A spectral approach for quenched limit theorems for random ...

arXiv:1705.02130v1 [math.DS] 5 May 2017

A spectral approach for quenched limit theorems for random expanding dynamical systems D. Dragiˇcević ∗, G. Froyland†, C. González-Tokman‡, S. Vaienti§ May 8, 2017

Abstract We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form Tσn−1 ω ◦ · · · ◦ Tσω ◦ Tω . An important aspect of our results is that we only assume ergodicity and invertibility of the random driving σ : Ω → Ω; in particular no expansivity or mixing properties are required.

Contents 1 Introduction

2

2 Preliminaries 2.1 Multiplicative ergodic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Notions of variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

7 7 11

School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. E-mail: [email protected]. † School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. E-mail: [email protected] . ‡ School of Mathematics and Physics, The University of Queensland, St Lucia QLD 4072, Australia. E-mail: [email protected]. § Sandro Vaienti, Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France and Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France. E-mail: [email protected].

1

2.3

Admissible cocycles of transfer operators . . . . . . . . . . . . . . . . . . . . 2.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Twisted transfer operator cocycles 3.1 The observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic properties of twisted transfer operator cocycles . . . . . . . 3.3 An auxiliary existence and regularity result . . . . . . . . . . . . . 3.4 A lower bound on Λ(θ) . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Quasicompactness of twisted cocycles and differentiability of Λ(θ) 3.6 Convexity of Λ(θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Choice of bases for top Oseledets spaces Yωθ and Yω∗θ . . . . . . . .

12 15

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17 18 18 20 22 25 28 29

. . . . . . .

30 31 32 35 36 39 43 44

A Technical results involving notions of volume growth A.1 Proof of Corollary 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Proof of Lemma 4.8, Step (1) . . . . . . . . . . . . . . . . . . . . . . . . . .

50 51 52

B Regularity of F B.1 First order regularity of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Second order regularity of F . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 52 58

C Differentiability of φθ , the top space for adjoint twisted cocycle Rθ∗

62

4 Limit theorems 4.1 Large deviations property . . . . . . . . . . . . . . . . 4.2 Central limit theorem . . . . . . . . . . . . . . . . . . . 4.3 Local central limit theorem . . . . . . . . . . . . . . . . 4.3.1 Proof of Theorem C . . . . . . . . . . . . . . . 4.3.2 Equivalent versions of the aperiodicity condition 4.3.3 Application to random Lasota–Yorke maps . . . 4.4 Local central limit theorem: periodic case . . . . . . .

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Introduction

The Nagaev-Guivarc’h spectral method for proving the central limit theorem (due to Nagaev [36, 37] for Markov chains and Guivarc’h [42, 24] for deterministic dynamics) is a powerful approach with applications to several other limit theorems, in particular large deviations and the local limit theorem. In the deterministic setting a map T : X → X on a state space X preserves a probability measure µ on X. An observable g : X → R generates µ-stationary process {g(T nx)}n≥0 and one studies the statistics of this process. Central to the spectral method is the transfer operator L : B , acting on a Banach space B ⊃ L1 (µ) of complexvalued functions with regularity properties compatible with the regularity of T 1 . A twist is 1

The transfer operator satisfies

R

X

f · g ◦ T dµ =

R

X

2

Lf · g dµ for f ∈ L1 (µ), g ∈ L∞ (µ).

introduced to form the twisted transfer operator Lθ f := L(eθg f ). The three key steps to the spectral approach are: Pn−1 g ◦ Ti S1. Representing the characteristic function of Birkhoff (partial) sums Sn g = i=0 as integrals of nth powers of twisted transfer operators. S2. Quasi-compactness (existence of a spectral gap) for the twisted transfer operators Lθ for θ near zero. S3. Regularity (e.g. twice differentiable for the CLT) of the leading eigenvalue of the twisted transfer operators Lθ with respect to the twist parameter θ, for θ near zero.

This spectral approach has been widely used to prove limit theorems for deterministic dynamics, including large deviation principles [26, 41], central limit theorems [42, 9, 26, 6], Berry-Esseen theorems [24, 21], local central limit theorems [42, 26, 21], and vector-valued almost-sure invariance principles [33, 22]. We refer the reader to the excellent review paper [23], which provides a broader overview of how to apply the spectral method to problems of these types, and the references therein. In this paper, we extend this spectral approach to the situation where we have a family of maps {Tω }ω∈Ω , parameterised by elements of a probability space (Ω, P). These maps are composed according to orbits of a driving system σ : Ω → Ω. The resulting dynamics takes the form of a map cocycle Tσn−1 ω ◦· · ·◦Tσω ◦Tω . In terms of real-world applications, we imagine that Ω is the class of underlying configurations that govern the dynamics on the (physical or state) space X. As time evolves, σ updates the current configuration and the dynamics Tω on X correspondingly changes. To retain the greatest generality for applications, we make minimal assumptions on the configuration updating (the driving dynamics) σ, and only assume σ is P-preserving, ergodic and invertible; in particular, no mixing hypotheses are imposed on σ. We will assume certain uniform-in-ω (eventual) expansivity conditions for the maps Tω . Our observable g : Ω × X → R can (and, in general, will) depend on the base configuration ω and will satisfy a fibrewise finite variation condition. One can represent the random dynamics by a deterministic skew product transformation τ (ω, x) = (σ(ω), Tω (x)), ω ∈ Ω, x ∈ X. It is well known that whenever σ is invertible and µ ˜ is a τ -invariant probability measure with marginal P on the base Ω, the disintegration of µ ˜ with respect to P produces conditional measures µω which are equivariant; namely µω ◦ Tω−1 = µσω . Our limit theorems will be established µω -almost surely and for P-almost all choices of ω; we therefore develop quenched limit theorems. In the much simpler case where σ is Bernoulli, which yields an i.i.d. composition of the elements of {Tω }ω∈Ω , one is often interested in the study of limit laws with respect to a measure µ ˆ which is invariant with respect to the averaged transfer operator, and reflects the outcomes of averaged observations [40, 4]. The corresponding limit laws with respect to µ ˆ are typically called annealed limit laws; see [2] and references therein for recent results in this framework. As is common in the quenched setting, we impose a fiberwise centering condition for the observable. Thus, limit theorems in this context deal with fluctuations about a timedependent mean. For example, if the observable is temperature, the limit theorems would 3

characterise temperature fluctuations about the mean, but this mean is allowed to vary with the seasons. The recent work [1] provides a discussion of annealed and quenched limit theorems, and in particular an example regarding the necessity of fibrewise centering the observable for the quenched case. Without such a condition, quenched limit theorems have been established exclusively in special cases where all maps preserve a common invariant measure [6, 38] (and where the centering is obviously identical on each fibre). In the quenched random setting we generalise the above three key steps of the spectral approach: R1. Representing the (ω-dependent) characteristic function of Birkhoff (partial) sums Sn g(ω, ·) = Pn−1 g(ω, ·) ◦ Tωi (·) as an integral of nth random compositions of twisted transfer opi=0 erators. R2. Quasi-compactness for the twisted transfer operator cocycle; equivalently, existence of θ,(n) a gap in the Lyapunov spectrum of the cocycle Lω := Lθσn−1 ω ◦ · · · ◦ Lθσω ◦ Lθω for θ near zero. R3. Regularity (e.g. twice differentiable for the CLT) of the leading Lyapunov exponent and Oseledets spaces of the twisted transfer operators cocycle with respect to the twist parameter θ, for θ near zero. At this point we note that the key steps S1–S3 in the deterministic spectral approach mean that one satisfies the requirement for a naive version of the Nagaev-Guivarc’h method [23]; namely E(eiθSn ) = c(θ)λ(θ)n + dn (θ) for c continuous at 0 and |dn |∞ → 0. In this case, λ(θ) is the leading eigenvalue of Lθ . Similarly, the key steps R1–R3 yield an analogue naive version of a random Nagaev-Guivarc’h method, where for all complex θ in a neighborhood of 0, and P-a.e. ω ∈ Ω, we have that 1 log |Eµω (eθSn g(ω,·) )| = Λ(θ), n→∞ n lim

where Λ(θ) is the top Lyapunov exponent of the random cocycle generated by Lθω (see Lemma 4.3). This condition is of course weaker than the asymptotic equivalence of [23], but together with the exponential decay of the norm of the projections to the complement of the top Oseledets space (see Section 4.2), which handles the error corresponding to quantity dn above, we are able to achieve the desired limit theorems. Under this analogy, we could consider our result as a new naive version of the Nagaev-Guivarc’h method, framed and adapted to random dynamical systems. The quasi-compactness of the twisted transfer operator cocycle (item 2 above) will be based on the works [16, 18], which have adapted multiplicative ergodic theory to the setting of cocycles of possibly non-injective operators; the non-injectivity is crucial for the study of endomorphisms Tω . These new multiplicative ergodic theorems, and in particular the quasi-compactness results, utilise random Lasota-Yorke inequalities in the spirit of Buzzi [10]. For the regularity of the leading Lyapunov exponent (item 3 above) we develop ab initio a cocycle-based perturbation theory, based on techniques of [26]. This is necessary 4

because in the random setting objects such as eigenvalues and eigenfunctions of individual transfer operators have no dynamical meaning and therefore one cannot simply apply standard perturbation results such as [27], as is done in [26] and all other spectral approaches for limit theorems. Multiplicative ergodic theorems do not provide, in general, a spectral decomposition with eigenvalues and eigenvectors as in the classical sense, but only a hierarchy of equivariant Oseledets spaces containing vectors which grow at a fixed asymptotic exponential rate, determined by the corresponding Lyapunov exponent. Let us now summarise the main results of the present paper, obtained with our new cocycle-based perturbation theory. These are limit theorems for random Birkhoff sums Sn g, associated to an observable g : Ω × X → R, and defined by Sn g(ω, x) :=

n−1 X i=0

i

g(τ (ω, x)) =

n−1 X

g(σ i ω, Tω(i)x),

i=0

(ω, x) ∈ Ω × X, n ∈ N.

(1)

The observable will be required to satisfy some regularity properties, which are made precise in Section 3.1. Moreover, we will suppose that g is fiberwise centered with respect to the invariant measure µ for τ . That is, Z g(ω, x) dµω (x) = 0 for P-a.e. ω ∈ Ω. (2) The necessary conditions on the dynamics are summarised in an admissibility notion, which is introduced in Definition 2.8. Our first results are quenched forms of the Large Deviations Theorem and the Central Limit Theorem. We remark that, while our results are all stated in terms of the fiber measures µω , in our examples, the same results hold true when µω is replaced by Lebesgue measure m. This is a consequence of a the result of Eagleson [14] combined with the fact that, in our examples, µω is equivalent to m. Theorem A (Quenched large deviations theorem). Assume the transfer operator cocycle R is admissible, and the observable g satisfies conditions (2) and (24). Then, there exists ǫ0 > 0 and a non-random function c : (−ǫ0 , ǫ0 ) → R which is nonnegative, continuous, strictly convex, vanishing only at 0 and such that 1 log µω (Sn g(ω, ·) > nǫ) = −c(ǫ), n→∞ n lim

for 0 < ǫ < ǫ0 and P-a.e. ω ∈ Ω.

Theorem B (Quenched central limit theorem). Assume the transfer operator cocycle R is admissible, and the observable g satisfies conditions (2) and (24). Assume also that the non-random variance Σ2 , defined in (47) satisfies Σ2 > 0. Then, for every bounded and continuous function φ : R → R and P-a.e. ω ∈ Ω, we have Z Z Sn g(ω, x) √ lim φ dµω (x) = φ dN (0, Σ2). n→∞ n (The discussion after (47) deals with the degenerate case Σ2 = 0). 5

Similar LDT and CLT results were previously obtained by Kifer [30, 31]. For the LDT, Kifer applied a small stochastic perturbation, additional to the randomness of the cocycle, and used variational principles to construct quantities analogous to our top Lyapunov exponent and top Oseledets spaces. He then showed that in the limit of vanishing stochastic perturbations one can recover limits of these objects for the original random dynamical system. In the present work we go further by avoiding auxiliary stochastic perturbations and directly developing a general and powerful spectral machinery for addressing quenched limit laws. For the CLT, Kifer used martingale techniques, however, to control the rate of mixing, strong conditions (φ-mixing and α-mixing) are assumed in [31], which are generally hard to check in the dynamical systems context. We use instead quenched decay of correlations on a space of regular observables, for example, bounded variation observables in one dimension or quasi-Hölder observables in higher dimensions. Finally, we note that in our recent article [13] we provide the first complete proof of the Almost Sure Invariance Principle for random transformations of the type covered in this paper using martingale techniques. In this work, we prove for the first time a Local Central Limit Theorem for random transformations. Theorem C presents the aperiodic version: This result relies on an assumpit,(n) tion concerning fast decay in n of the norm of the twisted operator cocycle kLω kB , for t ∈ R \ {0} and P-a.e. ω ∈ Ω. This hypothesis is made precise in (C5). Such an assumption is usually stated in the deterministic case (resp. in the random annealed situation), by asking that the twisted operator (resp. the averaged random twisted operator) Lit has spectral radius strictly less than one for t ∈ R \ {0}; this is called the aperiodicity condition. Theorem C (Quenched local central limit theorem). Assume the transfer operator cocycle R is admissible, and the observable g satisfies conditions (2) and (24). In addition, suppose the aperiodicity condition (C5) is satisfied. Then, for P-a.e. ω ∈ Ω and every bounded interval J ⊂ R, we have √ 1 − s2 2 lim sup Σ nµω (s + Sn g(ω, ·) ∈ J) − √ e 2nΣ |J| = 0. n→∞ s∈R 2π In the autonomous case, aperiodicity is equivalent to a co-boundary condition, which can be checked in particular examples [34]. We are also able to state an equivalence between the it,(n) decay of Lω and a (random) co-boundary equation, which opens the possibility to verify the hypotheses of the local limit theorem in specific examples. In addition, we establish a periodic version of the LCLT in Theorem 4.15. In summary, a main contribution of the present work is the development of the spectral method for establishing limit theorems for quenched (or ω fibre-wise) random dynamics. Our hypotheses are natural from a dynamical point of view, and we explicitly verify them in the framework of the random Lasota-Yorke maps, and more generally for random piecewise expanding maps in higher dimensions. The new spectral approach for the quenched random setting we present here has been specifically designed for generalisation and we are hopeful that this method will afford the same broad flexibility that continues to be exploited by work in the deterministic setting. While at present we have uniform-in-ω assumptions on time-asymptotic expansion and decay properties of the random dynamics, we hope that 6

in the future these assumptions can be relaxed to enable even larger classes of dynamical systems to be treated with our new spectral technique. For example, limit theorems for dynamical systems beyond the uniformly hyperbolic setting continues to be an active area of research, e.g. [21, 22, 23, 7, 11, 39, 32], and another interesting set of related results on limit theorems occur in the setting of homogenisation [20, 28, 29]. Our extension to the quenched random case opens up a wide variety of potential applications and future work will explore generalisation to random dynamical systems with even more complicated forms of behaviour.

2

Preliminaries

We begin this section by recalling several useful facts from multiplicative ergodic theory. We then introduce assumptions on the state space X; X will be a probability space equipped with a notion of variation for integrable functions. This abstract approach will enable us to simultaneously treat the cases where (i) X is a unit interval (in the context of LasotaYorke maps) and (ii) X is a subset of Rn (in the context of piecewise expanding maps in higher dimensions). We introduce several dynamical assumptions for the cocycle Lω , ω ∈ Ω of transfer operators under which our limit theorems apply. This section is concluded by constructing large families of examples of both Lasota-Yorke maps and piecewise expanding maps in Rn that satisfy all of our conditions.

2.1

Multiplicative ergodic theorem

In this subsection we recall the recently established versions of the multiplicative ergodic theorem which can be applied to the study of cocycles of transfer operators and will play an important role in the present paper. We begin by recalling some basic notions. A tuple R = (Ω, F , P, σ, B, L) will be called a linear cocycle, or simply a cocycle, if σ is an invertible ergodic measure-preserving transformation on a probability space (Ω, F , P), (B, k · k) is a Banach space and L : Ω → L(B) is a family of bounded linear operators such that log+ kL(ω)k ∈ L1 (P). Sometimes we will also use L to refer to the full cocycle R. In order to obtain sufficient measurability conditions in our setting of interest, we assume the following: (C0) σ is a homeomorphism, Ω is a Borel subset of a separable, complete metric space and L is P−continuous (that is, L is continuous on each of countably many Borel sets whose union is Ω). (n)

For each ω ∈ Ω and n ≥ 0, let Lω be the linear operator given by Lω(n) := Lσn−1 ω ◦ · · · ◦ Lσω ◦ Lω . (n)

Condition (C0) implies that the maps ω 7→ log kLω k are measurable. Thus, Kingman’s sub7

additive ergodic theorem ensures that the following limits exist and coincide for P-a.e. ω ∈ Ω: 1 log kL(n) ω k n→∞ n 1 κ(R) := lim log ic(L(n) ω ), n→∞ n

Λ(R) := lim

where n o ic(A) := inf r > 0 : A(BB ) can be covered with finitely many balls of radius r ,

and BB is the unit ball of B. The cocycle R is called quasi-compact if Λ(R) > κ(R). The quantity Λ(R) is called the top Lyapunov exponent of the cocycle and generalises the notion of (logarithm of) spectral radius of a linear operator. Furthermore, κ(R) generalises the notion of essential spectral radius to the context of cocycles. Let (B′ , | · |) be a Banach space such that B ⊂ B′ and that the inclusion (B, k · k) ֒→ (B′ , | · |) is compact. The following result, based on a theorem of Hennion [25], is useful to establish quasi-compactness. Lemma 2.1. ([18, Lemma C.5]) Let (Ω, F , P) be a probability space, σ an ergodic, invertible, P-preserving transformation on Ω and R = (Ω, F , P, σ, B, L) a cocycle. Assume Lω can be extended continuously to (B′ , | · |) for P-a.e. ω ∈ Ω, and that there exist measurable functions αω , βω , γω : Ω → R such that the following strong and weak Lasota-Yorke type inequalities hold for every f ∈ B, kLω f k ≤ αω kf k + βω |f | kLω k ≤ γω .

and

(3) (4)

In addition, assume

Then, κ(R) ≤

R

Z

log αω dP(ω) < Λ(R), and

Z

log γω dP(ω) < ∞.

log αω dP(ω). In particular, R is quasi-compact.

