert space. As a consequence, we obtain a volume growth interpretation for the Lyapunov exponents of a Banach space cocycle.


1.1.
Introduction. The purpose of this paper is to present a volume-based proof of the Multiplicative Ergodic Theorem (MET) for cocycles on a Banach space, from which a volume growth rate interpretation of Lyapunov exponents for such cocycles will be readily apparent.
By now, there are many proofs of the MET for cocycles on finite dimensional vector spaces, e.g., [1,2,3,4]; for additional treatments, see Section 3.4 of [5] or §1.5 of [6]. For applications, particularly for the purposes of smooth ergodic theory, the most successful interpretation of Lyapunov exponents is that of volume growth: the exponential growth rate of the q-dimensional volume of a 'generic' q-dimensional subspace is the sum of the first q Lyapunov exponents.
The volume growth interpretation for the Lyapunov exponents for cocycles of operators on finite dimensional spaces is implicit in Ruelle's proof of the MET [3]: the q-th Lyapunov exponent is defined as the exponential growth rate of the q-th singular value of high iterates of the cocycle, and it is shown that the 'slow' growing singular value decomposition subspaces for high iterates of the cocycle converge at an exponentially fast rate to the subspaces of 'slow' growing vectors for the cocycle. The very same approach is used to prove the MET for compact cocycles on Hilbert spaces in [7], and the volume growth interpretation is the same, since the finite dimensional subspaces of a Hilbert space carry notions of volume and determinant inherited from the inner product on the ambient Hilbert space.
For cocycles on Banach spaces, the MET was originally proved by Mañé for compact, injective cocycles satisfying a certain continuity condition [8]. Thieullen extended Mañé's result to include injective cocycles satisfying a quasicompactness condition [9], and Lian and Lu proved a version of the MET for injective cocycles on separable Banach spaces satisfying only a measurability condition [10]. The method in these papers is to show that the 'fast' growing subspaces (that is, subspaces of vectors achieving a sufficient rate of exponential contraction in backwards time) are almost surely finite dimensional. Then, Lyapunov exponents are defined to be the exponential rates at which the dimension of the 'fast' growing subspaces has a jump discontinuity, and it is shown that the 'slow' growing subspaces are the graphs of mappings given by a certain convergent infinite series of graphing maps.
Thieullen also considered non-injective cocycles [9], and deduced from the MET for injective cocycles that even when the cocycles aren't injective, the 'slow' growing subspaces are finitecodimensional and vary measurably, and the only possible growth rates of vectors are Lyapunov exponents; this result is referred to as the One-Sided MET, and holds even when the base dynamics are noninvertible. Thieullen's proof in [9] involves passing to an injective 'natural extension' cocycle over an augmented Banach space. Doan, in his PhD thesis [11], carried out an analogous deduction for strongly measurable cocycles on a separable Banach space, assuming the MET in [10].
Quas, Froyland and Lloyd [12] deduced from the One-Sided MET in [9] that even when the cocycle is not injective, there is still an invariant, measurably-varying distribution of 'fast' growing subspaces, as long as the base dynamics are invertible. Quas and González-Tokman [13] also deduced the existence of an invariant distribution of 'fast'-growing subspaces for cocycles on a separable Banach space satisfying the measurability condition in [10], assuming the One-Sided MET in [11].
In all known proofs of the Banach space MET, genuine finite-dimensional volumes are not used, and there is no apparent volume growth interpretation of the Lyapunov exponents. So, it is natural to ask whether there can be a volume growth interpretation for Lyapunov exponents of cocyles acting on a Banach space, noting that there is no trouble assigning volumes to the finite dimensional subspaces of a Banach space, using Haar measure, and that one can define determinants as volume ratios taken using these volumes.
Indeed, once the MET is proven by other means, it is possible to show that Lyapunov exponents have the very same volume growth interpretation as in the Hilbert space case. In light of Ruelle's approach to the MET in [3,7], though, this order of actions is strange: volume growth is used to prove the MET, and the volume growth interpretation of the Lyapunov exponents follows naturally.
The purpose of the present paper is to show how to deduce the Banach space MET using volume growth ideas, using arguments analogous to those in [7], from which the volume growth rate interpretation of Lyapunov exponents will be easily seen. The proof we present treats noninjective cocycles with no added effort, and yields immediately that vectors sufficiently inclined away from 'slow' growing subspaces grow uniformly quickly.
For the rest of this section, we will define and describe the volumes on finite dimensional subspaces that we use, state assumptions on the cocycles we will study, and state the MET. In Section 2, we will give preliminaries on the geometry of Banach spaces and the volumes on finite dimensional subspaces necessary for the proof, which will be given in Section 3.
1.2. Statement of Results. We now state our results precisely, beginning with an overview of volumes on finite dimensional subspaces of a Banach space.
1.2.1. Volumes and Determinants on Finite Dimensional Subspaces. We let (B, | · |) be an infinite dimensional Banach space, and let E ⊂ B be a a finite dimensional subspace. Treating E as a topological group with the group action given by addition of vectors, there exists a nonzero, translation invariant measure on E, unique up to a scalar factor: the Haar measure on E ( §2.2 of [14]). In the definition of volumes given below, we normalize the Haar measure on E by assigning a particular volume to the | · |-unit ball of E. Definition 1.1. Let q ∈ N and let E ⊂ B be a q-dimensional subspace. We define the induced volume m E on E to be the unique Haar measure on E for which where ω q is the q-dimensional volume of the Euclidean unit ball in R q .