Another result which will be useful in the sequel is the following comparison between Lyapunov exponents with respect to different norms. In what follows, we denote by λB (ω, f) the (n) Lyapunov exponent of f with respect to the norm k·kB . That is, λB (ω, f) = limn→∞ n1 log kLω f kB , where f ∈ B and (B, k · kB ) is a Banach space. Lemma 2.2 (Lyapunov exponents for different norms). Under the notation and hypotheses R of Lemma 2.1, let r := Ω log αω dP(ω) and assume that for some f ∈ B, λB′ (ω, f ) > r. Then, λB (ω, f ) = λB′ (ω, f ). Proof. The inequality λB (ω, f ) ≥ λB′ (ω, f ) is trivial, because k · k is stronger than | · | (i.e. because the embedding (B, k · k) ֒→ (B′ , | · |) is compact). In the other direction, the result essentially follows from Lemma C.5(2) in [18]. Indeed, this lemma establishes that if r < 0 and λB′ (ω, f) ≤ 0 then λB (ω, f) ≤ 0. The choice of 0 is irrelevant, because if the cocycle is rescaled by a constant C > 0, all Lyapunov exponents and r are shifted by log C. Thus, we conclude that if λB′ (ω, f ) > r then, λB (ω, f ) ≤ λB′ (ω, f ), as claimed. 8

A spectral-type decomposition for quasi-compact cocycles can be obtained via a multiplicative ergodic theorem, as follows. Theorem 2.3 (Multiplicative ergodic theorem, MET [16]). Let R = (Ω, F , P, σ, B, L) be a quasi-compact cocycle and suppose that condition (C0) holds. Then, there exists 1 ≤ l ≤ ∞ and a sequence of exceptional Lyapunov exponents Λ(R) = λ1 > λ2 > . . . > λl > κ(R)

(if 1 ≤ l < ∞)

or Λ(R) = λ1 > λ2 > . . .

and

lim λn = κ(R)

n→∞

(if l = ∞);

and for P-almost every ω ∈ Ω there exists a unique splitting (called the Oseledets splitting) of B into closed subspaces l M Yj (ω), (5) B = V (ω) ⊕ j=1

depending measurably on ω and such that:

(I) For each 1 ≤ j ≤ l, dim Yj (ω) is finite-dimensional (mj := dim Yj (ω) < ∞), Yj is equivariant i.e. Lω Yj (ω) = Yj (σω) and for every y ∈ Yj (ω) \ {0}, 1 log kL(n) ω yk = λj . n→∞ n lim

(Throughout this work, we will also refer to Y1 (ω) as simply Y (ω) or Yω .) (II) V is equivariant i.e. Lω V (ω) ⊆ V (σω) and for every v ∈ V (ω), 1 log kL(n) ω vk ≤ κ(R). n→∞ n lim

The adjoint cocycle associated to R is the cocycle R∗ := (Ω, F , P, σ −1 , B∗ , L∗ ), where (L )ω := (Lσ−1 ω )∗ . In a slight abuse of notation which should not cause confusion, we will often write L∗ω instead of (L∗ )ω , so L∗ω will denote the operator adjoint to Lσ−1 ω . ∗

Remark 2.4. It is straightforward to check that if (C0) holds for R, it also holds for R∗ . Furthermore, Λ(R∗ ) = Λ(R) and κ(R∗ ) = κ(R). The last statement follows from the equality, up to a multiplicative factor (2), ic(A) and ic(A∗ ) for every A ∈ L(B) [3, Theorem 2.5.1]. The following result gives an answer to a natural question on whether one can relate the Lyapunov exponents and Oseledets splitting of the adjoint cocycle R∗ with the Lyapunov exponents and Oseledets decomposition of the original cocycle R.

9

Corollary 2.5. Under the assumptions of Theorem 2.3, the adjoint cocycle R∗ has a unique, measurable, equivariant Oseledets splitting ∗

∗

B = V (ω) ⊕

l M

Yj∗ (ω),

(6)

j=1

with the same exceptional Lyapunov exponents λj and multiplicities mj as R. The proof of this result involves some technical properties about volume growth in Banach spaces, and is therefore deferred to Appendix A. Next, we establish a relation between Oseledets splittings of R and R∗ , which will be used in the sequel. Let the simplified Oseledets decomposition for the cocycle L (resp. L∗ ) be B = Y (ω) ⊕ H(ω) (resp. B∗ = Y ∗ (ω) ⊕ H ∗ (ω)), (7) where Y (ω) (resp. Y ∗ (ω)) is the top Oseledets subspace for L (resp. L∗ ) and H(ω) (resp. H ∗ (ω)) is a direct sum of all other Oseledets subspaces. For a subspace S ⊂ B, we set S ◦ = {φ ∈ B∗ : φ(f ) = 0 for every f ∈ S} and similarly for a subspace S ∗ ⊂ B∗ we define (S ∗ )◦ = {f ∈ B : φ(f ) = 0 for every φ ∈ S ∗ }.

Lemma 2.6 (Relation between Oseledets splittings of R and R∗ ). The following relations hold for P-a.e. ω ∈ Ω: H ∗ (ω) = Y (ω)◦

and H(ω) = Y ∗ (ω)◦ .

(8)

Proof. We first claim that lim sup n→∞

1 logkL∗,(n) |Y (ω)◦ k < λ1 , ω n

for P-a.e. ω ∈ Ω.

(9)

Let Πω denote the projection onto H(ω) along Y (ω) and take an arbitrary φ ∈ Y (ω)◦. We have (n)

φ)(f )| = sup |φ(Lσ−n ω (f ))| kL∗,(n) φkB∗ = sup |(L∗,(n) ω ω kf kB ≤1

= sup kf kB ≤1

kf kB ≤1

(n) |φ(Lσ−n ω (Πσ−n ω f ))|

and thus

(n)

≤ kφkB∗ · kLσ−n ω Πσ−n ω k,

(n)

kLω∗,(n) |Y (ω)◦ k ≤ kLσ−n ω Πσ−n ω k.

Hence, in order to prove (9) it is sufficient to show that lim sup n→∞

1 (n) logkLσ−n ω Πσ−n ω k < λ1 , n


However, it follows from results in [12] and [15, Lemma 8.2] that 1 (n) logkLσ−n ω |H(σ−n ω) k = λ2 n→∞ n lim

10

(10)

and

1 logkΠσ−n ω k = 0, n which readily imply (10). We now claim that lim

n→∞

B∗ = Y (ω)∗ ⊕ Y (ω)◦ ,


(11)

We first note that the sum on the right hand side of (11) is direct. Indeed, each nonzero vector in Y (ω)∗ grows at the rate λ1 , while by (9) all nonzero vectors in Y (ω)◦ grow at the rate < λ1 . Furthermore, since the codimension of Y (ω)◦ is the same as dimension of Y (ω)∗ , we have that (11) holds. Finally, by comparing decompositions (7) and (11), we conclude that the first equality in (8) holds. Indeed, each φ ∈ H ∗ (ω) can be written as φ = φ1 + φ2 , where φ1 ∈ Y (ω)∗ and φ2 ∈ Y (ω)◦ . Since φ and φ2 grow at the rate < λ1 and φ1 grows at the rate λ1 , we obtain that φ1 = 0 and thus φ = φ2 ∈ Y (ω)◦. Hence, H ∗ (ω) ⊂ Y (ω)◦ and similarly Y (ω)◦ ⊂ H ∗ (ω). The second assertion of the lemma can be obtained similarly.

2.2

Notions of variation

Let (X, G) be a measurable space endowed with a probability measure m and a notion of a variation var : L1 (X, m) → [0, ∞] which satisfies the following conditions: (V1) var(th) = |t| var(h); (V2) var(g + h) ≤ var(g) + var(h); (V3) khkL∞ ≤ Cvar (khk1 + var(h)) for some constant 1 ≤ Cvar < ∞; (V4) for any C > 0, the set {h : X → R : khk1 + var(h) ≤ C} is L1 (m)-compact; (V5) var(1X ) < ∞, where 1X denotes the function equal to 1 on X; (V6) {h : X → R+ : khk1 = 1 and var(h) < ∞} is L1 (m)-dense in {h : X → R+ : khk1 = 1}. (V7) for any f ∈ L1 (X, m) such that ess inf f > 0, we have var(1/f ) ≤

var(f ) . (ess inf f )2

(V8) var(f g) ≤ kf kL∞ · var(g) + kgkL∞ · var(f ). (V9) for M > 0, f : X → [−M, M] measurable and every C 1 function h : [−M, M] → C, we have var(h ◦ f ) ≤ kh′ kL∞ · var(f ). We define B := BV = BV (X, m) = {g ∈ L1 (X, m) : var(g) < ∞}. Then, B is a Banach space with respect to the norm kgkB = kgk1 + var(g). 11

From now on, we will use B to denote a Banach space of this type, and kgkB , or simply kgk will denote the corresponding norm. Well-known examples of this notion correspond to the case where X is a subset of Rn . In the one-dimensional case we use the classical notion of variation given by var(g) =

inf

sup

h=g(mod m) 0=s0 0 such that for every n ≥ 0, f ∈ B such that P-a.e. ω ∈ Ω. kLω(n) (f )kB ≤ K ′ e−λn kf kB .

R

f dm = 0 and

(C4) there exist N ∈ N, c > 0 such that for each a > 0 and any sufficiently large n ∈ N, ess inf Lω(N n) f ≥ c/2kf k1,

for every f ∈ Ca and P-a.e. ω ∈ Ω, R where Ca := {f ∈ B : f ≥ 0 and var(f ) ≤ a f dm}.

Admissible cocycles of transfer operators can be investigated via Theorem 2.3. Indeed, the following holds. Lemma 2.9. An admissible cocycle of transfer operators R = (Ω, F , P, σ, B, L) is quasicompact. Furthermore, the top Oseledets space is one-dimensional. That is, dim Y (ω) = 1 for P-a.e. ω ∈ Ω. Proof. The first statement follows readily from Lemma 2.1, (C2) and a simple observation that for a cocycle R of transfer operators we have that Λ(R) = 0. The fact that dim Y (ω) = 1 follows from (C3). The following result shows that, in this context, the top Oseledets space is indeed the unique random acim. That is, there exists a unique measurable function v 0 : Ω × X → R+ R such that for P-a.e. ω ∈ Ω, vω0 := v 0 (ω, ·) ∈ B, vω0 (x)dm = 1 and 0 Lω vω0 = vσω ,


(15)

Lemma 2.10 (Existence and uniqueness of a random acim). Let R = (Ω, F , P, σ, B, L) be an admissible cocycle of transfer operators, satisfying the assumptions of Theorem 2.3. Then, there exists a unique random absolutely continuous invariant measure for R.

13

Proof. Theorem 2.3 shows that the map ω 7→ Yω is measurable, where Yω is regarded as an element of the Grassmannian G of B. Furthermore, [16, Lemma 10] and an argument analogous to [18, Lemma 10] yields existence of a measurable selection of bases for Yω . Lemma 2.9 ensures that dim Y (ω) = 1. Hence, there exists a measurable map ω 7→ hω , with hω ∈ B such that hω spans Yω for P-a.e. ω ∈ Ω. Notice that Lebesgue measure m, when regarded as an element of B∗ , is a conformal measure for R. That is, m spans Yω∗ for P-a.e. ω ∈ Ω. In fact, it is straightforward to verify L∗ω m = m, because the Lω preserve integrals. Thus, the simplified Oseledets decomposition (7) in combination with the duality relations of Lemma 2.6 imply that m(hω ) 6= 0 for P-a.e. ω ∈ Ω. In particular we can consider the (still measurable) function ω 7→ vω0 := R hhωωdm . The equivariance property of Theorem 2.3 ensures that Lω vω0 ∈ Yσω and the fact that Lω preserves integrals, combined with the normalized choice of vω0 and the assumption that 0 dim Yσω = 1, implies that Lω vω0 = vσω . 0 The fact that vω ≥ 0 for P-a.e. ω ∈ Ω follows from the positivity and linearity properties of Lω , which ensure that the positive and negative parts, vω+ and vω− , are equivariant. Recall that vω+ , vω− , have non-overlapping supports. Thus, if vω+ 6= 0 6= vω− for a set of positive measure of ω ∈ Ω, the spaces Yω+ , Yω− spanned by vω+ , vω− , respectively, are subsets of Y (ω), contradicting the fact that dim Y (ω) = 1. Then, since the normalization condition implies vω+ 6= 0, we have vω− = 0 for P-a.e. ω ∈ Ω. The fact that the random acim is unique is also a direct consequence of the fact that dim Y (ω) = 1. For an admissible transfer operator cocycle R, we let µ be the invariant probability measure given by Z µ(A × B) = v 0 (ω, x) d(P × m)(ω, x), for A ∈ F and B ∈ G, (16) A×B

where v 0 is the unique random acim for R and G is the Borel σ-algebra of X. We note that µ is τ -invariant, because of (15). Furthermore, for each G ∈ L1 (Ω × X, µ) we have that Z Z Z G dµ = G(ω, x) dµω (x) dP(ω), Ω×X

Ω

X

where µω is a measure on X given by dµω = v 0 (ω, ·)dm. We now list several important consequences of conditions (C2), (C3) and (C4) established in [13, §2]. Lemma 2.11. The unique random acim v 0 of an admissible cocycle of transfer operators satiesfies the following: 1. ess supω∈Ω kvω0 kB < ∞;

(17)

2. ess inf vω0 (·) ≥ c/2 > 0, 14

for P-a.e. ω ∈ Ω;

(18)

3. there exists K > 0 and ρ ∈ (0, 1) such that Z Z Z (n) 0 n n ω ≤ Kρ khkL∞ · kf kB , L (f v )h dm − f dµ · h dµ ω σ ω ω X

X

(19)

X

for n ≥ 0, h ∈ L∞ (X, m), f ∈ B and P-a.e. ω ∈ Ω.

We emphasize that (19) is a special case of a more general decay of correlations result proved by Buzzi [10], but in this case with the stronger conclusion that the decay rates and coefficients K are uniform over ω ∈ Ω. 2.3.1

Examples

In order to be able to be in the setting of admissible transfer operators cocycles, we need to ensure that (C0) holds. To fulfill this requirement (see [16, Section 4.1] for a detailed discussion) in the rest of the paper we will assume (C0’) σ is a homeomorphism, Ω is a Borel subset of a separable, complete metric space, the map ω → Tω has a countable range T1 , T2 , . . . and for each j, {ω ∈ Ω : Tω = Tj } is measurable. Although this condition is somewhat restrictive, we emphasize that the assumptions on the structure of Ω are very mild and that the only requirements for σ are that it has to be an ergodic, measure-preserving homeomorphism. In particular, no mixing conditions are required. Furthermore, the Tω need only be chosen from a countable family. Following [13, §2], we present two classes of examples, one- and higher-dimensional piecewise smooth expanding maps, which yield admissible transfer operator cocycles. Random Lasota-Yorke maps. Let X = [0, 1], a Borel σ-algebra G on [0, 1] and the Lebesgue measure m on [0, 1]. Consider the notion of variation defined in (12). For a piecewise C 2 map T : [0, 1] → [0, 1], set δ(T ) = ess inf x∈[0,1] |T ′ | and let b(T ) denote the number of intervals of monoticity (branches) of T . Consider now a measurable map ω 7→ Tω , ω ∈ Ω of piecewise C 2 maps on [0, 1] such that b := ess supω∈Ω b(Tω ) < ∞, δ := ess inf ω∈Ω δ(Tω ) > 1, and D := ess supω∈Ω kTω′′ kL∞ < ∞. (20) For each ω ∈ Ω, let bω = b(Tω ), so that there are essentially disjoint sub-intervals ω Jω,k = I, so that Tω |Jω,k is C 2 for each 1 ≤ k ≤ bω . The Jω,1 , . . . , Jω,bω ⊂ I, with ∪bk=1 minimal such partition Pω := {Jω,1 , . . . , Jω,bω } is called the regularity partition for Tω . It is well known that whenever δ > 2, and ess inf ω∈Ω min1≤k≤bω m(Jω,k ) > 0, there exist α ∈ (0, 1) and K > 0 such that var(Lω f ) ≤ α var(f ) + Kkf k1 ,

15

for f ∈ BV and P-a.e. ω ∈ Ω.

More generally, when δ < 2, one can take an iterate N ∈ N so that δ N > 2. If the (N ) N regularity partitions PωN := {J1,ω , . . . , J N (N) } corresponding to the maps Tω also satisfy ω,bω

N ess inf ω∈Ω min1≤k≤b(N) m(Jω,k ) > 0, then there exist αN ∈ (0, 1) and K N > 0 such that ω N N var(LN ω f ) ≤ α var(f ) + K kf k1 ,

for f ∈ BV and P-a.e. ω ∈ Ω.

(21)

We assume that (21) holds for some N ∈ N. Finally, we suppose the following uniform covering condition holds: For every subinterval J ⊂ I, ∃k = k(J) s.t. for a.e. ω ∈ Ω, Tω(k) (J) = I.