Many identities involving volume on inner product spaces can be recovered for the induced volume as inequalities, with uniform bounds depending only on the dimension q of the subspace. The following is a prototypical example, and demonstrates that the induced volume really does 'see' the geometry on finite dimensional subspaces. Lemma 1.2. If E ⊂ B has dimension q < ∞, then for any v 1 , · · · , v q ∈ E, we have For vectors w 1 , · · · , w q ∈ B, we denote by w 1 , · · · , w k the subspace of B spanned by the vectors w 1 , · · · , w q . The symbol ≈ denotes equality up to a multiplicative constant depending only on q.
Using the tools developed in Section 2, the proof of Lemma 1.2 is straightforward, and is sketched in Remark 2.16.
With these volumes in place, the determinant is defined as a volume ratio: for a bounded linear map A : B → B and a finite dimensional subspace E ⊂ B, we write where O is any Borel subset of E with nonzero m E -volume. By the uniqueness of Haar measure, the right-hand side of (1.1) does not depend on the set O ⊂ E.
Define the maximal q-dimensional volume growth for bounded, linear maps A : B → B, and observe that since the determinant is multiplicative, being a Radon-Nikodym derivative, V q is submultiplicative, i.e., for bounded linear maps A, B : 1.2.2. Statement of the MET. Let (X, F, µ) be a probability space, and let f be a measure preserving transformation on (X, F, µ). Throughout the paper, we assume for simplicity that µ is ergodic; the only real difference in the nonergodic case is that the Lyapunov exponents and their multiplicities may now depend on the point in the base space, but this complication does not change the underlying arguments in any substantive way. A cocycle is a map T : X → L(B), where L(B) is the space of bounded linear operators on B, subject to the following measurability assumption.
There is a sequence of finite-valued maps T (n) : X → L(B) such that for any x ∈ X, T (n) x → T x in L(B) in the induced operator norm as n → ∞.
We write T n x = T f n−1 x • · · · • T x , and for λ ∈ R, we define the subspaces using the convention log 0 = −∞. For A ∈ L(B), we denote by |A| α the measure of noncompactness of A, which will be defined precisely in Definition 2.8.
We now state the main result of this paper, a version of the MET emphasizing the volume growth interpretation of the Lyapunov exponents of a Banach space cocycle. Note that we do not assume that T x is an injective operator. Theorem 1.3. Let T : X → L(B) satisfy (1.3), and assume moreover that Then, there is a µ-full, f -invariant set Γ ⊂ X with the following properties.
(1) For x ∈ Γ, the q-dimensional growth rates exist and are constant over Γ. Similarly, l α = lim n→∞ 1 n log |T n x | α exists and is constant over Γ.
(2) Defining the sequence {K q } q≥1 by K 1 = l 1 and K q = l q − l q−1 for q > 1, we have that K 1 ≥ K 2 ≥ · · · , and that K q → l α as q → ∞. We denote by λ i the distinct values of the sequence {K q } q with multiplicities m i . The Lyapunov exponents λ i satisfy the following.
(b) For any subsequent pair of Lyapunov exponents λ i > λ i+1 ≥ l α and for any (c) (Volume Growth) For any x ∈ Γ, any Lyapunov exponent λ i , and any complement E to F λ i (x), we have that Remark 1.4. The measurability assumption on the cocycle in [8,9] is called µ-continuity, and is stronger than (1.3): µ-continuity requires the base space X to be a compact topological space with Borel σ-algebra F, the base map f to be continuous, and the existence of a sequence of disjoint compact sets {K n } n in X with µ (∪ n K n ) = 1 for which T | Kn is norm continuous K n → L(B).
In [10], the Banach space B is assumed separable, and the measurability condition in that paper is strong measurability, which requires that the evaluation maps x → T x v for v ∈ B be measurable as maps (X, F) → B (Definition 3.5.4 in [15]). Strong measurability is weaker than (1.3); however, if T satisfies (1.3) and is almost separably-valued in L(B) with the operator norm, then T is strongly measurable (Theorem 3.5.5 of [15]).
On the other hand, we do not prove any measurability property of the map x → F λ i (x), nor do we prove the existence of an invariant distribution of 'fast'-growing subspaces.
1.3. Notation. We collect here the various notations used in the paper.
(1) We denote by (X, F, µ, f ) an ergodic m.p.t.; we do not assume that f is invertible.