(22)

The results of [13, §2] ensure that random Lasota-Yorke maps which satisfy the conditions of this section plus (C0’) are admissible. (While (C2) is not explicitely required by [13], it is established in the process of showing the remaining conditions.) Random piecewise expanding maps in higher dimensions. We now discuss the case of piecewise expanding maps in higher dimensions. Let X be a compact subset of RN which is the closure of its non-empty interior. Let X be equipped with a Borel σ-algebra G and Lebesgue measure m. We consider the notion of variation defined in (13) for suitable α and ǫ0 . We say that the map T : X → X is piecewise expanding if there exist finite families N N ˜ ˜ m ˜ A = {Ai }m i=1 and A = {Ai }i=1 of open sets in R , a family of maps Ti : Ai → R , i = 1, . . . , m and ǫ1 (T ) > 0 such that: S 1. A is a disjoint family of sets, m(X \ i Ai ) = 0 and A˜i ⊃ Ai for each i = 1, . . . , m; 2. there exists 0 < γ(Ti ) ≤ 1 such that each Ti is of class C 1+γ(Ti ) ;

3. For every 1 ≤ i ≤ m, T |Ai = Ti |Ai and Ti (A˜i ) ⊃ Bǫ1 (T ) (T (Ai )), where Bǫ (V ) denotes a neighborhood of size ǫ of the set V. We say that Ti is the local extension of T to the A˜i ; 4. there exists a constant C1 (T ) > 0 so that for each i and x, y ∈ T (Ai ) with dist(x, y) ≤ ǫ1 (T ), | det DTi−1 (x) − det DTi−1 (y)| ≤ C1 (T )| det DTi−1 (x)|dist(x, y)γ(T ) ; 5. there exists s(T ) < 1 such that for every x, y ∈ T (A˜i ) with dist(x, y) ≤ ǫ1 (T ), we have dist(Ti−1 x, Ti−1 y) ≤ s(T ) dist(x, y); 6. each ∂Ai is a codimension-one embedded compact piecewise C 1 submanifold and s(T )γ(T ) + where Z(T ) = sup x

P

4s(T ) ΓN −1 Z(T ) < 1, 1 − s(T ) ΓN

#{smooth pieces intersecting ∂Ai containing x} and ΓN is the

i

volume of the unit ball in RN . 16

Consider now a measurable map ω 7→ Tω , ω ∈ Ω of piecewise expanding maps on X such that ǫ1 := inf ǫ1 (Tω ) > 0, γ := inf γ(Tω ) > 0, C1 := sup C1 (Tω ) < ∞, s := sup s(Tω ) < 1 ω∈Ω

and

ω∈Ω

ω∈Ω

ω∈Ω

4s(Tω ) ΓN −1 < 1. sup s(Tω )γ(Tω ) + Z(Tω ) 1 − s(Tω ) ΓN ω∈Ω

Then, [43, Lemma 4.1] implies that there exist ν ∈ (0, 1) and K > 0 independent on ω such that var(Lω f ) ≤ ν var(f ) + Kkf k1 for each f ∈ B and ω ∈ Ω, (23) where var is given by (13) with α = γ and some ǫ0 > 0 sufficiently small (which is again independent on ω). We note that (23) readily implies that conditions (C1) and (C2) hold. Finally, we note that under additional assumption that for any open set J ⊂ X, there exists k = k(J) such that for a.e. ω ∈ Ω, Tωk (J) = X, the results in [13, §2] show that (C3) and (C4) also hold. Remark. We point out that while conditions (C1), (C3) and (C4) are stated in a uniform way, sometimes it is possible to recover them from non-uniform assumptions. For example, assuming that {Tω }ω∈Ω takes only finitely many values, one can recover a uniform version of (C3) from a non-uniform one, for example by compactness arguments (see the proof of Lemma 4.7 for a similar argument). Also, our results apply to cases where conditions (C1)– (C4), or the hypotheses which imply them (e.g. (20)), are only satisfied eventually; that is, (N ) for some iterate Tω , where N is independent of ω ∈ Ω.

3

Twisted transfer operator cocycles

We begin by introducing the class of observables to which our limit theorems apply. For a fixed observable and each parameter θ ∈ C, we introduce the twisted cocycle Lθ = {Lθω }ω∈Ω . We show that the cocycle Lθ is quasicompact for θ close to 0. Most of this section is devoted to the study of regularity properties of the map θ 7→ Λ(θ) on a neighborhood of 0 ∈ C, where Λ(θ) denotes the top Lyapunov exponent of the cocycle Lθ . In particular, we show that this map is of class C 2 and that its restriction to a neighborhood of 0 ∈ R is strictly convex. This is achieved by combining ideas from the perturbation theory of linear operators with our multiplicative ergodic theory machinery. As a byproduct of our approach, we explicitly construct the top Oseledets subspace of cocycle Lθ for θ close to 0.

17

3.1

The observable

Definition 3.1 (Observable). Let an observable be a measurable map g : Ω × X → R satisfying the following properties: • Regularity: kg(ω, x)kL∞ (Ω×X) =: M < ∞ and

ess supω∈Ω var(gω ) < ∞,

(24)

where gω = g(ω, ·), ω ∈ Ω. • Fiberwise centering: Z Z g(ω, x) dµω (x) = g(ω, x)vω0 (x) dm(x) = 0

for P-a.e. ω ∈ Ω,

(25)

where v 0 is the density of the unique random acim, satisfying (15). The main results of this paper will deal with establishing limit theorems for Birkhoff sums associated to g, Sn g, defined in (1).

3.2

Basic properties of twisted transfer operator cocycles

Throughout this section, R = (Ω, F , P, σ, B, L) will denote an admissible transfer operator cocycle. For θ ∈ C, the twisted transfer operator cocycle, or twisted cocycle, Rθ is defined as Rθ = (Ω, F , P, σ, B, Lθ ), where for each ω ∈ Ω, we define Lθω (f ) = Lω (eθg(ω,·) f ),

f ∈ B.

(26)

For convenience of notation, we will also use Lθ to denote the cocycle Rθ . For each θ ∈ C, set Λ(θ) := Λ(Rθ ), κ(θ) := κ(Rθ ) and (n) Lθ, = Lθσn−1 ω ◦ · · · ◦ Lθω , ω

for ω ∈ Ω and n ∈ N.

The next lemma provides basic information about the dependence of Lθω on θ. Lemma 3.2 (Basic regularity of θ 7→ Lθω ). 1. Assume (C1) holds. Then, there exists a continuous function K : C → (0, ∞) such that kLθω hkB ≤ K(θ)khkB , for h ∈ B, θ ∈ C and P-a.e. ω ∈ Ω. (27) 2. For ω ∈ Ω, θ ∈ C, let Mωθ be the linear operator on B given by Mωθ (h(·)) := eθg(ω,·) h(·). Then, θ 7→ Mωθ is continuous in the norm topology of B. Consequently, θ 7→ Lθω is also continuous in the norm topology of B.

18

Proof. Note that it follows from (24) that |eθg(ω,·) h|1 ≤ e|θ|M |h|1 . Furthermore, by (V8) we have var(eθg(ω,·) h) ≤ keθg(ω,·) kL∞ · var(h) + var(eθg(ω,·) ) · khkL∞ .

On the other hand, it follows from Lemma B.1 and (V9) that keθg(ω,·) kL∞ ≤ e|θ|M

and

var(eθg(ω,·) ) ≤ |θ|e|θ|M var(g(ω, ·))

and thus using (V3), keθg(ω,·) hkB = var(eθg(ω,·) h) + |eθg(ω,·) h|1

≤ e|θ|M khkB + |θ|e|θ|M var(g(ω, ·))khkL∞

(28)

≤ (e|θ|M + Cvar |θ|e|θ|M ess supω∈Ω var(g(ω, ·)))khkB. We now establish part 1 of the Lemma. It follows from (C1) that kLθω (h)kB = kLω (eθg(ω,·) h)kB ≤ Kkeθg(ω,·) hkB . Hence, (28) implies that (27) holds with K(θ) = K(e|θ|M + Cvar |θ|e|θ|M ess supω∈Ω var(g(ω, ·))).

(29)

For part 2 of the Lemma, we observe that |(Mωθ1 − Mωθ2 )hkB ≤ kMωθ1 kB k(I − Mωθ2 −θ1 )kB khkB . By (24) and the mean value theorem for the map z 7→ e(θ1 −θ2 )z , we have that for each x ∈ X, |e(θ1 −θ2 )g(ω,x) − 1| ≤ Me|θ1 −θ2 |M |θ1 − θ2 |. Thus, and

k1 − e(θ2 −θ1 )g(ω,·) kL∞ ≤ Me|θ1 −θ2 |M |θ1 − θ2 |

(30)

|(I − Mωθ2 −θ1 )h|1 ≤ Me|θ1 −θ2 |M |θ1 − θ2 | · |h|1 .

(31)

Assume that |θ2 − θ1 | ≤ 1. We note that conditions (V3) and (V8) together with (30) and Lemma B.2 imply var((I − Mωθ2 −θ1 )h) ≤ k1 − e(θ2 −θ1 )g(ω,·) kL∞ + Cvar var(1 − e(θ2 −θ1 )g(ω,·) ) khkB (32) ≤ C ′ |θ2 − θ1 |khkB , for some C ′ > 0. Hence, it follows from (31) and (32) that θ 7→ Mωθ is continuous in the norm topology of B. Continuity of θ 7→ Lθω then follows immediately from continuity of Lω and the definition of Lθω , in (26). The following lemma shows that the twisted cocycle naturally appears in the study of Birkhoff sums (1). 19

Lemma 3.3. The following statements hold: 1. for every φ ∈ B∗ , f ∈ B, ω ∈ Ω, θ ∈ C and n ∈ N we have that Lθ,(n) (f ) = Lω(n) (eθSn g(ω,·) f ), ω

and Lθ∗,(n) (φ) = eθSn g(ω,·) L∗(n) ω ω (φ),

(33)

where (eθSn g(ω,·) φ)(f ) := φ(eθSn g(ω,·) f ); 2. for every f ∈ B, ω ∈ Ω and n ∈ N we have that Z Z θ, (n) Lω (f ) dm = eθSn g(ω,·) f dm.

(34)

Proof. We establish the first identity in (33) by induction on n. The case n = 1 follows from the definition of Lθω . We recall that for every f, f˜ ∈ B, Lω(n) ((f˜ ◦ Tω(n) ) · f ) = f˜ · L(n) ω (f ).

(35)

Assuming the claim holds for some n ≥ 1, we get L(n+1) (eθSn+1 g(ω,·) f ) = Lσn ω Lω(n) (eθg(σ ω = Lσ n ω

n ω,·)◦T (n) ω

eθSn g(ω,·) f ) n θSn g(ω,·) eθg(σ ω,·) L(n) f ) = Lθσn ω Lθ,(n) (f ) = Lθ,(n+1) (f ). ω (e ω ω

The second identity in (33) follows directly from duality. Finally, we note that the second assertion of the lemma follows by integrating the first equality in (33) with respect to m and using the fact that Lnω preserves integrals with respect to m.

3.3

An auxiliary existence and regularity result

In this section we establish a regularity result, Lemma 3.5, which generalises a theorem of Hennion and Hervé [26] to the random setting. This result will be used later to show regularity of the top Oseledets space Yωθ := Y1θ (ω) of the twisted cocycle, for θ near 0. Let n S := V : Ω × X → C | V is measurable, V(ω, ·) ∈ B, Z o (36) ess supω∈Ω kV(ω, ·)kB < ∞, V(ω, x)dm = 0 for P-a.e. ω ∈ Ω ,

endowed with the Banach space structure defined by the norm kVk∞ := ess supω∈Ω kV(ω, ·)kB .

(37)

For θ ∈ C and W ∈ S, set F (θ, W)(ω, ·) = R

Lθσ−1 ω (W(σ −1 ω, ·) + vσ0−1 ω (·)) − W(ω, ·) − vω0 (·). 0 θ −1 Lσ−1 ω (W(σ ω, ·) + vσ−1 ω (·))dm 20

(38)

Lemma 3.4. There exist ǫ, R > 0 such that F : D → S is a well-defined map on D := {θ ∈ C : |θ| < ǫ} × BS (0, R), where BS (0, R) denotes the ball of radius R in S centered at 0. Proof. We define a map H by Z Z −1 θ −1 0 H(θ, W)(ω) = Lσ−1 ω (W(σ ω, ·) + vσ−1 ω (·)) dm = eθg(σ ω,·) (W(σ −1 ω, ·) + vσ0−1 ω (·)) dm. It is proved in Lemmas B.4 and B.5 of Appendix B.1 that H is a well-defined and differentiable function on a neighborhood of (0, 0) (and thus in particular continuous) with values in L∞ (Ω, P). Moreover, we observe that H(0, 0)(ω) = 1 for each ω ∈ Ω and therefore |H(θ, W)(ω)| ≥ 1 − |H(0, 0)(ω) − H(θ, W)(ω)| ≥ 1 − kH(0, 0) − H(θ, W)kL∞ , for P-a.e. ω ∈ Ω. Continuity of H implies that kH(0, 0) − H(θ, W)kL∞ ≤ 1/2 for all (θ, W) in a neighborhood of (0, 0) and hence, in such neighborhood, ess inf ω |H(θ, W)(ω)| ≥ 1/2. The above inequality together with Lemma 3.2 (1) and (17) yields the desired conclusion. Lemma 3.5. Let D = {θ ∈ C : |θ| < ǫ} × BS (0, R) be as in Lemma 3.4. Then, F : D → S is C 1 and the equation F (θ, W) = 0 (39) has a unique solution O(θ) ∈ S, for every θ in a neighborhood of 0. Furthermore, O(θ) is a C 2 function of θ. Proof. We notice that F (0, 0) = 0. Furthermore, Proposition B.12 of Appendix B ensures that F is C 2 on a neighborhood (0, 0) ∈ C × S, and (D2 F (0, 0)X )(ω, ·) = Lσ−1 ω (X (σ −1 ω, ·)) − X (ω, ·),

for ω ∈ Ω and X ∈ S.

We now prove that D2 F (0, 0) is bijective operator. For injectivity, we have that if D2 F (0, 0)X = 0 for Rsome nonzero X ∈ S, then Lω Xω = Xσω for P-a.e. ω ∈ Ω. Notice that Xω ∈ / hvω0 i because Xω (·)dm = 0 and Xω 6= 0. Hence, this yields a contradiction with the one-dimensionality of the top Oseledets space of the cocycle L, given by Lemma 2.9. Therefore, D2 F (0, 0) is injective. To prove surjectivity, take X ∈ S and let ∞ X (j) ˜ Lσ−j ω X (σ −j ω, ·). (40) X (ω, ·) := − j=0

It follows from (C3) that X˜ ∈ S and it is easy to verify that D2 F (0, 0)X˜ = X . Thus, D2 F (0, 0) is surjective. Combining the previous arguments, we conclude that D2 F (0, 0) is bijective. The conclusion of the lemma now follows directly from the implicit function theorem for Banach spaces (see, e.g. Theorem 3.2 [5]). 21

We end this section with a specialisation of the previous results to real valued θ. Proposition 3.6. There exists δ > 0 such that for each θ ∈ (−δ, δ), O(θ)(ω, ·) + vω0 is a density for P-a.e. ω ∈ Ω. We first show the following auxiliary result. Lemma 3.7. For θ ∈ R sufficiently close to 0, O(θ) is real-valued. Proof. We consider the space n ˜ S := V : Ω × X → R | V is measurable, V(ω, ·) ∈ B, Z o ess supω∈Ω kV(ω, ·)kB < ∞, V(ω, x)dm = 0 for P-a.e. ω ∈ Ω .

Hence, S˜ consists of real-valued functions V ∈ S. We note that S˜ is a Banach space with the norm k·k∞ defined by (37). Moreover, we can define a map F˜ on a neighborhood of (0, 0) in R × S˜ with values in S˜ by the RHS of (38). Proceeding as in Appendix B.1, one can show that F˜ is a differentiable map on a neighborhood of (0, 0). Moreover, arguing as in the proof of Lemma 3.5 one can conclude that for θ sufficiently close to 0, there exists a unique ˜ ˜ ˜ O(θ) ∈ S˜ such that F˜ (θ, O(θ)) = 0 and that O(θ) is differentiable with respect to θ. Since ˜ S ⊂ S and from the uniqueness property in the implicit function theorem, we conclude that ˜ O(θ) = O(θ) for θ sufficiently close to 0 which immediately implies the conclusion of the lemma. Proof of Proposition 3.6. for θ sufficiently close to 0, O(θ)(ω, ·) + vω0 (·) is R By Lemma 3.7, 0 real-valued. Moreover, (O(θ)(ω, ·) + vω (·)) dm = 1 for a.e. ω ∈ Ω. It remains to show that O(θ)(ω, ·) + vω0 (·) ≥ 0 for P-a.e. ω ∈ Ω. Since the map θ 7→ O(θ) is continuous, there exists δ > 0 such that for all θ ∈ (−δ, δ), O(θ) belongs to a ball of radius c/(4Cvar ) centered at 0 in S. In particular, ess supω∈Ω kO(θ)(ω, ·)kB < c/(4Cvar ) and therefore, ess supω∈Ω kO(θ)(ω, ·)kL∞ < c/4. By (18), ess inf(O(θ)(ω, ·) + vω0 (·)) ≥ c/4,

for a.e. ω ∈ Ω,

which completes the proof of the proposition.