(3) Let q ∈ N. For a, b ∈ [0, ∞), we will write a b when a ≤ C q b, and a ≈ b if a b and b a. It will be clear from context which constant , ≈ depend on. (4) For normed vector spaces V 1 , V 2 , we denote by L(V 1 , V 2 ) the space of bounded linear operators V 1 → V 2 . (5) B will always refer to an infinite dimensional Banach space with norm | · |, and we always refer to the induced operator norm on L(B) := L(B, B) by the same symbol | · |. (6) For a subspace E ⊂ B, we let B E := {v ∈ E : |v| ≤ 1} denote the closed unit ball and S E := {v ∈ E : |v| = 1} denote the unit sphere.
and let E ⊂ B be a subspace. We denote by A| E ∈ L(E, B) the restriction of A to E. The codimension of a closed subspace F ⊂ B is defined by codim F = dim F • . (11) For a vector space V , we write V = E ⊕ F when V is the direct sum of E and F , i.e., each vector in V can be written as the unique sum of an element of E and an element of F , and we call such a splitting of V an algebraic splitting; we write π E//F for the projection operator onto E parallel to F . (12) When V is a normed vector space, V = E ⊕ F is an algebraic splitting, and π E//F is a bounded linear operator, we call V = E ⊕ F a topological splitting. For a closed subspace E ⊂ B, if F ⊂ B is a closed subspace for which B = E ⊕ F is a topological splitting, we call F a topological complement to E. (13) G(B) refers to the Grassmanian of closed subspaces of B, and G q (B), G q (B) refer to the subsets of q-dimensional and q-codimensional subspaces of B, respectively. d H refers to the Hausdorff distance on G(B), defined in (2.2). (14) For A ∈ L(B) and q ∈ N, c q (A) denotes the q-th Gelfand number of A, defined in (2.17), and c(A) is defined as the limit lim q→∞ c q (A). We denote by |A| α the measure of compactness of A, as in Definition 2.8.

Induced Volumes and Banach Space Geometry
In this section, we collect some results on the geometry of Banach spaces and discuss the geometric properties of the induced volumes defined in Section 1.
For our purposes, there are two echelons for the geometry for Banach spaces: the 'local' geometry of finite dimensional subspaces and the induced volumes on them, and the 'global' geometry of the Banach space at large.

Global Geometry of Banach Spaces.
We begin by defining a concept of 'angle' in Banach space, and showing its relation to the projection maps associated to splittings. Next, we recall some facts about the geometry of the Grassmanian of closed subspaces of B. Finally, we recall the definition and some basic properties of the measure of noncompactness | · | α .
2.1.1. Angles, Splittings and Complements. Key to volume-growth based approaches to the MET is a concept of angle between vectors and subspaces. The following notion of angle is well-suited to our purposes. Definition 2.1 is adapted from Section II of Mañé's paper [8], and is also very similar to the minimal angle defined in Section 5 of [16].
Roughly speaking, the minimal angle θ(E, F ) will be small whenever E is inclined towards F . Note that θ(E, F ) may not equal θ(F, E). The minimal angle has an interpretation in terms of splittings of Banach space.
where π E//F is the projection to E parallel to F .
Proof. Let P = π E//F , which is well-defined and possibly unbounded when B = E ⊕ F is not topological. One computes where we interpret the RHS of this formula as ∞ whenever θ(E, F ) = 0.
It is crucial to also have access to complements for subspaces of finite dimension and codimension, as there is no assignment of an 'orthogonal complement' to closed subspaces of a Banach space. Lemma 2.3 (III.B.10 and III.B.11 of [17]). Fix q ∈ N.
For any subspace We will also need the following lemma regarding complements.
One computes that P | E 1 is the identity on E 1 , and so P is a bounded projection operator. Define F = ker P , which is a topological complement to E 1 . With F 1 := {v ∈ B : Av ∈ F 2 }, we will show that F = F 1 , as desired.
First, let The Grassmanian on Banach Space. We denote by G(B) the space of closed subspaces of B, endowed with the Hausdorff distance of unit spheres, where S U = {u ∈ U : |u| = 1}, and the Hausdorff distance between closed subsets A, B of a metric space is defined by The following are some basic properties of the Hausdorff distance d H . ( It is actually somewhat inconvenient to compute directly with d H , and so frequently it will be easier to use the gap, defined for E, E ′ ∈ G(B) by (3) Let q ∈ N, and assume either that .
Proof. Item 1 is proven in Chapter IV, §2.1 of [18], and Item 2 is Lemma 2.8 in Chapter IV, §2.3 of [18]. For Item 3, if U, V ∈ G q (B), let {u 1 , · · · , u q } be a of U : for any α > 1, we can obtain a basis , and so we can apply the q-dimensional version of (2.3) to conclude in this case.
When B = E ⊕ F is a topological splitting, it is important to know when a subspace E ′ ⊂ B nearby to E is also complemented to F . Using intuition from finite dimensional geometry, it is reasonable to suggest that this involves the distance d H (E, E ′ ) and the inclination of E towards F , namely sin θ(E, F ).
The proof of Proposition 2.7 is given in full detail in the Appendix. We end this part with another relation between the Hausdorff distance and splittings.
2.1.3. Measure of Noncompactness. We close this section with a brief review of the measure of noncompactness | · | α . Let C ⊂ B be a bounded set. We define q(C) to be the infimum over the set of all r > 0 for which C admits a finite open cover by | · |-balls of radius r.
Definition 2.8. Let B, B ′ be two Banach spaces, and let A ∈ L(B, B ′ ). We define the measure of noncompactness of A to be |A| α := q(AB B ).

2.2.