3.4

A lower bound on Λ(θ)

ˆ The goal of this section is to establish a differentiable lower bound (Λ(θ)) on Λ(θ), the top Lyapunov exponent of the twisted cocycle, for θ ∈ C in a neighborhood of 0. In Section 3.5, we will show that this lower bound in fact coincides with Λ(θ), and hence all the results of this section will immediately translate into properties of Λ. 22

Let 0 < ǫ < 1 be as in Lemma 3.4 and O(θ) be as in Lemma 3.5. Let vωθ (·) := vω0 (·) + O(θ)(ω, ·). We notice that us define

and

R

(41)

vωθ (·) dm = 1 and by Lemma 3.5, θ 7→ v θ is continuously differentiable. Let Z Z ˆ Λ(θ) := log eθg(ω,x) vωθ (x) dm(x) dP(ω), (42) λθω

:=

Z

eθg(ω,x) vωθ (x) dm(x)

=

Z

Lθω vωθ (x) dm(x),

(43)

where the last identity follows from (34). Notice also that ω 7→ λθω is an integrable function. ˆ Lemma 3.8. For every θ ∈ BC (0, ǫ) := {θ ∈ C : |θ| < ǫ}, Λ(θ) ≤ Λ(θ). Proof. Recall that O(θ) satisfies the equation F (θ, O(θ)) = 0, for θ ∈ {θ ∈ C : |θ| < ǫ}. θ Hence, for P-a.e. ω ∈ Ω, vωθ (·) satisfies the equivariance equation Lθω vωθ (·) = λθω vσω (·) Thus, using Birkhoff’s ergodic theorem to go from the first to the second line below, we get n−1 1 1 1X θ,(n) θ θ,(n) θ Λ(θ) ≥ lim log kLω vω kB ≥ lim log kLω vω k1 ≥ lim log |λθσj ω | n→∞ n n→∞ n n→∞ n j=0 Z Z Z ˆ = log |λθω |dP(ω) = log eθg(ω,x) vωθ (·) dm(x) dP(ω) = Λ(θ).

ˆ The rest of the section deals with differentiability properties of Λ(θ). From now on we shall also use the notation O(θ)ω for O(θ)(ω, ·). ˆ is differentiable on a neighborhood of 0, and Lemma 3.9. We have that Λ ! Z θ R θg(ω,·) 0 θg(ω,·) ′ λ ( g(ω, ·)e (O(θ) (·) + v (·)) + e O (θ) (·) dm) ω ω ω ω ′ ˆ (θ) = ℜ Λ dP(ω) , |λθω |2 where ℜ(z) denotes the real part of z and z the complex conjugate of z. Proof. Write ˆ Λ(θ) = where Z(θ, ω) :=

log |λθω |

Z

Z(θ, ω) dP(ω),

Z = log eθg(ω,x) (O(θ)ω (x) + vω0 (x)) dm(x) .

Note that Z(θ, ω) = log |H(θ, O(θ))(σω)|, where H is as in Lemma 3.4. Since H(0, 0) = 1 and both H and O are continuous (by Lemma 3.5), there is a neighborhood U of 0 in C on which kH(θ, O(θ)) − H(0, 0)kL∞ < 1/2. In particular, Z is well defined and Z(θ, ω) ∈ [log 12 , log 23 ] 23

for every θ ∈ U ∩ BC (0, ǫ) and P-a.e. ω ∈ Ω. Thus, the map ω 7→ Z(θ, ω) is P-integrable for every θ ∈ U ∩ BC (0, ǫ). It follows from Lemma 3.10 below that for P-a.e. ω ∈ Ω, the map θ 7→ Zω (θ) := Z(θ, ω) is differentiable in a neighborhood of 0, and R ℜ λθω ( g(ω, ·)eθg(ω,·)(O(θ)ω (·) + vω0 (·)) + eθg(ω,·) O ′ (θ)ω (·) dm) , Zω′ (θ) = |λθω |2 where ℜ(z) denotes the real part of z and z the complex conjugate of z. In particular, R (g(ω, x)eθg(ω,x)(O(θ)ω (x) + v 0 (x)) + eθg(ω,x) O ′(θ)ω (x)) dm(x)| ω ′ R |Zω (θ)| ≤ . θg(ω,x) | e (O(θ)ω (x) + vω0 (x)) dm(x)|

We claim that there exists an integrable function C : Ω → R such that |Zω′ (θ)| ≤ C(ω),

for all θ in a neighborhood of 0 and P-a.e. ω ∈ Ω.

(44)

Once this is established, the conclusion of the lemma follows from Leibniz rule for exchanging the order of differentiation and integration. To complete the proof, let us show (44). For θ ∈ U we have 1 Z eθg(ω,x) (O(θ)ω (x) + vω0 (x)) dm(x) ≥ . 2

Also, recall that ǫ < 1, so that for θ ∈ BC (0, ǫ) one has Z Z g(ω, x)eθg(ω,x)(O(θ)ω (x) + v 0 (x)) dm(x) ≤ g(ω, x)eθg(ω,x)(O(θ)ω (x) + v 0 (x)) dm(x) ω ω ≤ MeM |O(θ)ω + vω0 |1 ≤ MeM (1 + kO(θ)ω kB ) ≤ MeM (1 + kO(θ)k∞).

Finally, Z θg(ω,x) ′ e O (θ)ω (x) dm(x) ≤ eM |O ′ (θ)ω |1 ≤ eM kO ′(θ)ω kB ≤ eM kO ′(θ)k∞ ,

for P-a.e. ω ∈ Ω. Since O and O ′ are continuous by Lemma 3.5, the terms on the RHS of the above inequalities are uniformly bounded for θ in a (closed) neighborhood of 0. Hence, (44) holds for a constant function C. Lemma 3.10. For P-a.e. ω ∈ Ω, and θ in a neighborhood of 0, the map θ 7→ Zω (θ) := Z(θ, ω) is differentiable. Moreover, R ℜ λθω ( g(ω, ·)eθg(ω,·)(O(θ)ω (·) + vω0 (·)) + eθg(ω,·) O ′ (θ)ω (·) dm) , Zω′ (θ) = |λθω |2 where ℜ(z) denotes the real part of z and z the complex conjugate of z. 24

Proof. First observe that if θ 7→ f (θ) ∈ C, has polar decomposition f (θ) = r(θ)eiφ(θ) , then, ′ (θ)) ¯ whenever |f |(θ) 6= 0, d|fdθ|(θ) = ℜ(f (θ)f , where f ′ denotes differentiation with respect to θ. r(θ) Thus, by the chain rule, it is sufficient to prove that the map λθω is differentiable with respect to θ and that Z θ θg(ω,x) 0 θg(ω,x) ′ Dθ λ ω = g(ω, x)e (O(θ)ω (x) + vω (x)) + e O (θ)ω (x) dm(x). (45) Using the same notation as in Lemma 3.4, we can write λθω = H(θ, O(θ))σω . The differentiability of the map θ 7→ H(θ, W) implies the differentiability of the map θ 7→ H(θ, W)ω for a.e. ω. Hence, differentiability of λθω with respect to θ and (45), for P-a.e. ω ∈ Ω, follow directly from the differentiability of O, Lemma B.5 (in particular (94)) and the chain rule. ˆ ′ (0) = 0. Lemma 3.11. We have that Λ Proof. Let F be as in Lemma 3.5. By identifying D1 F (0, 0) with its value at 1, it follows from the implicit function theorem that O ′ (0) = −D2 F (0, 0)−1 (D1 F (0, 0)). It is shown in Lemma 3.5 that D2 F (0, 0) : S → S is bijective. Thus, D2 F (0, 0)−1 : S → S and therefore O ′(0) ∈ S which implies that Z O ′(0)ω dm(x) = 0 for P-a.e. ω ∈ Ω. (46) The conclusion of the lemma follows directly from Lemma 3.9 and the centering condition (25).

3.5

Quasicompactness of twisted cocycles and differentiability of Λ(θ)

In this section we establish quasicompactness of the twisted transfer operator cocycle, as well as differentiability of the top Lyapunov exponent with respect to θ, for θ ∈ C near 0. Theorem 3.12 (Quasi-compactness of twisted cocycles, θ near 0). For θ ∈ C sufficiently close to 0, we have that the twisted cocycle Lθ is quasi-compact. Furthermore, for such θ, the top Oseledets space of Lθ is one-dimensional. That is, dim Y θ (ω) = 1 for P-a.e. ω ∈ Ω. The following Lasota-Yorke type estimate will be useful in the proof.

25

Lemma 3.13. Assume conditions (C1) and (C2) hold. Then, we have kLωθ,(N ) f kB ≤ α ˜ θ,N (ω)kf kB + β N (ω)kf k1, where α ˜

θ,N

N

|θ|M

(ω) = α (ω) + C|θ|e

N −1 X

K N −1−j K(θ)j ,

j=0

for some constant C > 0 where K(θ) is given by Lemma 3.2 and K is given by ((C1)). Proof. It follows from (C2) that ) ) kLωθ,(N ) f kB ≤ kLω(N ) f kB + kLθ,(N − L(N ω ω kB · kf kB

) ) ≤ αN (ω)kf kB + β N (ω)kf k1 + kLθ,(N − L(N ω ω kB · kf kB .

On the other hand, we have that ) Lθ,(N ω

−

Lω(N )

=

N −1 X j=0

θ,(j)

LσN−j ω (LθσN−1−j ω − LσN−1−j ω )Lω(N −1−j) .

It follows from (C1) and (27) that θ,(j)

and kLσN−j ω kB ≤ K(θ)j .

kLω(N −1−j) kB ≤ K N −1−j

Furthermore, using (V3) and Lemma B.2, we have that for any h ∈ B, k(Lθω − Lω )(h)kB = kLω (eθg(ω,·) h − h)kB ≤ Kk(eθg(ω,·) − 1)hkB

= K var((eθg(ω,·) − 1)h) + Kk(eθg(ω,·) − 1)hk1

≤ Kkeθg(ω,·) − 1kL∞ · var(h) + K var(eθg(ω,·) − 1) · khkL∞ + Kkeθg(ω,·) − 1kL∞ ·khk1

≤ K|θ|e|θ|M MkhkB + KCvar |θ|e|θ|M ess supω∈Ω (var g(ω, ·))khkB , where Lemma 2.1, (C2), and (V3) are used to obtain the final inequality. Therefore, kLωθ,(N )

−

Lω(N ) kB

|θ|M

≤ C|θ|e

N −1 X

K(θ)j K N −1−j ,

j=0

where C = KM + KCvar ess supω∈Ω (var g(ω, ·)) and the conclusion of the lemma follows by combining the above estimates. Theorem 3.12 may now be established as follows. 26

Proof of Theorem 3.12. It follows from Lemma 3.13 and the dominated convergence theorem that Z Z θ,N log α ˜ (ω) dP(ω) → log αN (ω) dP(ω) < 0 when θ → 0. Ω

Ω

Thus, there exists δ > 0 such that Z Z 1 θ,N log α ˜ (ω) dP(ω) ≤ log αN (ω) dP(ω), 2 Ω Ω

for θ ∈ BC (0, δ).

ˆ in a neighborhood Lemma 3.8 implies that Λ is bounded below by a continuous function Λ ˆ of 0, and Λ(0) = Λ(0) = 0. Hence, by decreasing δ if necessary, we can assume that Z 1 NΛ(θ) > log αN (ω) dP(ω) for θ ∈ BC (0, δ), 2 Ω noting that NΛ(θ) is the top Lyapunov exponent of the cocycle over σ N with generator θ,(N ) ω 7→ Lω . Indeed, the inequality NΛ(θ) ≥ Λ(Rθ(N ) ) is straightforward from subadditivity, and the reverse inequality essentially follows from existence of the limits and basic facts from ergodic theory (see e.g. [18, Appendix C]). By Lemma 2.1, we conclude that this cocycle is quasi-compact, which immediately implies the first statement of the Theorem. Now we show dim Y θ := dim Y1θ = 1. Let λθ1 = µθ1 ≥ µθ2 ≥ · · · ≥ µθLθ > κ(θ) be the exceptional Lyapunov exponents of twisted cocycle Lθω , enumerated with multiplicity. That is, mθj = dim Yjθ (ω) denotes the multiplicity of the Lyapunov exponent λθj . As in Theorem 2.3, let Mjθ := mθ1 + · · · + mθj . Therefore, Λ(θ) = λθ1 = µθi for every 1 ≤ i ≤ M1θ θ and λθj = µθi for every Mj−1 + 1 ≤ i ≤ Mjθ and for every finite 1 < j ≤ lθ . By Lemma 3.2(2) the map θ 7→ Lθω is continuous in the norm topology of B for every ω ∈ Ω and also that the functions ω 7→ log+ kLθω k are dominated by an integrable function whenever θ is restricted to a compact set. Thus, Lemma A.3 of Appendix A shows that θ 7→ µθ1 + µθ2 is upper-semicontinuous. Hence, 0 > µ01 + µ02 ≥ lim sup(µθ1 + µθ2 ), θ→0

where the first inequality follows from the one-dimensionality of the top Oseledets subspace ˆ of the cocycle Lω . We note that Lemmas 3.8 and 3.9, ensure that lim supθ→0 µθ1 ≥ Λ(0) = 0. θ θ Therefore lim supθ→0 µ2 < 0 and dim Y1 = 1, as claimed. ˆ Corollary 3.14. For θ ∈ C near 0, we have that Λ(θ) = Λ(θ). In particular, Λ(θ) is ′ differentiable near 0 and Λ (0) = 0. ˆ ˆ is differentiable near 0, by Lemma 3.9. In addition, Proof. We recall that Λ(0) = 0 and Λ θ vω (·), defined in (41), gives a one-dimensional measurable equivariant subspace of B which ˆ grows at rate Λ(θ) (see (42)). Theorem 3.12 shows that lim supθ→0 µθ2 < 0. In particular, ˆ µθ2 < Λ(θ) for θ sufficiently close to 0. Combining this information with the multiplicative ˆ ergodic theorem (Theorem 2.3) and Lemma 3.8, we get that Λ(θ) = Λ(θ) and Y1θ (ω) = hvωθ i, for all θ ∈ C near 0. Thus, lemma 3.11 implies that Λ′ (0) = 0. 27

3.6

Convexity of Λ(θ)

We continue to denote by µ the invariant measure for the skew product transformation τ defined in (16). Furthermore, let Sn g be given by (1). By expanding the term [Sn g(ω, x)]2 it is straightforward to verify using standard computations and (19) that 1 lim n→∞ n

Z

2

[Sn g(ω, x)] dµ(ω, x) = Ω×X

Z

2

g(ω, x) dµ(ω, x)+2

Ω×X

∞ Z X n=1

g(ω, x)g(τ n(ω, x)) dµ(ω, x)

Ω×X

and that the right-hand side of the above equality is finite. Set Z ∞ Z X 2 2 Σ := g(ω, x) dµ(ω, x) + 2 g(ω, x)g(τ n(ω, x)) dµ(ω, x). Ω×X

n=1

(47)

Ω×X

Obviously, Σ2 ≥ 0 and from now on we shall assume that Σ2 > 0. This is equivalent to a non-coboundary condition on g; we refer the interested reader to [13] for a precise statement characterising the degenerate case Σ2 = 0. Lemma 3.15. We have that Λ is of class C 2 on a neighborhood of 0 and Λ′′ (0) = Σ2 . Proof. Using the notation in subsection 3.4, it follows from Lemma 3.9 and Corollary 3.14 that Z λθ (R g(ω, ·)eθg(ω,·)(O(θ) (·) + v 0 (·)) + eθg(ω,·) O ′ (θ) (·) dm) ω ω ω ω ′ Λ (θ) = ℜ dP(ω) . |λθω |2 Proceeding as in the proof of Lemma 3.9, one can show that Λ is of class C 2 on a neighborhood of 0 and that Z λθ ′′ (λθ ′ )2 ω ω ′′ − dP(ω) , (48) Λ (θ) = ℜ λθω (λθω )2

where we have used ′ to denote derivative with respect to θ. We recall that λ0ω = 1, λθω ′ is given by (45), and in particular λθω ′ |θ=0 = 0 for P-a.e. ω ∈ Ω. It is then straightforward, using (48), the chain rule and the formulas in Appendices B.1 and B.2, to verify that Z Z ′′ Λ (0) = ℜ g(ω, x)2vω0 (x) + 2g(ω, x)O ′(0)ω (x) + O ′′(0)ω (x) dm(x) dP(ω) .

Moreover, since θ 7→ O ′ (θ) is a map on a neighborhood of 0 with values in S we can regard O ′′ (0) as an element of S which implies that Z O ′′ (0)ω (x) dm(x) = 0 for a.e. ω and thus ′′

Λ (0) = ℜ

Z Z

(g(ω, x)2vω0 (x) + 2g(ω, x)O ′(0)ω (x)) dm(x) dP(ω) . 28

(49)

On the other hand, by the implicit function theorem, O ′(0)ω = −(D2 F (0, 0)−1 (D1 F (0, 0)))ω . Furthermore, (40) implies that −1

(D2 F (0, 0) W)ω = −

∞ X j=0

(j)

Lσ−j ω (Wσ−j ω ),

for each W ∈ S. This together with Proposition B.7 gives that ′

O (0)ω =

∞ X j=1

(j)

Lσ−j ω (g(σ −j ω, ·)vσ0−j ω (·)).

(50)

Using (49), (50), the duality property of transfer operators, as well as the fact that σ preserves P, we have that Z Z ∞ Z X (j) ′′ 2 0 −j 0 g(ω, x)Lσ−j ω (g(σ ω, ·)vσ−j ω ) dm(x) dP(ω) Λ (0) = g(ω, x) vω dm(x) + 2 =

Z Z

=

Z

g(ω, x)2 dµ(ω, x) + 2

=

Z

g(ω, x)2 dµ(ω, x) + 2

2

g(ω, x) dµω (x) + 2

j=1 ∞ XZ

j=1 ∞ XZ Z j=1 ∞ Z X

(j) g(ω, Tσ−j ω x)g(σ −j ω, x) dµσ−j ω (x)

dP(ω)

g(σ j ω, Tω(j)x)g(ω, x) dµω (x) dP(ω)

g(ω, x)g(τ j (ω, x)) dµ(ω, x) = Σ2 .

j=1

The following result is a direct consequence of the previous lemma. Corollary 3.16. Λ is strictly convex on a neighborhood of 0.

3.7

Choice of bases for top Oseledets spaces Yωθ and Yω∗θ

We recall that Yωθ and Yω∗θ are top Oseledets subspaces for twisted and adjoint twisted cocycle, Lθ and Lθ∗ , respectively. The Oseledets decomposition for these cocycles can be written in the form B = Yωθ ⊕ Hωθ and B∗ = Yω∗ θ ⊕ Hω∗ θ , (51) L θ Yjθ (ω) is the equivariant complement to Yωθ := Y1θ (ω), and Hω∗ θ is where Hωθ = V θ (ω) ⊕ lj=2 defined similarly. Furthermore, Lemma 2.6 shows that the following duality relations hold: ψ(y) = 0 whenever y ∈ Yωθ and ψ ∈ Hω∗ θ , φ(f ) = 0 whenever φ ∈

Yω∗ θ

29

and f ∈

Hωθ .

and

(52)

Let us fix convenient choices for elements of the one-dimensional top Oseledets spaces Yωθ R and Yω∗ θ , for θ ∈ C close to 0. Let vωθ ∈ Yωθ be as in (41), so that vωθ (·)dm = 1. (In view of Proposition 3.6, when θ ∈ R close to 0, the operators Lθω are positive, so we can additionally assume vωθ ≥ 0 and so kvωθ k1 = 1). Since dim Yωθ = 1, vωθ is defined uniquely for P-a.e. ω ∈ Ω. Theorem 2.3 ensures that, for P-a.e. ω ∈ Ω, there exists λθω ∈ C (λθω > 0 if θ ∈ R) such that θ Lθω vωθ = λθω vσω .

Integrating (53), and using (43), we obtain Z θ λω = eθg(ω,x) vωθ (x) dm(x),

(53)

(54)

and thus λθω coincides with the quantity introduced in (43). By (42) and Corollary 3.14, Z Λ(θ) = log |λθω | dP(ω). (55) Next, let us fix φθω ∈ Yω∗ θ so that φθω (vωθ ) = 1. This selection is again possible and unique, because of (52). Furthermore, this choice implies that (Lθω )∗ φθσω = λθω φθω ,

(56)

because Yω∗θ is one-dimensional and equivariant. Indeed, if Cωθ is the constant such that (Lθω )∗ φθσω = Cωθ φθω , then θ ) = φθσω (Lθω vωθ ) = ((Lθω )∗ φθσω )(vωθ ) = Cωθ φθω (vωθ ) = Cωθ . λθω = λθω φθσω (vσω

4

Limit theorems

In this section we establish the main results of our paper. To obtain the large deviation principle (Theorem A), we first link the asymptotic behaviour of moment generating (and characteristic) functions associated to Birkhoff sums with the Lyapunov exponents Λ(θ). Then, we combine the strict convexity of the map θ 7→ Λ(θ) on a neighborhood of 0 ∈ R with the classical Gärtner-Ellis theorem. We establish the central limit theorem (Theorem B) by applying Levy’s continuity theorem and using the C 2 -regularity of the map θ 7→ Λ(θ) on a neighborhood of 0 ∈ C. Finally, we demonstrate the full power of our approach by proving for the first time random versions of the local central limit theorem, both under the so-called aperiodic and periodic assumptions (Theorems C and 4.15). In addition, we present several equivalent formulations of the aperiodicity condition.

30

4.1

Large deviations property

In this section we establish Theorem A. The main tool in establishing this large deviations property will be the following classical result. Theorem 4.1. (Gärtner-Ellis [26]) For n ∈ N, let Pn be a probability measure on a measurable space (Y, T ) and let En denote the corresponding expectation operator. Furthermore, let Sn be a real random variable on (Ω, T ) and assume that on some interval [−θ+ , θ+ ], θ+ > 0, we have 1 (57) lim log En (eθSn ) = ψ(θ), n→∞ n where ψ is a strictly convex continuously differentiable function satisfying ψ ′ (0) = 0. Then, there exists ǫ+ > 0 such that the function c defined by c(ǫ) = sup {θǫ − ψ(θ)}

(58)

|θ|≤θ+

is nonnegative, continuous, strictly convex on [−ǫ+ , ǫ+ ], vanishing only at 0 and such that 1 log Pn (Sn > nǫ) = −c(ǫ), n→∞ n lim

for every ǫ ∈ (0, ǫ+ ).

We will also need the following results, linking the asymptotic behaviour of characteristic functions associated to Birkhoff sums with the numbers Λ(θ). Lemma 4.2. Let θ ∈ C be sufficiently close to 0, so that the results of Section 3.7 apply. Let f ∈ B be such that f ∈ / Hωθ . That is, φθω (f ) 6= 0. Then, Z 1 lim log eθSn g(ω,x) f dm = Λ(θ). n→∞ n

Proof. Given f ∈ B, we may write (see (51)) f = φθω (f )vωθ + hθω , where hθω ∈ Hωθ . Using this decomposition and applying repeatedly (53), we get ! n−1 Y Lθ,(n) f= λθσi ω φθω (f )vσθ n−1 ω + Lωθ,(n) hθω . (59) ω i=0

Theorem 2.3 ensures that

1 log kLθ,(n) |Hωθ k < Λ(θ). (60) ω n→∞ n Thus, the second term in (59) grows asymptotically with n at an exponential rate strictly slower than Λ(θ). By (34) and (59), we have that for P-a.e. ω ∈ Ω Z Z 1 1 θSn g(ω,x) lim log e f dm = lim log Lθ,(n) f dm ω n→∞ n n→∞ n " # n−1 θ,(n) θ Z L h 1 1X ω log |λθσi ω | + lim log φθω (f )vσθ n−1 ω + Qn−1 θω dm , = lim n→∞ n n→∞ n i=0 |λσi ω | i=0 lim

31

whenever the RHS limits exist. The first limit in the previous line equals Λ(θ) by (55). The second limit is zero, because the choice of vσθ n−1 ω ensures the integral of the first term in the square brackets is φθω (f ) 6= 0 (by assumption), which is independent of n, and the second term in the square brackets goes to zero as n → ∞ by (60). The conclusion follows. Lemma 4.3. For all complex θ in a neighborhood of 0, and P-a.e. ω ∈ Ω, we have that Z 1 lim log eθSn g(ω,x) dµω (x) = Λ(θ). n→∞ n Proof. Since

Z Z 1 1 θSn g(ω,x) lim log e dµω (x) = lim log eθSn (ω,x) vω0 (x) dm(x) , n→∞ n n→∞ n

θ 0 0 0 by R 0Lemma 4.2 it is sufficient to show that φω (vω ) 6=θ0 for θ near 0. We know that φω (vω ) = vω dm = 1. Hence, the differentiability of θ 7→ φ at θ = 0, established in Appendix C, together with the uniform bound on kvω0 kB provided by (17), ensure that for θ ∈ C sufficiently close to 0 and P-a.e. ω ∈ Ω, φθω (vω0 ) 6= 0 as required.

Proof of Theorem A. The proof follows directly from Theorem 4.1 when applied to the case when (Y, T ) = (X, B), Pn = µω Sn = Sn g(ω, ·) and ψ(θ) = Λ(θ).

Indeed, we note that (57) holds by Lemma 4.3 (the absolute values are irrelevant when θ ∈ R). Furthermore, it follows from Corollary 3.14 that Λ is continuously differentiable on a neighborhood of 0 in R satisfying Λ′ (0) = 0 and by Corollary 3.16, we have that Λ is strictly convex on a neighborhood of 0 in R. Finally, c does not depend on ω by (58).

4.2

Central limit theorem

The goal of section is to establish Theorem B. We start with the following lemma, which will be useful in the proofs of the both central limit theorem and local central limit theorem. Lemma 4.4. There exist C > 0, 0 < r < 1 such that for every θ ∈ C sufficiently close to 0, every n ∈ N and P-a.e. ω ∈ Ω, we have Z 1 θ,(n) 0 θ 0 θ L (v − φ (v )v ) dm (61) ≤ Cr n . ω ω ω ω ω 0

Proof. The following argument generalises [26, Lemma III.9] to the random setting. For each θ near 0 and ω ∈ Ω, let Qθω f := Lθω (f − φθω (f )vωθ ). Note that, in view of Lemma 3.2 and differentiability of θ 7→ v θ and θ 7→ φθ (established in Lemma 3.5 (see (41)) and Appendix C, respectively), we get that there exists N > 1 such that kQθω k < N for every ω ∈ Ω, provided θ is sufficiently close to 0. 32

In addition, since f − φθω (f )vωθ is the projection of f onto Hωθ along the top Oseledets space Yωθ , we get that, for every n ≥ 1, Qωθ,(n) f = Lθ,(n) (f − φθω (f )vωθ ). ω R Furthermore, since f − φ0ω (f )vω0 = f − ( f dm)vω0 , condition (C3) and Lemma 2.11(1) ensure 0,(n) that there exist K ′ , λ > 0 such that for every n ≥ 0 and P-a.e. ω ∈ Ω, kQω k ≤ K ′ e−λn . Let 1 > r > e−λ , and let n0 ∈ N be such that K ′ e−λn0 < r n0 . Lemma 3.2 together with differentiability of θ 7→ v θ and θ 7→ φθ ensure that θ 7→ Qθω is continuous in the norm topology of B. In fact, the uniform control over ω ∈ Ω, guaranteed by the aforementioned differentiability conditions, along with Condition (C1), ensure that one can choose ǫ > 0 so θ,(n ) that if |θ| < ǫ, then kQω 0 k < r n0 for every ω ∈ Ω. Writing n = kn0 + ℓ, with 0 ≤ ℓ < n0 , we get kQθ,(n) k ω with c =

N n0 . r Z

k−1 Y

≤

j=0

Thus,

1

0

θ,(ℓ)

θ,(n )

kQσjn00ω k(kQσkn0 ω k) < r n (N/r)ℓ ≤ cr n ,

Lθ,(n) (vω0 ω

−

φθω (vω0 )vωθ ) dm

≤ kLθ,(n) (vω0 − φθω (vω0 )vωθ )k1 ω

≤ kLθ,(n) (vω0 − φθω (vω0 )vωθ )kB = kQωθ,(n) (vω0 )kB ≤ cr n kvω0 kB . ω

˜ > 0 such that kv 0 kB ≤ K ˜ for P-a.e. ω ∈ Ω, so the proof of the By (17), there exists K ω lemma is complete. Proof of Theorem B. We recall that Σ2 > 0 is given by (47). It follows from Levy’s continuity theorem that it is sufficient to prove that, for every t ∈ R, Z S g(ω,·) t2 Σ2 it n√n e lim dµω = e− 2 , for P-a.e. ω ∈ Ω. n→∞

it √

it √

Assume n is sufficiently large so that dim Y1 n = 1 and vω n can be chosen as in (41). In parit √ R 1 √it Qn−1 √itn √itn ,(n) √it ticular, 0 vω n dm = 1 and Lω n vω n = ( j=0 λσj ω )vσn ω , for P-a.e. ω ∈ Ω. Furthermore, using (34), Z Z Z it g(ω,·) g(ω,·) √ ,(n) it Sn√ it Sn√ 0 n n e dµω = e vω dm = Lω n vω0 dm Z 1 it √it it it it √ ,(n) √ √ √ = Lω n φω n (vω0 )vω n + (vω0 − φω n (vω0 )vω n ) dm 0

it √ n

= φω

(vω0 )

·

n−1 Y

λ

it √ n σj ω

+

j=0

33

Z

0

1

it √

Lω n

,(n)

it √

it √

(vω0 − φω n (vω0 )vω n ) dm.

Lemma 4.4 shows that the second term converges to 0 as n → ∞. Also, differentiability of it √

θ 7→ φθ , established in Appendix C, ensures that limn→∞ φω n (vω0 ) = φ0ω (vω0 ) = 1. Thus, to conclude the proof of the theorem, we need to prove that lim

n→∞

n−1 Y

it √

λσjnω = e−

t2 Σ2 2

,

j=0

for P-a.e. ω ∈ Ω,

(62)

which is equivalent to lim

n→∞

n−1 X j=0

it √

log λσjnω = −

t2 Σ2 , 2


Using the notation of Lemmas 3.4 and 3.5, we have that λθω = H(θ, O(θ))(σω) and thus we need to prove that n−1 X it it t2 Σ2 log H √ , O( √ ) (σ j+1 ω) = − lim for P-a.e. ω ∈ Ω. (63) n→∞ 2 n n j=0 ˜ be a map defined in a neighborhood of 0 in C with values in L∞ (Ω) by H(θ) ˜ Let H = 2 ˜ ′ ˜ ˜ log H(θ, O(θ)). It will be shown in Lemma 4.5 that H is of class C , H(0)(ω) = 0, H (0)(ω) = 0 and Z ′′ ˜ H (0)(ω) = (g(σ −1 ω, ·)2vσ0−1 ω + 2g(σ −1ω, ·)O ′(0)σ−1 ω ) dm. ˜ in a Taylor series around 0, we have that Developing H

˜ ′′ (0)(ω) H ˜ θ2 + R(θ)(ω), H(θ)(ω) = log H(θ, O(θ))(ω) = 2 where R denotes the remainder. Therefore, ˜ ′′ (0)(σ j+1ω) √ it it t2 H log H √ , O( √ ) (σ j+1ω) = − + R(it/ n)(σ j+1 ω), n n 2n which implies that n−1 n−1 n−1 X X √ it t2 1 X ˜ ′′ it j+1 j+1 H (0)(σ ω) + R(it/ n)(σ j+1ω). (64) log H √ , O( √ ) (σ ω) = − · n n 2 n j=0 j=0 j=0 The asymptotic behaviour of the first term is governed by Birkhoff’s ergodic theorem, so using (49) second equality, and Lemma 3.15 fot the third one, we get: n−1

t2 t2 1 X ˜ ′′ H (0)(σ j+1 ω) = − lim − n→∞ 2 n j=0 2

Z

2 2 ˜ ′′ (0)(ω) dP(ω) = − t Λ′′ (0) = − t Σ2 H 2 2

for P-a.e. ω ∈ Ω. (65)

34

˜ ˜ Now we deal with the last term of (64). Writing R(θ) = θ2 R(θ) with limθ→0 R(θ) = 0, we ˜ conclude that for each ǫ > 0 and t ∈ R \ {0}, there exists δ √ > 0 such that kR(θ)kL∞ ≤ tǫ2 for all |θ| ≤ δ. We note that there exists n0 ∈ N such that |it/ n| ≤ δ for each n ≥ n0 . Hence, n−1 n−1 t2 X X √ √ t2 nǫ j+1 j+1 ˜ ≤ | R(it/ · 2 = ǫ, n)(σ ω) n)(σ ω)| ≤ R(it/ n n t j=0 j=0

for every n ≥ n0 , which implies that the second term on the right-hand side of (64) converges to 0 and thus (63) holds. The proof of the theorem is complete. ˜ ˜ Lemma 4.5. The map H(θ) = log H(θ, O(θ)) is of class C 2 . Moreover, H(0)(ω) = 0, ′ ˜ H (0)(ω) = 0 and Z ′′ ˜ H (0)(ω) = (g(σ −1 ω, ·)2vσ0−1 ω + 2g(σ −1ω, ·)O ′(0)σ−1 ω ) dm.

Proof. The regularity of T follows directly from the results in Appendices B.1 and B.2. ˜ Moreover, we have H(0)(ω) = log H(0, O(0))(ω) = log 1 = 0. Furthermore, 1 ˜ ′ (θ)(ω) = H [D1 H(θ, O(θ))(ω) + (D2 H(θ, O(θ))O ′(θ))(ω)]. H(θ, O(θ))(ω) Taking into account formulas in Appendix B.1, (25) and (46), we have Z Z ′ −1 0 ˜ H (0)(ω) = g(σ ω, ·)vσ−1 ω dm + O ′ (0)σ−1 ω dm = 0. Finally, taking into account that D22 H = 0 (see Appendix B.2) we have

˜ ′′ (θ)(ω) = −D1 H(θ, O(θ))(ω) [D1 H(θ, O(θ))(ω) + (D2 H(θ, O(θ))O ′(θ))(ω)] H [H(θ, O(θ))(ω)]2 1 + [D11 H(θ, O(θ))(ω) + (D21 H(θ, O(θ))O ′(θ))(ω)] H(θ, O(θ))(ω) 1 [(D12 H(θ, O(θ))O ′(θ))(ω) + (D2 H(θ, O(θ))O ′′(θ))(ω)]. + H(θ, O(θ))(ω) ˜ ′′ (0). Using formulas in Appendices B.1 and B.2, we obtain the desired expression for H

4.3

Local central limit theorem

In order to obtain a local central limit theorem, we introduce an additional assumption related to aperiodicity, as follows. (C5) For P-a.e. ω ∈ Ω and for every compact interval J ⊂ R \ {0} there exist C = C(ω) > 0 and ρ ∈ (0, 1) such that kLωit,(n) kB ≤ Cρn ,

for t ∈ J and n ≥ 0.