Finite dimensional subspaces and the induced volume. In this part, we collect results on the induced volume in Definition 1.1.
We begin this subsection by enumerating the most basic properties of the induced volume. Second, we will approximate the norm on a given finite dimensional subspace by an inner product, using John's Theorem, obtaining access to the 'exact' identities and relations between volumes and angles for inner product spaces. Translating what we do in the inner product setting back into the normed vector space vocabulary, we will obtain estimates with errors depending only on the dimension. Third, we will consider the measurability properties of the maximal q-dimensional growths V q defined in (1.2). Fourth, we will relate V q with the measure of noncompactness | · | α .
2.2.1. Basic Properties of the Induced Volume and Determinants. Recall from Section 1 that the induced volume m E for a q-dimensional subspace E ⊂ B is the Haar measure on E normalized so that m E {v ∈ E : |v| ≤ 1} = ω q , the mass of the unit ball in R q . We now give the most basic properties of m E . Lemma 2.10. Let q ∈ N and let E ⊂ B be a q-dimensional subspace. The induced volume m E satisfies the following.
(1) For any v ∈ E and any Borel Proof. These all follow from the uniqueness of Haar measure [14], except for Item 4, which has a simple proof following from Proposition 2.14.
Remark 2.11. The normalization in Definition 1.1 for the induced volume was introduced by Busemann [20] as a way of assigning volume elements to Finsler manifolds, where the tangent spaces are normed vector spaces, not inner product spaces. For more on this, see Chapter I, §8 of [21].
Recall now the definition of the determinant. otherwise.
The following are some basic properties of this determinant.
Proof. All these follow, once again, from the uniqueness of Haar measure.

2.2.2.
Approximation by Inner Products. Every pair of norms on a finite dimensional vector space is equivalent, and so given any finite dimensional subspace E ⊂ B, one can always compare the norm on E with any choice of inner product (·, ·) on E. There is no control, however, on how bad this approximation can be, and so we must take care to ensure our approximating inner product is not too far off from the original norm. The approximation we use here comes from the following version of John's theorem.
Proposition 2.14. Let q ∈ N, and let E ⊂ B be a q-dimensional subspace. Then, there exists an inner product (·, ·) E on E for which the following hold.
(1) The norm · E induced by (·, ·) E satisfies the inequality so that the induced volume m E coincides with the Lebesgue volume on E arising from the inner product (·, ·) E .
Proof. This is merely a re-telling of John's theorem (Theorem 15 in Chapter 4 of [22]), which asserts that given any convex body C ⊂ R n which contains 0 in its interior and is symmetric about the origin, there is a unique ellipsoid D containing C, having minimal volume among all ellipsoids containing C; additionally, D satisfies the inclusion 1 √ n D ⊂ C. Noting that ellipsoids centered at 0 in R n are in one-to-one correspondence with inner products on R n , we let (·, ·) refer to the inner product from John's Theorem applied to the unit ball B E of (E, | · |). We modify (·, ·) by a scalar to define the inner product (·, ·) E : the scalar is chosen so that the Lebesgue volume of (·, ·) E and the induced volume m E coincide. By Item 2 of Lemma 2.10, this is ensured by the condition The bound in (2.5) follows from the fact that the norm · induced by the John's theorem inner product (·, ·) satisfies v ≤ |v| ≤ √ q v for all v ∈ E.
Notation. For the remainder of this subsection, q ∈ N is fixed, and if a, b > 0 are real numbers, we will write a b if there is a constant C q depending only on q for which a ≤ C q b, and similarly for a b. We write a ≈ b if a b and a b hold. For example, (2.5) in Proposition 2.14 can be written as v E ≈ |v| for v ∈ E. We now pursue the program of using (·, ·) E , · E to deduce approximate identities for the induced volume m E . There are two applications we will cover in this section: the first is a way of estimating 'block' determinants, and the second involves estimating determinants in terms of special bases.

Application 1: Estimating 'Block Determinants'.
Let V = E ⊕ F be a splitting of a finite dimensional subspace V ⊂ B, where dim V = q and dim E = k = q − dim F , and assume k < q. For A ∈ L(B), our goal is to estimate det(A|V ) in terms of the product det(A|E) · det(A|F ).
The point of departure for us is the explicit formula for the volume of a parallelepiped in an inner product space: when v 1 , · · · , v q ∈ V , where d V refers to the minimal distance taken using the · V norm.
Let v 1 , · · · , v k and v k+1 , · · · , v q be | · |-unit vector bases for E, F respectively, and observe that because of this choice, holds by formula (2.6) for m V P [v k+1 , · · · , v q ], where we abuse notation and let m V refer also to the volume on F arising from the restriction of · V to F . On the other hand, v V ≈ |v| ≈ v F by (2.5) for all v ∈ F , from which we deduce m V P [v k+1 , · · · , v q ] ≈ m F P [v k+1 , · · · , v q ] by Lemma 2.10, Items 2 and 3. Collecting this, An application of (2.6) to m E P [v 1 , · · · , v k ] now implies as long as the denominator of the central term in (2.8) is nonzero.
Having estimated in this way, one can now estimate the determinant det(A|V ) by considering (2.8) applied to the parallelepipeds P [v 1 , · · · , v q ] and P [Av 1 , · · · , Av q ] = AP [v 1 , · · · , v q ]. We collect the results in the following lemma.