(66)

The proof of Theorem C is presented in Section 4.3.1. In Section 4.3.2, we show that (C5) can be phrased as a so-called aperiodicity condition, resembling a usual requirement for autonomous versions of the local CLT. Examples are presented in Section 4.3.3. 35

4.3.1

Proof of Theorem C

Using the density argument (see [34]), it is sufficient to show that Z Z √ 1 − s2 2 sup Σ n h(s + Sn g(ω, ·)) dµω − √ e 2nΣ h(u) du → 0, 2π s∈R R

(67)

ˆ has compact support. Moreover, when n → ∞ for every h ∈ L1 (R) whose Fourier transform h we recall the following inversion formula Z 1 itx ˆ h(t)e dt. (68) h(x) = 2π R By (34), (68) and Fubini’s theorem, √ Z Z Z 1 √ Σ n 1 ˆ Σ n h(t)eit(s+Sn g(ω,·)) dt dµω h(s + Sn g(ω, ·)) dµω = 2π 0 √ Z0 R Z 1 Σ n its ˆ e h(t) eitSn g(ω,·) dµω dt = 2π R 0 √ Z Z 1 Σ n ˆ = eits h(t) eitSn g(ω,·) vω0 dm dt 2π R 0 √ Z Z 1 Σ n ˆ eits h(t) Lωit,(n) vω0 dm dt = 2π R 0 Z 1 it Z √ ,(n) its Σ t √ ˆ n = Lω n vω0 dm dt. e h( √ ) 2π R n 0 Considering the √RHS of (67) and recalling that the Fourier transform of f (x) = e− 2 2 given by fˆ(t) = Σ2π e−t /2Σ we have s2 1 √ e− 2nΣ2 2π

Z

Σ2 x2 2

is

ˆ s2 h(0) h(u) du = √ e− 2nΣ2 2π R ˆh(0)Σ √ fˆ(−s/ n) = 2π ˆh(0)Σ Z √its Σ2 t2 e n · e− 2 dt. = 2π R

Hence, we need to prove that Z Z Z 1 it ˆ Σ √ ,(n) its its Σ2 t2 h(0)Σ t √ √ n 0 − ˆ √ ) vω dm dt − sup e n h( e n · e 2 dt → 0, Lω 2π n 0 s∈R 2π R R

(69)

ˆ is contained in when n → ∞, for P-a.e. ω ∈ Ω. Choose δ > 0 such that the support of h Qn−1 θ,(n) θ θ θ [−δ, δ]. Recall that Lω vω = ( j=0 λσj ω )vσn ω for P-a.e. ω ∈ Ω, and for all θ near 0. Then, 36

for any δ˜ ∈ (0, δ), we have, Z Z Z 1 it ˆ √ ,(n) its its Σ2 t2 h(0)Σ Σ t √ √ n 0 ˆ n e h( √ ) e n · e− 2 dt vω dm dt − Lω 2π R n 0 2π R Z n−1 it Y 2 2 √ its Σ √ n − Σ 2t ˆ √t ) ˆ n h( dt e = λ − h(0)e j 2π |t| 1−ǫ have that Fj1 (Litω ) ≤ 1 for every j. Also, if ω ∈ Aǫ , then Fd1 (Litω |Yωit ) < 1 − ǫ and therefore Πdj=1 Fj1 (Litω |Yωit ) < 1 − ǫ. Thus, for P-a.e. ω ∈ Ω, lim

1 1 0 = lim log Πdj=1 Fj1 (Lωit,(n) |Yωit ) = lim log Vd1 (Lωit,(n) |Yωit ) n→∞ n Z Z n→∞ n (87) ≤ log Vd1 (Litω |Yωit )dP(ω) ≤ C log Πdj=1 Fj1 (Litω |Yωit )dP(ω) ≤ CP(Aǫ ) log(1 − ǫ) ≤ 0, where C > 0 is such that for every A : L1 → L1 , Vd1 (A) ≤ CΠdj=1 Fj1 (A), as guaranteed by Lemma A.2. Thus, all inequalities in (87) must be equalities and therefore P(Aǫ ) = 0 for every ǫ > 0, which means that k(Litω |Yωit )−1 k1 = 1 for P-a.e. ω ∈ Ω. Thus, inf v∈Yωit ∩S1 kLitω vk1 = 1, as claimed.

B

Regularity of F

In this section, we establish regularity properties of the map F defined in (38).

B.1

First order regularity of F

Let S ′ be the Banach space of all functions V : Ω × X → C such that Vω := V(ω, ·) ∈ B and ess supω∈Ω kVω kB < ∞. Note that S, defined in (36), consists of those V ∈ S ′ such that 52

Vω dm = 0 for P-a.e. ω ∈ Ω. We define G : BC (0, 1)×S → S ′ and H : BC (0, 1)×S → L∞ (Ω) by Z θ 0 G(θ, W)ω = Lσ−1 ω (Wσ−1 ω + vσ−1 ω ) and H(θ, W)(ω) = Lθσ−1 ω (Wσ−1 ω + vσ0−1 ω ) dm, R

where vω0 is defined in (15). It follows easily from Lemmas 2.11 and 3.2 (together with (29) which implies sup|θ| 0 such that var(eθ1 g(σ

−1 ω,·)

− eθ2 g(σ

−1 ω,·)

) ≤ Ce|θ1 −θ2 |M |θ1 − θ2 |(e|θ2 |M + |θ2 |e|θ2 |M ),

for ω ∈ Ω.

(88)

Proof. We note that it follows from (V8) that var(eθ1 g(σ

−1 ω,·)

− eθ2 g(σ

−1 ω,·)

) = var(eθ2 g(σ ≤ keθ2 g(σ

−1 ω,·)

−1 ω,·)

+ var(eθ2 g(σ

(e(θ1 −θ2 )g(σ

−1 ω,·)

− 1))

kL∞ · var(e(θ1 −θ2 )g(σ

−1 ω,·)

) · ke(θ1 −θ2 )g(σ

−1 ω,·)

−1 ω,·)

− 1)

− 1kL∞ .

−1

Moreover, observe that it follows from (24) that keθ2 g(σ ω,·) kL∞ ≤ e|θ2 |M . On the other hand, by applying (V9) for f = g(σ −1 ω, ·) and h(z) = e(θ1 −θ2 )z − 1, we obtain var(e(θ1 −θ2 )g(σ

−1 ω,·)

− 1) ≤ |θ1 − θ2 |e|θ1 −θ2 |M var(g(σ −1ω, ·)). −1

Finally, we want to estimate ke(θ1 −θ2 )g(σ ω,·) − 1kL∞ . By applying the mean value theorem for the map z 7→ e(θ1 −θ2 )z , we have that for each x ∈ [0, 1], |e(θ1 −θ2 )g(σ

−1 ω,x)

− 1| ≤ e|θ1 −θ2 |M |θ1 − θ2 | · |g(σ −1ω, x)| ≤ Me|θ1 −θ2 |M |θ1 − θ2 |,

and consequently ke(θ1 −θ2 )g(σ

−1 ω,·)

− 1kL∞ ≤ Me|θ1 −θ2 |M |θ1 − θ2 |.

The conclusion of the lemma follows directly from the above estimates together with (24) and Lemma B.1. Lemma B.3. D2 G exists and is continuous on BC (0, 1) × S.

53

Proof. Since G is an affine map in the second variable W, we conclude that (D2 G(θ, W)H)ω = Lθσ−1 ω Hσ−1 ω ,

for ω ∈ Ω and H ∈ S.

(89)

We now establish the continuity of D2 G. Take an arbitrary (θi , W i ) ∈ BC (0, 1)×S, i ∈ {1, 2}. We have kD2 G(θ1 , W 1 ) − D2 G(θ2 , W 2 )k = sup kD2 G(θ1 , W 1 )(H) − D2 G(θ2 , W 2 )(H)k∞ kHk∞ ≤1

= sup ess supω∈Ω kLθσ1−1 ω Hσ−1 ω − Lθσ2−1 ω Hσ−1 ω kB . kHk∞ ≤1

Observe that kLθσ1−1 ω Hσ−1 ω − Lθσ2−1 ω Hσ−1 ω kB = kLσ−1 ω ((eθ1 g(σ ≤ Kk(eθ1

−1 ω,·)

g(σ−1 ω,·)

= K var((eθ1 + Kk(eθ1

− eθ2 g(σ

− eθ2

g(σ−1 ω,·)

g(σ−1 ω,·)

−1 ω,·)

g(σ−1 ω,·)

− eθ2

− eθ2

)Hσ−1 ω )kB

)Hσ−1 ω kB

g(σ−1 ω,·)

g(σ−1 ω,·)

)Hσ−1 ω )

)Hσ−1 ω k1 .

Take an arbitrary x ∈ X. By applying the mean value theorem for the map z 7→ ezg(σ and using (24), we conclude that |eθ1 g(σ

−1 ω,x)

− eθ2 g(σ

−1 ω,x)

−1 ω,x)

| ≤ MeM |θ1 − θ2 |

(90)

and thus ess supω∈Ω k(eθ1 g(σ

−1 ω,·)

− eθ2 g(σ

−1 ω,·)

)Hσ−1 ω k1 ≤ MeM |θ1 − θ2 | ess supω∈Ω kHσ−1 ω k1

≤ MeM |θ1 − θ2 | ess supω∈Ω kHσ−1 ω kB

(91)

M

≤ Me kHk∞ · |θ1 − θ2 |. Furthermore, var((eθ1 g(σ

−1 ω,·)

− eθ2 g(σ

−1 ω,·)

)Hσ−1 ω ) ≤ var(eθ1 g(σ

−1 ω,·)

θ1 g(σ−1 ω,·)

+ ke

− eθ2 g(σ

−1 ω,·)

θ2 g(σ−1 ω,·)

−e

) · kHσ−1 ω kL∞

kL∞ · var(Hσ−1 ω ),

which, using (90), implies that var((eθ1 g(σ

−1 ω,·)

−eθ2 g(σ

−1 ω,·)

)Hσ−1 ω ) ≤ Cvar var(eθ1 g(σ

−1 ω,·)

−eθ2 g(σ

It follows from Lemma B.2 that

−1 ω,·)

)+MeM |θ1 −θ2 | kHk∞ . (92)

kD2 G(θ1 , W 1 ) − D2 G(θ2 , W 2 )k ≤ (KC + 2KMeM )|θ1 − θ2 |, which implies (Lipschitz) continuity of D2 G on BC (0, 1) × S. 54

Lemma B.4. D2 H exists and is continuous on a neighborhood of (0, 0) ∈ C × S. Proof. We first note that H is also an affine map in the variable W which implies that Z (D2 H(θ, W)H)(ω) = Lθσ−1 ω Hσ−1 ω dm, for ω ∈ Ω and H ∈ S. (93) Moreover, using (90) we have that 1

2

kD2 H(θ1 , W )H − D2 H(θ2 , W )HkL∞

Z Z θ θ = ess supω∈Ω Lσ1−1 ω Hσ−1 ω dm − Lσ2−1 ω Hσ−1 ω dm Z −1 −1 = ess supω∈Ω (eθ1 g(σ ω,·) − eθ2 g(σ ω,·) )Hσ−1 ω dm ≤ MeM |θ1 − θ2 | ess supω∈Ω kHσ−1 ω k1

≤ MeM |θ1 − θ2 | ess supω∈Ω kHσ−1 ω kB

= MeM |θ1 − θ2 |·kHk∞ ,

for every (θ1 , W 1 ), (θ2 , W 2 ) that belong to a sufficiently small neighborhood of (0, 0) on which H is defined. We conclude that D2 H is continuous. Lemma B.5. D1 H exists and is continuous on a neighborhood of (0, 0) ∈ C × S. Proof. We first note that H(θ, W)(ω) =

Z

eθg(σ

−1 ω,·)

(Wσ−1 ω + vσ0−1 ω ) dm.

We claim that for ω ∈ Ω and h ∈ BC (0, 1), Z −1 (D1 H(θ, W)h)(ω) = hg(σ −1 ω, ·)eθg(σ ω,·) (Wσ−1 ω + vσ0−1 ω ) dm =: (L(θ, W)h)ω .

(94)

Note that L(θ, W) : BC (0, 1) → L∞ (Ω) is a bounded linear operator. We first note that for each ω ∈ Ω, (H(θ + h, W) − H(θ, W) − L(θ, W)h)(ω) Z −1 −1 −1 = (e(θ+h)g(σ ω,·) − eθg(σ ω,·) − hg(σ −1 ω, ·)eθg(σ ω,·) )(Wσ−1 ω + vσ0−1 ω ) dm. For each ω ∈ Ω and x ∈ X, it follows from Taylor’s remainder theorem applied to the −1 function z 7→ ezg(σ ω,x) that for |θ|, |h| ≤ 21 , |e(θ+h)g(σ

−1 ω,x)

− eθg(σ

−1 ω,x)

− hg(σ −1 ω, x)eθg(σ

−1 ω,x)

1 | ≤ M 2 eM |h|2 . 2

Hence, 1 kH(θ + h, W) − H(θ, W) − L(θ, W)hkL∞ ≤ M 2 eM |h|2 (kWk∞ + kv 0 k∞ ), 2 55

(95)

and therefore 1 kH(θ + h, W) − H(θ, W) − L(θ, W)hkL∞ → 0, |h|

when h → 0.

We conclude that (94) holds. Furthermore, (D1 H(θ1 , W 1 )h)(ω) − (D1 H(θ2 , W 2 )h)(ω) Z −1 = hg(σ −1 ω, ·)eθ1g(σ ω,·) (Wσ1−1 ω + vσ0−1 ω ) dm Z −1 − hg(σ −1 ω, ·)eθ2 g(σ ω,·) (Wσ2−1 ω + vσ0−1 ω ) dm Z −1 = hg(σ −1 ω, ·)eθ1g(σ ω,·) (Wσ1−1 ω + vσ0−1 ω ) dm Z −1 − hg(σ −1 ω, ·)eθ1 g(σ ω,·) (Wσ2−1 ω + vσ0−1 ω ) dm Z −1 + hg(σ −1 ω, ·)eθ1g(σ ω,·) (Wσ2−1 ω + vσ0−1 ω ) dm Z −1 − hg(σ −1 ω, ·)eθ2 g(σ ω,·) (Wσ2−1 ω + vσ0−1 ω ) dm Z −1 = hg(σ −1 ω, ·)eθ1g(σ ω,·) (Wσ1−1 ω − Wσ2−1 ω ) dm Z −1 −1 + hg(σ −1 ω, ·)(eθ1g(σ ω,·) − eθ2 g(σ ω,·) )(Wσ2−1 ω + vσ0−1 ω ) dm. Note that Z −1 ω,·) −1 θ g(σ 1 2 (Wσ−1 ω − Wσ−1 ω ) dm ≤ |h|MeM kW 1 − W 2 k∞ ess supω∈Ω hg(σ ω, ·)e 1

and, using (90),

Z −1 ω,·) −1 ω,·) −1 θ g(σ θ g(σ 2 0 1 2 −e )(Wσ−1 ω + vσ−1 ω ) dm ess supω∈Ω hg(σ ω, ·)(e ≤ |h|M 2 eM |θ1 − θ2 |(R + kv 0 k∞ ),

if W 2 ∈ BS (0, R). Hence, kD1 H(θ1 , W 1 ) − D1 H(θ2 , W 2 )k ≤ MeM kW 1 − W 2 k∞ + M 2 eM |θ1 − θ2 |(R + kv 0 k∞ ), which implies the continuity of D1 H. Lemma B.6. D1 G exists and is continuous on a neighborhood of (0, 0) ∈ C × S.

56

Proof. We claim that for ω ∈ Ω and h ∈ BC (0, 1), (D1 G(θ, W)h)ω = Lσ−1 ω (hg(σ −1 ω, ·)eθg(σ

−1 ω,·)

(Wσ−1 ω + vσ0−1 ω )) =: (L(θ, W)h)ω .

(96)

Note that L(θ, W) : BC (0, 1) → S ′ is a bounded linear operator. We note that (G(θ + h, W) − G(θ, W) − L(θ, W)h)ω = Lσ−1 ω ((e(θ+h)g(σ

−1 ω,·)

− eθg(σ

−1 ω,·)

− hg(σ −1 ω, ·)eθg(σ

−1 ω,·)

)(Wσ−1 ω + vσ0−1 ω )),

and therefore k(G(θ + h, W) − G(θ, W) − L(θ, W)h)ω kB ≤ Kk(e(θ+h)g(σ

−1 ω,·)

= K var((e(θ+h)g(σ + Kk(e(θ+h)g(σ

− eθg(σ

−1 ω,·)

−1 ω,·)

−1 ω,·)

− eθg(σ

− eθg(σ


−1 ω,·)

−1 ω,·)

−1 ω,·)


− hg(σ −1ω, ·)eθg(σ

)(Wσ−1 ω + vσ0−1 ω )kB

−1 ω,·)

−1 ω,·)

)(Wσ−1 ω + vσ0−1 ω ))

)(Wσ−1 ω + vσ0−1 ω )k1 .

In the proof of Lemma B.5 we have showed that ke(θ+h)g(σ

−1 ω,·)

− eθg(σ

−1 ω,·)


−1 ω,·)

1 kL∞ ≤ M 2 eM |h|2 . 2

Moreover, by applying (V9) for f = g(σ −1 ω, ·) and h(z) = e(θ+h)z − eθz − hzeθz , one can conclude that var((e(θ+h)g(σ

−1 ω,·)

− eθg(σ

−1 ω,·)


−1 ω,·)

) ≤ C|h|2 .

(97)

The last two inequalities combined with (V8) readily imply that 1 kG(θ + h, W) − G(θ, W) − L(θ, W)hk∞ → 0, |h|

when h → 0,

which implies (96). Moreover, (D1 G(θ1 , W 1 )h − D1 G(θ2 , W 2 )h)ω = hLσ−1 ω (g(σ −1ω, ·)(eθ1g(σ

−1 ω,·)

− hLσ−1 ω (g(σ −1 ω, ·)eθ2g(σ

− eθ2 g(σ

−1 ω,·)

−1 ω,·)

)(Wσ1−1 ω + vσ0−1 ω ))

(Wσ2−1 ω − Wσ1−1 ω )).