Remark 2.16. Observe that Lemma 1.2 for a q-dimensional subspace E ⊂ B follows straightaway from (2.6) applied to the norm · E , using (2.5) to compare | · | with · E .

Application 2: Estimating Determinants from Bases.
Let q ∈ N, A ∈ L(B), and let V ⊂ B be a q-dimensional subspace. We will give here an estimate for det(A|V ) in terms of a special kind of basis for V . To begin, form a (·, ·) V -orthonormal set v 1 , · · · , v q of V and recall the Hadamard bound for the determinant: More refined information is realized using the Singular Value Decomposition (SVD): if {v i } is an orthonormal SVD basis for A| V on (V, (·, ·) V ) (that is, a (·, ·) V -orthonormal basis of eigenvectors of (A| V ) * A| V , where the adjoint (A| V ) * is taken with respect to (V, (·, ·) V ) and (V ′ , (·, ·) V ′ )), then and when A| V is injective, w i = Av i / Av i V ′ form a (·, ·) V ′ -orthonormal set (as the w i are a set of eigenvectors for (A| V )(A| V ) * ).
All this motivates the following definition.
Definition 2.17. Let V ⊂ B be a subspace of dimension q. A basis of | · |-unit vectors v 1 , · · · , v q for V is called an almost orthonormal basis if it is (·, ·) V -orthogonal.
We now apply (2.5) to each of (2.9) and (2.10), obtaining the following. (1) (Hadamard Bound) If v 1 , · · · , v q is an almost orthonormal basis of V , then (2) (Singular Value Decomposition) There exists an almost orthonormal basis v 1 , · · · , v q for V such that The following is a simple corollary of Proposition 2.18, relating the minimal expansion of a linear map on a finite dimensional subspace to volume growth on that subspace. ( Proof. We begin by proving the following. Exchanging the roles of A and B and taking a logarithm, we arrive at where a ∨ b := max{a, b} for a, b ∈ R. We will also make use of the elementary estimate which holds when the parenthetical quantity on the RHS of (2.12) is nonnegative. Applying (2.11) to the situation B = A n , and taking n sufficiently large so that |A −1 n | ≤ 2|A −1 |, by (2.12), we see now that |A n − A| → 0 implies | log det(A −1 n • A|V )| → 0, i.e., det(A n |V ) → det(A|V ), as desired.
Proof of Item 1. If det(T |E) = 0, let v ∈ E be a unit vector for which T v = 0. By Corollary 2.19, det(T n |E) ≤ C q |T n | E | q−1 |T n v|, where C q > 0 depends on q alone. The right hand side goes to zero as n → ∞ when T n | E → T | E in norm, and so det(T n |E) → 0. So, from here on we assume det(T |E) = 0.
To complete the proof of Item 1 in this case, one can compute using Item 1 of Lemma 2.6 that It now follows by Proposition 2.7 that if F ⊂ B is a fixed topological complement to T E, then T n E is complemented to F for n sufficiently large. Since π TnE//F | T E • π T E//F | TnE = Id TnE , the identity on T n E, we may now decompose Note that the parenthetical term on the RHS of (2.14) goes to 1 as n → ∞ by Claim 2.21, T E). So, we estimate the remaining term on the RHS of (2.14); however, since T n E → T E in the Hausdorff distance, it follows by (2.4) that π F//TnE | T E → 0 in norm as n → ∞. Because π TnE//F | T E = Id T E −π F//TnE | T E , it follows by a simple argument involving Lemma 2.10 that det(π TnE//F |T E) → 1, as desired.
Proof of Item 2. If V q (T ) = 0, then an argument similar to the beginning of the proof of Item 1 lets us conclude V q (T n ) → V q (T ). Hereafter we assume that V q (T ) > 0.
Note that For each n, let E n ⊂ B be a subspace of dimension q for which V q (T n ) ≤ (1 + 1 n ) det(T n |E n ). We take n large enough so that V q (T ) ≤ 2V q (T n ), using (2.15). Now, by Corollary 2.19, where C q > 0 depends on q alone, and so taking n large enough so that |T n | ≤ 2|T |, we have that |(T n | En ) −1 | ≤ D, where D is a constant independent of n. Applying (2.12), we also obtain |(T | En ) −1 | ≤ 2D for n sufficiently large. Because we can bound |(T n | En ) −1 |, |(T | En ) −1 | from above independently of n, we can apply (2.13) to the subspace E n in place of E, obtaining d H (T n E n , T E n ) → 0 as n → ∞. For each n, let F n be a complement to E n with |π T En//Fn | ≤ √ q (Lemma 2.3). So, by Proposition 2.7, we deduce that for n sufficiently large, T n E n complements F n . We now decompose analogously to (2.14). The estimate (2.11) and the fact that |(T n | En ) −1 |, |(T | En ) −1 | ≤ 2D imply that the parenthetical term on the RHS of (2.16) goes to 1 as n → ∞. The fact that π Fn//TnEn | T En → 0 in norm (following from (2.4) and that d H (T n E n , T E n ) → 0) implies that the remaining RHS term of (2.16) goes to 1, analogously to the proof of Item 1. Therefore, the LHS of (2.16) goes to 1 as n → ∞, and we conclude that lim sup n→∞ V q (T n ) ≤ V q (T ), as desired.

2.2.4.
Global Geometry and the Induced Volume. We will now discuss properties of the induced volume which depend on the Banach space at large. We will give a relation between V q and the Gelfand numbers, and a relation to the measure of noncompactness | · | α . The Gelfand Numbers and Maximal Volume Growth.
For an operator A on a Hilbert space, V q (A) (defined by (1.2)) is equal to the product of the first q singular values of A, i.e., the product of the highest q eigenvalues of √ A * A (Proposition 1.4 in Chapter V of [23]). As a consequence, one can think of the q-th Lyapunov exponent (with multiplicity) of a cocycle as the generic asymptotic growth rate of the q-th singular value of high iterates of the cocycle [2,3]. In this section, we will recover this identification for a suitable generalization of 'singular value' to the Banach space setting, a fact to be put to use in Section 3.
The idea of 'singular values' has been generalized in several nonequivalent ways for Banach spaces: for an account, see [24]. The definition we shall employ here is that of the Gelfand numbers c q (·), q ∈ N, defined for A ∈ L(B) by (2.17) with c 1 (A) := |A| (Chapter 2, Section 4 of [24]). Notice that {c q (A)} q is a decreasing sequence, and so we define c(A) := lim m c m (A) = inf m c m (A). Observe that c(A) = inf{|A| R | : R closed, codim R < ∞}; this is a different kind of measure of noncompactness from | · | α (2.4.10 in [19]), and as we will see in Lemma 2.23, is equivalent to | · | α .
We will now show that V q (A) is approximated by the product of the first q Gelfand numbers c i (A).
where ≈ depends on q alone.
Proof. We prove , separately. Assume V q−1 (A) > 0, since otherwise, A is a finite rank operator of rank ≤ q − 2 and (2.18) is trivial.
To prove , let V ⊂ B be a subspace of dimension q for which det(A|V ) ≥ 1 2 V q (A), and let R ⊂ B have codimension q − 1 such that |A| R | ≤ 2c q (A).
Note that dim R ∩ V ≥ 1 by the definition of codimension; fix w ∈ R ∩ V with |w| = 1. Let V 0 ⊂ V be a complement to w in V for which |π w //V | = 1 (by Lemma 2.3). We now estimate using the 'block determinant' estimate in Lemma 2.15: which is what we wanted.
Let w ∈ F 1 , |w| = 1. Using the lower bound in Lemma 2.15, we estimate This inequality holds for any w ∈ F 1 , |w| = 1, and so implies an upper bound on |A| F 1 |, which is ≥ c q (A) by definition. This completes the estimate.

Measures of Noncompactness and Maximal Volume Growth.
Like singular values of operators in Hilbert space, the Gelfand numbers can detect compactness and estimate the measure of noncompactness | · | α in the following sense. Lemma 2.23 (2.5.5 in [19]). Let A ∈ L(B). Then, The maximal volume growths V q also have a relation to | · | α . Proof. For x ∈ B and r > 0, we denote B(x, r) = {v ∈ B : |v − x| < r}.
If A is an operator of finite rank, then (2.19) holds vacuously, and so we can assume without loss that V q (A) > 0 for all q ∈ N.
Fix r > |A| α and let {B(x i , r)} Cr i=1 be a finite cover of A B(0, 1) by balls of radius r centered at points x i ∈ B. For each q ∈ N, let E q ⊂ B be a q-dimensional subspace for which V q (A) ≤ 2 det(A|E q ). Writing AE q = E ′ q , note that dim E ′ q = q, and that As one can easily check, for Hilbert spaces we have that m E ′ q E ′ q ∩ B(x i , r) ≤ r q ω q , with equality when x i ∈ E ′ q . For Banach spaces, the following can be recovered.
From Claim 2.25, we obtain Proof of Claim 2.25. Let F be a topological complement to E in B (Lemma 2.3), and decompose x = e + f . If f = 0, then x ∈ E and Claim 2.25 is obvious: so, we may assume f = 0. Moreover, without loss, we can take e = 0 by the the translation invariance of m E on E.

Lyapunov Exponents for Banach Space Cocycles
In this section, we prove our main result, Theorem 1.3, by emulating Ruelle's proof in [3,7]. We will begin by using the measurability assumption (1.3) and Lemma 2.20 to obtain the Lyapunov exponents and show they may accumulate only at the asymptotic exponential growth rate l α of |T n x | α . Then, we will state our primary tool, Proposition 3.4, which should be thought of as a 'trajectory-wise' version of the MET which extracts the 'slow' growing subspace corresponding to the second Lyapunov exponent, and show using an induction procedure (Lemma 3.6) how to complete the proof. Remaining at that point will be to prove Proposition 3.4 and Lemma 3.6, and to prove the volume growth (1.5) in Theorem 1.3, which we formulate as Lemma 3.9.