Proceeding as in the previous lemmas and using (90) and Lemma B.2 together with a simple observation that var(g(σ −1ω, ·)(eθ1g(σ

−1 ω,·)

− eθ2 g(σ

−1 ω,·)

)) ≤ M var(eθ1 g(σ

−1 ω,·)

− eθ2 g(σ

+ var(g(σ −1ω, ·))keθ1

we easily obtain the continuity of D1 G. 57

−1 ω,·)

g(σ−1 ω,·)

)

− eθ2 g(σ

−1 ω,·)

k L∞ ,

The following result is a direct consequence of the previous lemmas. Proposition B.7. The map F defined by (38) is of class C 1 on a neighborhood (0, 0) ∈ C×S. Moreover, R θ Lσ−1 ω Hσ−1 ω dm 1 θ (D2 F (θ, W)H)ω = Lσ−1 ω Hσ−1 ω − G(θ, W)ω − Hω , H(θ, W)(ω) [H(θ, W)(ω)]2 for ω ∈ Ω and H ∈ S and 1 −1 Lσ−1 ω (g(σ −1ω, ·)eθg(σ ω,·) (Wσ−1 ω + vσ0−1 ω )) H(θ, W)(ω) R −1 g(σ −1ω, ·)eθg(σ ω,·) (Wσ−1 ω + vσ0−1 ω ) dm θ − Lσ−1 ω (Wσ−1 ω + vσ0−1 ω ), [H(θ, W)(ω)]2

(D1 F (θ, W))ω =

for ω ∈ Ω, where we have identified D1 F (θ, W) with its value at 1, and G is as defined at the beginning of Section B.1.

B.2

Second order regularity of F

Lemma B.8. D12 H and D22 H exist and are continuous on a neighborhood of (0, 0) ∈ C×S. Proof. We first note that it follows directly from (93) that D22 H = 0. We claim that Z −1 ((D12 H(θ, W)h)H)(ω) = h g(σ −1 ω, ·)eθg(σ ω,·) Hσ−1 ω dm, for ω ∈ Ω, H ∈ S and h ∈ C.

(98)

Indeed, we note that ((D2 H(θ + h, W) − D2 H(θ, W))H)(ω) =

Z

(e(θ+h)g(σ

−1 ω,·)

− eθg(σ

−1 ω,·)

)Hσ−1 ω dm.

Hence, using (95), Z −1 ω,·) −1 θg(σ ((D2 H(θ + h, W) − D2 H(θ, W))H)(ω) − h g(σ ω, ·)e Hσ−1 ω dm Z −1 ω,·) −1 ω,·) −1 ω,·) (θ+h)g(σ θg(σ −1 θg(σ −e − hg(σ ω, ·)e )Hσ−1 ω dm = (e 1 1 ≤ M 2 eM |h|2 kHσ−1 ω k1 ≤ M 2 eM |h|2 kHσ−1 ω kB . 2 2

Thus, Z −1 ess supω∈Ω ((D2 H(θ + h, W) − D2 H(θ, W))H)(ω) − h g(σ −1 ω, ·)eθg(σ ω,·) Hσ−1 ω dm 58

1 ≤ M 2 eM |h|2 kHk∞ , 2 which readily implies (98). We now establish the continuity of D12 H. By (90), we have that |((D12 H(θ1 , W 1 )h)H)(ω) − ((D12 H(θ2 , W 2 )h)H)(ω)| Z −1 −1 = |h| g(σ −1ω, ·)(eθ1 g(σ ω,·) − eθ2 g(σ ω,·) )Hσ−1 ω dm ≤ |h|M 2 eM |θ1 − θ2 | · kHσ−1 ω kB .

Thus, kD12 H(θ1 , W 1 ) − D12 H(θ2 , W 2 )k ≤ M 2 eM |θ1 − θ2 |,

which implies the continuity of D12 H.

Lemma B.9. D11 H and D21 H exist and are continuous on a neighborhood of (0, 0) ∈ C×S. Proof. By identifying D1 H(θ, W) with its value in 1, it follows from (94) that Z −1 (D1 H(θ, W))(ω) = g(σ −1ω, ·)eθg(σ ω,·) (Wσ−1 ω + vσ0−1 ω ) dm. We claim that (D11 H(θ, W)h)(ω) = h

Z

g(σ −1 ω, ·)2eθg(σ

−1 ω,·)

(Wσ−1 ω + vσ0−1 ω ) dm,

for ω ∈ Ω and h ∈ C.

(99)

Indeed, observe that (D1 H(θ + h, W))(ω) − (D1 H(θ, W))(ω) Z −1 −1 = g(σ −1ω, ·)(e(θ+h)g(σ ω,·) − eθg(σ ω,·) )(Wσ−1 ω + vσ0−1 ω ) dm. Hence, using (95), we obtain that Z −1 ω,·) −1 2 θg(σ 0 (Wσ−1 ω + vσ−1 ω ) dm ess supω∈Ω (D1 H(θ + h, W))(ω) − (D1 H(θ, W))(ω) − h g(σ ω, ·) e Z −1 (θ+h)g(σ−1 ω,·) θg(σ−1 ω,·) −1 θg(σ−1 ω,·) 0 −e − hg(σ ω, ·)e )(Wσ−1 ω + vσ−1 ω ) dm = ess supω∈Ω g(σ ω, ·)(e

1 ≤ M 3 eM |h|2 (kWk∞ + kv 0 k∞ ), 2

which readily implies that (99) holds. We now establish the continuity of D11 H. It follows from (90) that |(D11 H(θ1 , W 1 )h)(ω) − (D11 H(θ2 , W 2 )h)(ω)| 59

Z Z −1 −1 = |h| · g(σ −1ω, ·)2eθ1 g(σ ω,·) (Wσ1−1 ω + vσ0−1 ω ) dm − g(σ −1 ω, ·)2eθ2 g(σ ω,·) (Wσ2−1 ω + vσ0−1 ω ) dm Z Z −1 2 θ1 g(σ−1 ω,·) 1 0 −1 2 θ2 g(σ−1 ω,·) 1 0 ≤ |h| · g(σ ω, ·) e (Wσ−1 ω + vσ−1 ω ) dm − g(σ ω, ·) e (Wσ−1 ω + vσ−1 ω ) dm Z Z −1 2 θ2 g(σ−1 ω,·) 1 0 −1 2 θ2 g(σ−1 ω,·) 2 0 (Wσ−1 ω + vσ−1 ω ) dm − g(σ ω, ·) e (Wσ−1 ω + vσ−1 ω ) dm + |h| · g(σ ω, ·) e 3 M 1 0 2 M 1 2 ≤ |h| · M e |θ1 − θ2 |(kW k∞ + kv k∞ ) + M e kW − W k∞ , for P-a.e. ω ∈ Ω, which implies the continuity of D11 H. Furthermore, we note that D1 H is affine in W, which implies that Z −1 (D21 H(θ, W)H)(ω) = g(σ −1 ω, ·)eθg(σ ω,·) Hσ−1 ω dm. Continuity of D21 H follows easily from (90). Lemma B.10. D22 G and D12 G exist and are continuous on a neighborhood of (0, 0) ∈ C×S. Proof. It follows directly from (89) that D22 G = 0. We claim that (D12 G(θ, W)h(H))ω = hLσ−1 ω (g(σ −1 ω, ·)eθg(σ

−1 ω,·)

Hσ−1 ω ) for ω ∈ Ω, H ∈ S and h ∈ C. (100)

Indeed, we first note that (D2 G(θ + h, W) − D2 G(θ, W))(H)ω = Lσ−1 ω ((e(θ+h)g(σ

−1 ω,·)

− eθg(σ

−1 ω,·)

)Hσ−1 ω ).

We have that kLσ−1 ω ((e(θ+h)g(σ

−1 ω,·)

(θ+h)g(σ−1 ω,·)

≤ Kk(e

− eθg(σ

θg(σ−1 ω,·)

−e

(θ+h)g(σ−1 ω,·)

= K var((e

(θ+h)g(σ−1 ω,·)

+ Kk(e

(θ+h)g(σ−1 ω,·)

+ Kke

(θ+h)g(σ−1 ω,·)

+ Kke


−e

− hg(σ −1 ω, ·)e

θg(σ−1 ω,·)

θg(σ−1 ω,·)

−e

θg(σ−1 ω,·)

−e

)Hσ−1 ω )kB

)Hσ−1 ω kB

θg(σ−1 ω,·)

− hg(σ −1 ω, ·)e

θg(σ−1 ω,·)

− hg(σ −1ω, ·)e

θg(σ−1 ω,·)

−e

−1 ω,·)

θg(σ−1 ω,·)

θg(σ−1 ω,·)

−e

(θ+h)g(σ−1 ω,·)

≤ K var(e

−1 ω,·)

θg(σ−1 ω,·)

− hg(σ −1 ω, ·)e

θg(σ−1 ω,·)

− hg(σ −1 ω, ·)e

)Hσ−1 ω k1

θg(σ−1 ω,·)

− hg(σ −1 ω, ·)e

)Hσ−1 ω )

) · kHσ−1 ω kL∞

kL∞ · var(Hσ−1 ω ) kL∞ · kHσ−1 ω k1

It follows from (95) and (97) that 1 −1 −1 −1 sup kLσ−1 ω ((e(θ+h)g(σ ω,·) − eθg(σ ω,·) − hg(σ −1 ω, ·)eθg(σ ω,·) )Hσ−1 ω )kB → 0, |h| kHk∞ ≤1 when h → 0, which establishes (100). It remains to establish the continuity of D12 G. We have k(D12 G(θ1 , W 1 )h(H))ω − (D12 G(θ2 , W 2 )h(H))ω kB 60

= |h| · kLσ−1 ω (g(σ −1 ω, ·)(eθ1g(σ ≤ K|h| · kg(σ −1ω, ·)(eθ1g(σ

−1 ω,·)

= K|h| · var(g(σ −1ω, ·)(eθ1 g(σ + K|h| · kg(σ −1ω, ·)(eθ1 + KM|h| · keθ1

+ KM|h| · keθ1

g(σ−1 ω,·)

−1 ω,·)

− eθ2

g(σ−1 ω,·)

)Hσ−1 ω )kB

)Hσ−1 ω kB

−1 ω,·)

g(σ−1 ω,·)

− eθ2

g(σ−1 ω,·)

−1 ω,·)

−1 ω,·)

− eθ2 g(σ

− eθ2

g(σ−1 ω,·)

− eθ2

− eθ2 g(σ

− eθ2 g(σ

g(σ−1 ω,·)

≤ K|h| · var(g(σ −1 ω, ·)(eθ1 g(σ−1 ω,·)

−1 ω,·)

)Hσ−1 ω )

)Hσ−1 ω k1

g(σ−1 ω,·)

)) · kHσ−1 ω k1

kL∞ · var(Hσ−1 ω ) kL∞ · kHσ−1 ω k1 .

Moreover, var(g(σ −1ω, ·)(eθ1g(σ

−1 ω,·)

− eθ2 g(σ

−1 ω,·)

)) ≤ M var(eθ1 g(σ

−1 ω,·)

− eθ2 g(σ

−1 ω,·)

θ1 g(σ−1 ω,·)

−1

+ var(g(σ ω, ·)) · ke

) − eθ2 g(σ

−1 ω,·)

k L∞ ,

which together with (90) and Lemma B.2 gives the continuity of D12 G. Lemma B.11. D11 G and D21 G exist and are continuous on a neighborhood of (0, 0) ∈ C×S. Proof. By identifying D1 G(θ, W) with its value in 1, it follows from (96) that D1 G(θ, W)ω = Lσ−1 ω (g(σ −1ω, ·)eθg(σ

−1 ω,·)

(Wσ−1 ω + vσ0−1 ω )),

ω ∈ Ω.

We claim that (D11 G(θ, W)h)ω = hLσ−1 ω (g(σ −1 ω, ·)2eθg(σ

−1 ω,·)

(Wσ−1 ω + vσ0−1 ω )).

(101)

Indeed, we have kD1 G(θ + h, W)ω − D1 G(θ, W)ω − hLσ−1 ω (g(σ −1 ω, ·)2eθg(σ = kLσ−1 ω (g(σ −1 ω, ·)(e(θ+h)g(σ ≤ Kkg(σ −1ω, ·)(e(θ+h)g(σ

−1 ω,·)

= var(g(σ −1ω, ·)(e(θ+h)g(σ + kg(σ −1ω, ·)(e(θ+h)g(σ

≤ var(g(σ −1ω, ·)(e(θ+h)g(σ + Mke(θ+h)g(σ

−1 ω,·)

− eθg(σ

−1 ω,·)

− eθg(σ

−1 ω,·)



−1 ω,·)

)(Wσ−1 ω + vσ0−1 ω ))

)(Wσ−1 ω + vσ0−1 ω )k1

−1 ω,·)

−1 ω,·)

)(Wσ−1 ω + vσ0−1 ω ))kB

)(Wσ−1 ω + vσ0−1 ω )kB

−1 ω,·)

−1 ω,·)


−1 ω,·)

−1 ω,·)



−1 ω,·)

(Wσ−1 ω + vσ0−1 ω ))kB



−1 ω,·)

− eθg(σ

−1 ω,·)

−1 ω,·)

−1 ω,·)

− eθg(σ

− eθg(σ

−1 ω,·)

−1 ω,·)

− eθg(σ

− eθg(σ

−1 ω,·)

−1 ω,·)

+ kg(σ −1ω, ·)(e(θ+h)g(σ

−1 ω,·)

−1 ω,·)

)) · kWσ−1 ω + vσ0−1 ω kL∞

)kL∞ · var(Wσ−1 ω + vσ0−1 ω )

kL∞ · kWσ−1 ω + vσ0−1 ω k1 ,

and therefore (101) follows directly from (95) and (97). We now establish the continuity of D11 G. Observe that k(D11 G(θ1 , W 1 )h)ω − (D11 G(θ2 , W 2 )h)ω kB 61

= |h| · kLσ−1 ω (g(σ −1ω, ·)2eθ1 g(σ ≤ K|h| · kg(σ −1 ω, ·)2eθ1 g(σ

≤ K|h| · kg(σ −1 ω, ·)2eθ1 g(σ

−1 ω,·)

−1 ω,·)

−1 ω,·)

+ K|h| · kg(σ −1 ω, ·)2eθ2 g(σ

(Wσ1−1 ω + vσ0−1 ω ) − g(σ −1ω, ·)2eθ2 g(σ

(Wσ1−1 ω + vσ0−1 ω ) − g(σ −1 ω, ·)2eθ2 g(σ

(Wσ1−1 ω + vσ0−1 ω ) − g(σ −1 ω, ·)2eθ2 g(σ

−1 ω,·)

−1 ω,·)

−1 ω,·)

−1 ω,·)

(Wσ1−1 ω + vσ0−1 ω ) − g(σ −1ω, ·)2 eθ2 g(σ

(Wσ2−1 ω + vσ0−1 ω ))kB

(Wσ2−1 ω + vσ0−1 ω )kB

(Wσ1−1 ω + vσ0−1 ω )kB

−1 ω,·)

(Wσ2−1 ω + vσ0−1 ω )kB .

The continuity of D11 G now follows easily from (90) and Lemma B.2. Finally, we note that D1 G is an affine map in W and therefore (D21 G(θ, W)H)ω = Lσ−1 ω (g(σ −1 ω, ·)eθg(σ

−1 ω,·)

Hσ−1 ω ),

which can be showed to be continuous by using (90) and Lemma B.2 again. The following result is a direct consequence of the previous lemmas. Proposition B.12. The function F defined by (38) is of class C 2 on a neighborhood (0, 0) ∈ C × S.

C

Differentiability of φθ , the top space for adjoint twisted cocycle Rθ∗

We begin with some auxiliary results. Lemma C.1. There exists C > 0 such that kLθω1 − Lθω2 k ≤ C|θ1 − θ2 |,

for θ1 , θ2 ∈ BC (0, 1) and ω ∈ Ω.

(102)

Proof. For any f ∈ B we have that kLθω1 f − Lθω2 f kB = kLω (eθ1 g(ω,·) f − eθ2 g(ω,·) f )kB ≤ Kk(eθ1 g(ω,·) − eθ2 g(ω,·) )f kB = K var((eθ1 g(ω,·) − eθ2 g(ω,·) )f ) + Kk(eθ1 g(ω,·) − eθ2 g(ω,·) )f k1

The claim of the lemma now follows directly from (88) and (90). Lemma C.2. The following statements hold: 1. There exists K ′′ > 0 such that for φ ∈ B∗ such that φ(vω0 ) = 0 and ω ∈ Ω,

kL∗,(n) φkB∗ ≤ K ′′ e−λn kφkB∗ ω

(103)

with λ > 0 as in (C3); 2. Let φ0ω ∈ B∗ be as in (56). Then, ess supω∈Ω kφ0ω kB∗ < ∞. 62

(104)

Proof. Let Πω denote the projection on B onto the subspace B0 of functions of zero mean along the subspace spanned by vω0 . Furthermore, set γ(ω) = inf{kf + gkB : f ∈ B0 , g ∈ span{vω0 }, kf kB = kgkB = 1}. 2 . Take now arbitrary f ∈ B0 , g ∈ span{vω0 } As in Lemma 1 in [12] we have that kΠω k ≤ γ(ω) such that kf kB = kgkB = 1. It follows from (C1) that

kf + gkB ≥

1 1 (n) kL(n) (kL(n) ω (f + g)kB ≥ ω gkB − kLω f kB ). n K Kn

(105)

Writing g = λvω0 with |λ| = 1/kvω0 kB , it follows from (17) that kLω(n) gkB = |λ| · kvσ0n ω kB =

kvσ0n ω k1 1 kvσ0n ω kB ≥ ≥ , 0 0 ˜ kvω kB kvω kB K

˜ = ess supω∈Ω kv 0 kB < ∞, in view of (17). By (C3) and (105), where K ω kf + gkB ≥

1 ˜ − K ′ e−λn ). (1/K Kn

Then, we can choose n, independently of ω, such that ǫ :=

1 ˜ − K ′ e−λn ) > 0, (1/K Kn

which implies that γ(ω) ≥ ǫ and thus ess supω∈Ω kΠω k ≤ 2/ǫ < ∞.