3.1. Lyapunov Exponents for T . In this section, we find the Lyapunov exponents for the cocycle T using growth rates, and prove their basic properties. To begin, the following is an immediate consequence of Item 2 in Lemma 2.20. Then, for any n, q ≥ 1, the map x → V q (T n x ) is measurable as a map (X, F) → (R, Bor(R)). The following lemma identifies the Lyapunov exponents of the cocycle T in terms of volume growth. Then, the following hold.
(1) For any q ≥ 1, the exponential growth rates l q , defined by exist and are constant µ-almost surely.
(2) Writing K 1 = l 1 , K q = l q − l q−1 for q > 1, the sequence {K q } q≥1 is nonincreasing, i.e., K 1 ≥ K 2 ≥ · · · . (3) Defining l α = lim q→∞ K q , we have that µ-almost surely, We write λ 1 > λ 2 > · · · for the distinct values of the sequence {K q } q . There may be finitely many of these, in which case the last of these values is equal to l α , or infinitely many, in which case λ i → l α as i → ∞. We write m i for the multiplicity of the value λ i amongst the sequence {K q } q and M 1 := 0, M i := m 1 + · · · + m i−1 for i ≥ 2.
Proof. Item 1. The almost sure convergence of 1 n log V q (T n x ) follows immediately from the Kingman Subadditive Ergodic Theorem (KSET) ( §1.5 of [6]), in light of the integrability hypothesis (3.1) and the measurability of , where ≈ depends only on q, and so one can see directly from the convergence of the sequences 1 n log V q (T n x ) as n → ∞ that the growth rates of the Gelfand numbers 1 n log c q (T n x ) converge to K q as n → ∞ for any q ≥ 1. Obviously, c q (·) ≥ c q+1 (·) for any q ≥ 1, and so K 1 ≥ K 2 ≥ · · · follows immediately.
Item 3. Recall that c(A) = inf m c m (A). Note that almost surely, By Lemma 2.23, taking q → ∞ lets us deduce l α ≥ lim sup n→∞ 1 n log |T n x | α . For the other direction, the limit as q → ∞ of the Cesaro averages 1 q l q equals l α . Recall that 1 q l q is decreasing, and that by the KSET, the limit in (3.2) is almost surely an infimum ( §1.5 of [6]). Now, using Lemma 2.24, which completes the estimate.

3.2.
Proof of Theorem 1.3. For the remainder of this section, we give the volume-based proof of Theorem 1.3.

3.2.1.
A 'trajectory-wise' version of the MET. Below, we state a version of the MET, to be applied one trajectory at a time: this the analogue of Proposition 2.1 in Section 2 of Ruelle's paper [7]. We refer to the norms on the spaces V i below with the same symbol | · |.
Proposition 3.4. Let V 0 , V 1 , V 2 , · · · be Banach spaces and let T i : V i → V i+1 be a sequence of bounded linear maps. Write T n = T n−1 • · · · • T 0 , and assume the following of {T n }.
For any η ∈ (0, 1), the former convergence occurs uniformly over vectors in the cone C η = {v ∈ B : d(v, F ) ≥ η|v|} in the following sense: In particular, for any complement E to F .
We now show how to derive all of Theorem 1.3 from Proposition 3.4, except for (1.5). Below, we assume that there are infinitely many distinct Lyapunov exponents-the proof for when there are finitely many distinct exponents is virtually identical.
Proof of Theorem 1.3 from Proposition 3.4. We define Γ ⊂ X to be the set of all x ∈ X such that the limit lim n→∞ 1 n log V q (T n x ) exists and equals l q , and for which The condition (3.5) is µ-generic by the Birkhoff Ergodic Theorem and the integrability hypothesis For each x ∈ Γ, the sequence T n := T f n x satisfies the hypotheses of Proposition 3.4, and so we obtain the following, which can be thought of as the MET for the first Lyapunov exponent. Below, with F λ (x) as in (1.4), we define F i (x) := F λ i (x) for Lyapunov exponents λ i as in Remark 3.3. The former convergence occurs uniformly over vectors in the cone C η (x) = {v ∈ B : d(v, F 2 (x)) ≥ η|v|}, in the following sense. |T n x v| |v| = λ 1 .
As a consequence, for any complement E to F 2 (x), we have that We now formulate an induction step, to show that the hypotheses of Proposition 3.4 are satisfied for the sequence T 0 := T x | F 2 (x) , T n = T f n x , with the Lyapunov exponents 'shifted down' so as to eliminate the top exponent λ 1 .
Lemma 3.6 (Exponent Extraction Lemma). For any x ∈ Γ, we have that lim n→∞ 1 n log V q (T n x | F 2 (x) ) exists and equals l q+m 1 − l m 1 for any q ≥ 1.
We will prove this in Section 3.2.3. One now has that the sequence of operators defined by T 0 := T x | F 2 (x) , T n := T f n x , n > 0, satisfies the hypotheses of Proposition 3.4 with λ = λ 2 , λ = λ 3 , m = m 2 , and the subspace F ⊂ F 2 (x) as in the statement of Proposition 3.4 is F 3 (x). We obtain that the codimension of F 3 (x) in F 2 (x) is m 2 , and so codim F 3 (x) = M 3 = m 1 + m 2 , and Inductively, assume that we have already shown that F i (x) has codimension M i , and that for Under this condition, one can show, repeating the argument in the proof of Lemma 3.6, that for any x ∈ Γ, the limit lim n→∞ 1 n log V q (T n x | F i (x) ) exists for any q ≥ 1 and equals l q+M i − l M i . Therefore, T 0 := T x | F i (x) , T n := T f n x , n > 0 satisfies the hypotheses of Proposition 3.4 with λ = λ i , λ = λ i+1 , m = m i , and F = F i+1 (x). We obtain that the codimension of This completes the induction argument.
We now give the proofs of Proposition 3.4 and Lemma 3.6, and the proof of the volume growth formula in (1.5), formulated as Lemma 3.9.

Proof of Proposition 3.4.
Proof of Proposition 3.4. For each n, let E 1 n be an m-dimensional subspace of V 0 for which det(T n |E 1 n ) ≥ 1 2 V m (T n ), and define E 2 n = T n E 1 n . Let F 2 n be a complement to E 2 n , as in Lemma 2.3, for which P 2 n := π E 2 n //F 2 this is a complement to E 1 n by Lemma 2.4. We write P 1 n = π E 1 n //F 1 n . To carry out our argument, we need to show that T n expands vectors on E 1 n by a factor of approximately e nλ , that |P 1 n | does not grow too quickly in n, and that |T n | F 1 n | is bounded from above by approximately e nλ on F 1 n . Growth on E 1 n . Let v ∈ E 1 n , |v| = 1. Then, by Corollary 2.19, there is a constant C m depending only on m for which Controlling |P 1 n |. Using (3.7) and the formula (2.1) for |P 1 n | in Lemma 2.4, we have that the projection P 1 n := π E 1 n //F 1 n satisfies where C ′ m is again a constant depending only on m. Bounding |T n | F 1 n |. Let v ∈ F 1 n , |v| = 1. We will estimate |T n v| by estimating the growth rate of the m + 1-dimensional subspace E 1 n ⊕ v under T n . Treating E 1 n ⊕ v as a splitting and using Lemma 2.15, we have that , and since det(T n | v ) = |T n v| and det(T n |E 1 where C ′′ m again depends only on m. The RHS of (3.9) is approximately e nλ for n large, as desired. Key to this approach to the MET is showing that the subspaces F 1 n converge in the Hausdorff distance at a sufficiently fast exponential rate.
Proof of Lemma 3.6. We will assume that l m 1 +q > −∞. The proof is similar otherwise, and we omit it.
Let q ∈ N. We will show separately that lim sup n→∞ 1 n log V q (T n x | F 2 (x) ) ≤ l m 1 +q − l m 1 and that lim inf n→∞ 1 n log V q (T n x | F 2 (x) ) ≥ l m 1 +q − l q . Lower bound on lim inf n→∞ 1 n log V q (T n x | F 2 (x) ). For each n, let H n ⊂ B be an (m 1 +q)-dimensional subspace for which det(T n x |H n ) ≥ 1 2 V m 1 +q (T n x ). Observe that H n ∩ F 2 (x) has dimension ≥ q; let G n ⊂ H n ∩ F 2 (x) be any q-dimensional subspace. Let J n be a complement to G n inside H n of dimension m 1 for which |π Gn//Jn | ≤ √ q (Lemma 2.3). Using Lemma 2.15, we now estimate: for constants C q , C ′ q depending on q alone, yielding lim inf n→∞ 1 n log V q (T n x | F 2 (x) ) ≥ l m 1 +q − l m 1 . Upper bound on lim sup n→∞ 1 n log V q (T n x | F 2 (x) ). For each n, let G n ⊂ F 2 (x) be a q-dimensional subspace for which det(T n x |G n ) ≥ 1 2 V q (T n x | F 2 (x) ). Observe that T n x G n is always q-dimensional because V q (T n x | F 2 (x) ) > 0 for any n, which follows from (3.16) and the assumption l m 1 +q > −∞. Let E ⊂ B be any topological complement to F 2 (x), so that dim(E ⊕ G n ) = m 1 + q for any n. As |π T n n log V q (T n x | F 2 (x) ).
The second term of the LHS equals zero by Lemma 3.8, and the lim inf in the third term is a limit, equalling l m 1 by (3.6).

3.2.4.
Proof of the Volume Growth Clause in Theorem 1.3. We finish the paper by proving (1.5) from Theorem 1.3. Proof. The ideas for the proof are already present in the case when E is a complement to F 3 (x) (that is, i = 2), and so we concentrate on that case.
The first step is to find a splitting of E of the form E = E 1 ⊕ E 2 , where E 1 is a complement to F 2 (x) in B and E 2 ⊂ F 2 (x). We set E 2 := F 2 (x) ∩ E, and let E 1 be a complement to E 2 in E; it is not hard to check that because F 2 (x) ⊃ F 3 (x) and B = E ⊕ F 3 (x), one has that dim E 2 = dim E − codim F 2 (x) = m 1 , so that B = E 1 ⊕ F 2 (x) and F 2 (x) = E 2 ⊕ F 3 (x) hold.
By (3.4) in Proposition 3.4, we know that for j = 1, 2, lim n→∞ 1 n log det(T n x |E j ) = m j λ j .