(106)

Therefore, for φ that belongs to annihilator of vω0 , using (C3) and (106) we have (n)

(n)

kL∗,(n) φkB∗ = sup |φ(Lσ−n ω f )| = sup |φ(Lσ−n ω Πσ−n ω f )| ω kf k≤1

kf k≤1

≤ K ′ e−λn kφkB∗ · kΠσ−n ω k 2K ′ −λn ≤ e kφkB∗ , ǫ

for every n ≥ 0. We conclude that (103) holds. with K ′′ = 2K ′ /ǫ. Finally, R (104) is follows directly from the straightforward fact that for P-a.e. ω ∈ Ω, 0 φω (f ) = f dm.

Next, we consider B∗ with the norm topology, and associated Borel σ−algebra. Let 0 ∗ N = Φ : Ω → B : Φ is measurable, ess supω∈Ω kΦω kB∗ < ∞, Φω (vω ) = 0 for P-a.e. ω ∈ Ω 63

and

∗ N = Φ : Ω → B : Φ is measurable, ess supω∈Ω kΦω kB∗ < ∞ , ′

where Φω := Φ(ω). We note that N and N ′ are Banach spaces with respect to the norm kΦk∞ = ess supω∈Ω kΦω kB∗ . We define G1 : BC (0, 1) × N → N ′ by G1 (θ, Φ)ω = (Lθω )∗ (Φσω + φ0σω ),

ω ∈ Ω.

It follows readily from (27) and (104) that G1 is well-defined. G2 : BC (0, 1) × N → L∞ (Ω) by G2 (θ, Φ)(ω) = (Φσω + φ0σω )(Lθω vω0 ),

Furthermore, we define

ω ∈ Ω.

Again, it follows from (17), (27) and (104) that G2 is well-defined. Lemma C.3. D2 G1 exists and is continuous on BC (0, 1) × N . Proof. We first note that G1 is an affine map in the variable Φ which implies that (D2 G1 (θ, Φ)Ψ)ω = (Lθω )∗ Ψσω ,

for ω ∈ Ω and Ψ ∈ N .

Moreover, using (102) we have kD2 G1 (θ1 , Φ1 ) − D2 G1 (θ2 , Φ2 )k = sup kD2 G1 (θ1 , Φ1 )Ψ − D2 G1 (θ2 , Φ2 )Ψk∞ kΨk∞ ≤1

= sup ess supω∈Ω k(Lθω1 )∗ Ψσω − (Lθω2 )∗ Ψσω kB∗ kΨk∞ ≤1

≤ C|θ1 − θ2 |, for any (θ1 , Φ1 ), (θ2 , Φ2 ) ∈ BC (0, 1) × N . Hence, D2 G1 is continuous on BC (0, 1) × N . Lemma C.4. D1 G1 exists and is continuous on a neighborhood of (0, 0) ∈ C × N . Proof. We claim that (D1 G1 (θ, Φ)h)ω (f ) = (Φσω + φ0σω )(Lω (hg(σ −1 ω, ·)eθg(σ

−1 ω,·)

f )),

(107)

for f ∈ B, ω ∈ Ω and h ∈ C. Denote the operator on the right hand side of (107) by L(θ, Φ). We note that (G1 (θ + h, Φ)ω − G1 (θ, Φ)ω − hL(θ, Φ)ω )(f ) = (Φσω + φ0σω )(Lω ((e(θ+h)g(σ

−1 ω,·)

− eθg(σ 64

−1 ω,·)


−1 ω,·)

)f )).

Therefore, it follows from (C1) that kG1 (θ + h, Φ) − G1 (θ, Φ) − hL(θ, Φ)k∞

= ess supω∈Ω sup |(Φσω + φ0σω )(Lω ((e(θ+h)g(σ

−1 ω,·)

kf kB ≤1

≤ K(kΦk∞ + kφ0 k∞ ) ess supω∈Ω sup k(e(θ+h)g(σ

− eθg(σ

−1 ω,·)

kf kB ≤1

−1 ω,·)

− eθg(σ


−1 ω,·)

−1 ω,·)


)f ))|

−1 ω,·)

)f kB .

By (95) and (97), we conclude that 1 lim kG1 (θ + h, Φ) − G1 (θ, Φ) − hL(θ, Φ)k∞ = 0, h→0 h and thus (107) holds. Moreover, (D1 G1 (θ1 , Φ1 )h)ω (f ) − (D1 G1 (θ2 , Φ2 )h)ω (f ) = (Φ1σω − Φ2σω )(Lω (hg(σ −1ω, ·)eθ1 g(σ + (Φ2σω + φ0σω )(Lω (hg(σ −1 ω, ·)(eθ1

−1 ω,·)

f ))

g(σ−1 ω,·)

− eθ2 g(σ

−1 ω,·)

)f )),

which in view of (C1), (24), (88) and (90) easily implies that D1 G1 is continuous. Lemma C.5. D2 G2 exists and is continuous on a neighborhood of (0, 0) ∈ C × N . Proof. We note that G2 is affine map in the variable Φ and hence (D2 G2 (θ, Φ)Ψ)(ω) = Ψσω (Lθω vω0 ),

ω ∈ Ω.

It follows from (102) that kD2 G2 (θ1 , Φ1 ) − D2 G2 (θ2 , Φ2 )k = sup kD2 G2 (θ1 , Φ1 )Ψ − D2 G2 (θ2 , Φ2 )ΨkL∞ kΨk∞ ≤1

= sup ess supω∈Ω |Ψσω (Lθω1 vω0 − Lθω2 vω0 )| kΨk∞ ≤1

≤ C|θ1 − θ2 | · ess supω∈Ω kvω0 kB , and thus (in a view of (17)) we conclude that D2 G2 is continuous. Lemma C.6. D1 G2 exists and is continuous on a neighborhood of (0, 0) ∈ C × N . Proof. We claim that (D1 G2 (θ, Φ)h)(ω) = (Φσω + φ0σω )(Lω (g(σ −1ω, ·)eθg(σ

−1 ω,·)

vω0 )),

h ∈ C, ω ∈ Ω.

Let us denote the operator on the right hand side of (108) by R(θ, Φ). We have that (G2 (θ + h, Φ) − G2 (θ, Φ) − hR(θ, Φ))(ω) 65

(108)

= (Φσω + φ0σω )(Lω ((e(θ+h)g(σ

−1 ω,·)

− eθg(σ

−1 ω,·)


−1 ω,·)

)vω0 )).

Therefore, it follows from (C1) that kG2 (θ + h, Φ) − G2 (θ, Φ) − hR(θ, Φ)kL∞

= ess supω∈Ω |(Φσω + φ0σω )(Lω ((e(θ+h)g(σ

−1 ω,·)

≤ K(kΦk∞ + kφ0 k∞ ) ess supω∈Ω k(e(θ+h)g(σ

− eθg(σ

−1 ω,·)

−1 ω,·)

− eθg(σ


−1 ω,·)

−1 ω,·)


)vω0 ))|

−1 ω,·)

)vω0 kB .

By (17), (95) and (97), we conclude that 1 lim kG2 (θ + h, Φ) − G2 (θ, Φ) − hR(θ, Φ)kL∞ = 0. h→0 h Thus, (108) holds. Moreover, (D1 G2 (θ1 , Φ1 )h)(ω) − (D1 G2 (θ2 , Φ2 )h)(ω) = (Φ1σω − Φ2σω )(Lω (hg(σ −1 ω, ·)eθ1g(σ

−1 ω,·)

+ (Φ2σω + φ0σω )(Lω (hg(σ −1ω, ·)(eθ1 g(σ

vω0 ))

−1 ω,·)

− eθ2 g(σ

−1 ω,·)

)vω0 )),

which in view of (C1), (24), (88) and (90) easily implies that D1 G2 (θ1 , Φ1 ) → D1 G2 (θ2 , Φ2 ) when (θ1 , Φ1 ) → (θ2 , Φ2 ). Hence, D1 G2 is continuous. Let G(θ, Φ)ω =

(Lθω )∗ (Φσω + φ0σω ) − Φω − φ0ω . (Φσω + φ0σω )(Lθω vω0 )

(109)

Proposition C.7. The map G is of class C 1 on a neighborhood of (0, 0) ∈ C × N . Furthermore, ((D2 G(θ, Φ))Ψ)ω =

(Lθω )∗ Ψσω Ψσω (Lθω vω0 ) − G1 (θ, Φ)ω − Ψω , G2 (θ, Φ)(ω) [G2 (θ, Φ)(ω)]2

ω ∈ Ω, Ψ ∈ N .

(110)

Proof. The desired conclusion follows directly from Lemmas C.3, C.4, C.5 and C.6 after we note that G2 (0, 0)(ω) = 1 for ω ∈ Ω. Lemma C.8. D2 G(0, 0) is invertible. Proof. By (110), (D2 G(0, 0)Ψ)ω = L∗ω Ψσω − Ψω ,

for ω ∈ Ω and Ψ ∈ N .

Now one can proceed as in the proof of Lemma 3.5 to show that (103) implies the desired conclusion.

66

It follows from Proposition C.7, Lemma C.8 and the implicit function theorem that there exists a neighborhood U of 0 ∈ C and a smooth function F : U → N such that F (0) = 0 and G(θ, F (θ)) = 0, for θ ∈ U. (111) Finally, set Ψ(θ)ω =

F (θ)ω + φ0ω , (F (θ)ω + φ0ω )(vωθ )

for ω ∈ Ω and θ ∈ U.

Using the differentiability of θ 7→ v θ , we observe that there exists a neighborhood U ′ ⊂ U of 0 ∈ C such that Ψ(θ) is well-defined and differentiable for θ ∈ U ′ . Furthermore, we note that Ψ(θ)ω (vωθ ) = 1. Finally, it follows from (109) and (111) that (Lθω )∗ Ψ(θ)σω = Cωθ Ψ(θ)ω , for some scalar Cωθ . The arguments in Subsection 3.7 imply that φθω = Ψ(θ)ω . Therefore, we have established the differentiability of θ → φθ .

Acknowledgements We thank Yuri Kifer for raising the possibility of proving a quenched random LCLT. The research of DD was supported by the Australian Research Council Discovery Project DP150100017 and in part by the Croatian Science Foundation under the project IP-2014-09-228. The research of GF was supported by the Australian Research Council Discovery Project DP150100017 and a Future Fellowship. CGT was supported by ARC DE160100147. SV was supported by the project APEX “Systèmes dynamiques: Probabilités et Approximation Diophantienne PAD” funded by the Région PACA, by the Labex Archiméde (AMU University), by the Leverhulme Trust for support thorough the Network Grant IN-2014-021 and by the project Physeco, MATH-AMSud. CGT thanks Jacopo de Simoi and Carlangelo Liverani for their hospitality in Rome and for conversations related to this topic. Parts of this work were completed when (some or all) the authors met at AIM (San Jose), CPT & CIRM (Marseille), SUSTC (Shenzhen), the University of New South Wales and the University of Queensland. We are thankful to all of these institutions for their support and hospitality.

References [1] M. Abdelkader and R. Aimino. On the quenched central limit theorem for random dynamical systems. J. Phys. A, 49(24):244002, 13, 2016. [2] R. Aimino, M. Nicol, and S. Vaienti. Annealed and quenched limit theorems for random expanding dynamical systems. Probability Theory and Related Fields, 162(1-2):233–274, 2015.

67

[3] R. R. Akhmerov, M. I. Kamenski˘ı, A. S. Potapov, A. E. Rodkina, and B. N. Sadovski˘ı. Measures of noncompactness and condensing operators, volume 55 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1992. Translated from the 1986 Russian original by A. Iacob. [4] L. Arnold. Random dynamical systems. Springer Monographs in Mathematics. SpringerVerlag, Berlin, 1998. [5] A. Avez. Differential calculus. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1986. Translated from the French by D. Edmunds. [6] A. Ayyer, C. Liverani, and M. Stenlund. Quenched CLT for random toral automorphism. Discrete Contin. Dyn. Syst., 24(2):331–348, 2009. [7] W. Bahsoun and C. Bose. Mixing rates and limit theorems for random intermittent maps. Nonlinearity, 29(4):1417–1433, 2016. [8] A. Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete and Continuous Dynamical Systems, 36(5):2377–2403, 2016. [9] A. Broise. Transformations dilatantes de l’intervalle et théorèmes limites. Astérisque, ´ (238):1–109, 1996. Etudes spectrales d’opérateurs de transfert et applications. [10] J. Buzzi. Exponential Decay of Correlations for Random Lasota–Yorke Maps. Communications in Mathematical Physics, 208:25–54, 1999. [11] J. de Simoi and C. Liverani. Fast-slow partially hyperbolic systems: beyond averaging. Part I (Limit Theorems). ArXiv e-prints, Aug. 2014. [12] D. Dragiˇcević and G. Froyland. Hölder continuity of Oseledets splittings for semiinvertible operator cocycles. Ergodic Theory and Dynamical Systems. To appear. Published online: 09 September 2016. http://dx.doi.org/10.1017/etds.2016.55. [13] D. Dragiˇcević, G. Froyland, C. González-Tokman, and S. Vaienti. Almost sure invariance principle for random Lasota–Yorke maps. Preprint. https://arxiv.org/abs/1611.04003. [14] G. K. Eagleson. Some simple conditions for limit theorems to be mixing. Teor. Verojatnost. i Primenen., 21(3):653–660, 1976. [15] G. Froyland, S. Lloyd, and A. Quas. Coherent structures and isolated spectrum for Perron-Frobenius cocycles. Ergodic Theory Dynam. Systems, 30:729–756, 2010. [16] G. Froyland, S. Lloyd, and A. Quas. A semi-invertible Oseledets theorem with applications to transfer operator cocycles. Discrete Contin. Dyn. Syst., 33(9):3835–3860, 2013.

68

[17] G. Froyland and O. Stancevic. Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems. Stochastics and Dynamics, 13(4):1350004, 2013. [18] C. González-Tokman and A. Quas. A semi-invertible operator Oseledets theorem. Ergodic Theory and Dynamical Systems, 34:1230–1272, 8 2014. [19] C. González-Tokman and A. Quas. A concise proof of the multiplicative ergodic theorem on banach spaces. Journal of Modern Dynamics, 9(01):237–255, 2015. [20] G. A. Gottwald and I. Melbourne. Homogenization for deterministic maps and multiplicative noise. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469(2156):20130201, 16, 2013. [21] S. Gouëzel. Berry–esseen theorem and local limit theorem for non uniformly expanding maps. In Annales de l’Institut Henri Poincare (B) Probability and Statistics, volume 41, pages 997–1024, 2005. [22] S. Gouëzel. Almost sure invariance principle for dynamical systems by spectral methods. The Annals of Probability, 38(4):1639–1671, 2010. [23] S. Gouëzel. Limit theorems in dynamical systems using the spectral method. In Hyperbolic dynamics, fluctuations and large deviations, volume 89 of Proc. Sympos. Pure Math., pages 161–193. Amer. Math. Soc., Providence, RI, 2015. [24] Y. Guivarc’h and J. Hardy. Théorèmes limites pour une classe de chaˆınes de markov et applications aux difféomorphismes d’anosov. Annales de l’IHP Probabilités et statistiques, 24(1):73–98, 1988. [25] H. Hennion. Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc., 118(2):627–634, 1993. [26] H. Hennion and L. Hervé. Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness, volume 1766 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2001. [27] T. Kato. Perturbation theory for linear operators. Classics in Mathematics. SpringerVerlag, Berlin, 1995. Reprint of the 1980 edition. [28] D. Kelly and I. Melbourne. Smooth approximation of stochastic differential equations. Ann. Probab., 44(1):479–520, 2016. [29] D. Kelly and I. Melbourne. Deterministic homogenization for fast–slow systems with chaotic noise. J. Funct. Anal., 272(10):4063–4102, 2017. [30] Y. Kifer. Perron-Frobenius theorem, large deviations, and random perturbations in random environments. Mathematische Zeitschrift, 222(4):677–698, 1996. 69

[31] Y. Kifer. Limit theorems for random transformations and processes in random environments. Transactions of the American Mathematical Society, 350(4):1481–1518, 1998. [32] J. Leppänen and M. Stenlund. Quasistatic dynamics with intermittency. Mathematical Physics, Analysis and Geometry, 19(2):8, 2016. [33] I. Melbourne and M. Nicol. A vector-valued almost sure invariance principle for hyperbolic dynamical systems. The Annals of Probability, pages 478–505, 2009. [34] T. Morita. A generalized local limit theorem for Lasota-Yorke transformations. Osaka J. Math., 26(3):579–595, 1989. [35] T. Morita. Correction to: “A generalized local limit theorem for Lasota-Yorke transformations” [Osaka J. Math. 26 (1989), no. 3, 579–595; MR1021432 (91a:58176)]. Osaka J. Math., 30(3):611–612, 1993. [36] S. V. Nagaev. Some limit theorems for stationary markov chains. Theory of Probability & Its Applications, 2(4):378–406, 1957. [37] S. V. Nagaev. More exact statement of limit theorems for homogeneous markov chains. Theory of Probability & Its Applications, 6(1):62–81, 1961. [38] P. Nándori, D. Szász, and T. Varj´ u. A central limit theorem for time-dependent dynamical systems. J. Stat. Phys., 146(6):1213–1220, 2012. [39] M. Nicol, A. Török, and S. Vaienti. Central limit theorems for sequential and random intermittent dynamical systems. Ergodic Theory and Dynamical Systems, pages 1–27, 2016. [40] T. Ohno. Asymptotic behaviors of dynamical systems with random parameters. Publications of the Research Institute for Mathematical Sciences, 19(1):83–98, 1983. [41] L. Rey-Bellet and L.-S. Young. Large deviations in non-uniformly hyperbolic dynamical systems. Ergodic Theory and Dynamical Systems, 28(02):587–612, 2008. [42] J. Rousseau-Egele. Un théoreme de la limite locale pour une classe de transformations dilatantes et monotones par morceaux. The Annals of Probability, 11:772–788, 1983. [43] B. Saussol. Absolutely continuous invariant measures for multidimensional expanding maps. Israel J. Math., 116:223–248, 2000.

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