1 Introduction

The multiplicative ergodic theorem (MET) is a powerful tool with various applications in different fields of mathematics, including analysis, probability theory, and geometry, and a cornerstone in smooth ergodic theory. It was first proved by Oseledets [18] for matrix cocycles. Since then, the theorem attracted many researchers to provide new proofs and formulations with increasing generality [2, 6, 11, 15, 17, 19,20,21,22,23].

In [12], the authors gave a proof for an MET for cocycles acting on measurable fields of Banach spaces. Let us quickly recall the setting here: If \((\Omega ,\mathcal {F},\mathbb {P})\) denotes a probability space, we call a family of Banach spaces \(\{E_{\omega } \}_{\omega \in \Omega }\) a measurable field if there exists a linear subspace \(\Delta \) of all sections \(\Pi _{\omega \in \Omega } E_{\omega }\) and a countable subset \(\Delta _0 \subset \Delta \) such that \(\{g(\omega )\, :\, g \in \Delta _0\}\) is dense in \(E_{\omega }\) for every \(\omega \in \Omega \) and \(\omega \mapsto \Vert g(\omega ) \Vert _{E_{\omega }}\) is measurable for every \(g \in \Delta \). Note that this definition implies that every Banach space \(E_{\omega }\) is separable. On the other hand, every separable Banach space defines a field of Banach spaces by simply setting \(E_{\omega } = E\). This structure is similar to a measurable version of a Banach bundle with base \(\Omega \) and total space \(\Pi _{\omega \in \Omega } E_{\omega }\) in which every space \(E_{\omega }\) is a fiber. However, the fundamental difference is that we do not put any measurable (or topological) structure on the bundle \(\Pi _{\omega \in \Omega } E_{\omega }\) itself! In fact, the existence of the set \(\Delta \) is a substitute for the measurable structure and will help to prove measurability for functionals defined on \(\Pi _{\omega \in \Omega } E_{\omega }\) as we will see many times in this work. If \((\Omega ,\mathcal {F},\mathbb {P},\theta )\) is a measure preserving dynamical systems, a cocycle acting on the field \(\{E_{\omega }\}_{\omega \in \Omega }\) consists of a family of maps \(\varphi _{\omega } :E_{\omega } \rightarrow E_{\theta \omega }\). Setting \(\varphi ^n_{\omega } := \varphi _{\theta ^{n-1} \omega } \circ \cdots \circ \varphi _{\omega }\), we furthermore claim that \(\omega \mapsto \Vert \varphi ^n_{\omega }(g(\omega )) \Vert _{E_{\theta ^n \omega }}\) is measurable for every \(g \in \Delta \) and every \(n \in \mathbb {N}\).

There are numerous examples in which it is natural to study cocycles on random spaces. In [12], our motivation was to study dynamical properties of singular stochastic delay differential equations in which the spaces \(E_{\omega }\) are (essentially) spaces of controlled Brownian paths known in rough paths theory [8]. In the finite dimensional case, linearizing a \(C^1\)-cocycle on a manifold yields a linear cocycle acting on the tangent bundle [1, Chapter 4.2]. In the context of stochastic partial differential equations (SPDE), cocycles on random metric spaces were studied, for instance, when uniqueness of the equation is unknown and one has to work with a measurable selection instead, cf. [9] in the case of the 3D stochastic Navier–Stokes equation. Other examples in the situation of SPDE can be found in [3, 4]. In the deterministic case, a similar structure appears when studying the flow on time-dependent domains [14]. More recently, scales of time-dependent Banach spaces where introduced to study dynamical properties of non-autonomous PDEs in [5, 7].

We will now restate the MET [12, Theorem 4.17] in a slightly simplified version.

Theorem 0.1

Let \((\Omega ,\mathcal {F},\mathbb {P},\theta )\) be an ergodic measurable metric dynamical system and \(\varphi \) be a compact linear cocycle acting on a measurable field of Banach spaces \(\{E_{\omega }\}_{\omega \in \Omega }\). For \(\mu \in \mathbb {R}\cup \{-\infty \}\) and \(\omega \in {\Omega }\), define

$$\begin{aligned} F_{\mu }(\omega ) := \big \lbrace x\in E_{\omega }\, :\, \limsup _{n\rightarrow \infty } \frac{1}{n} \log \Vert \varphi ^n_{\omega }(x) \Vert \leqslant \mu \big \rbrace . \end{aligned}$$

Assume that

$$\begin{aligned} \log ^+ \Vert \varphi _{\omega } \Vert \in L^{1}(\Omega ). \end{aligned}$$

Then there is a measurable forward invariant set \(\tilde{\Omega } \subset \Omega \) of full measure and a decreasing sequence \(\{\mu _i\}_{i \ge 1}\), \(\mu _i \in [-\infty , \infty )\) with the properties that \(\lim _{n \rightarrow \infty } \mu _n = - \infty \) and either \(\mu _i > \mu _{i+1}\) or \(\mu _i = \mu _{i+1} = -\infty \) such that for every \(\omega \in \tilde{\Omega }\),

$$\begin{aligned} x\in F_{\mu _{i}}(\omega ){\setminus } F_{\mu _{i+1}}(\omega ) \quad \text {if and only if}\quad \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \varphi ^n_{\omega }(x) \Vert = \mu _{i}. \end{aligned}$$
(0.1)

Moreover, there are numbers \(m_1, m_2, \ldots \) such that \({\text {codim}} F_{\mu _j}(\omega ) = m_1 + \ldots + m_{j-1}\) for every \(\omega \in \tilde{\Omega }\).

Let us mention here that, motivated by our example of a stochastic delay equation, we proved this theorem for compact cocycles only, but it should be straightforward to generalize it to the quasi-compact case as Thieullen did in [22]. Consequently, we believe that all our results in this work will hold for quasi-compact cocycles, too.

The numbers \(\{\mu _i\}\) are the Lyapunov exponents, the subspaces \(F_{\mu }(\omega )\) are sometimes called slow-growing subspaces and the resulting filtration

$$\begin{aligned} E_{\omega } = F_{\mu _1}(\omega ) \supset F_{\mu _2}(\omega ) \supset \cdots \end{aligned}$$

is called Oseledets filtration. Is is easily seen that the slow-growing spaces are equivariant, meaning that \(\varphi _{\omega }(F_{\mu _i}(\omega )) \subset F_{\mu _i}(\theta \omega )\). In the proof of this theorem, no invertibility of \(\theta \) or \(\varphi \) is assumed, in which case a filtration of slow-growing subspaces is the best one can hope for. However, things change when we assume that the base \(\theta \) is invertible. In this case, it is possible to deduce a splitting of the spaces \(E_{\omega }\) consisting of fast-growing subspaces which are invariant under \(\varphi \). Such a splitting is called Oseledets splitting, and the corresponding theorem is called semi-invertible MET. Let us emphasize that we only need to assume invertibility of the base \(\theta \) and no invertibility of the cocyle \(\varphi \). In the context of SPDE or stochastic delay equations, these assumptions are quite natural: \(\theta \) usually denotes the shift of a random trajectory (which can be shifted forward and backward in time) and the cocycle denotes the solution map, which is not injective if the equation can be solved forward in time only.

Our first main result is a semi-invertible MET on a measurable field of Banach spaces. We state a simplified version here, the full statement can be found in Theorem 1.21 below.

Theorem 0.2

In addition to the assumptions made in Theorem 0.1, assume that \(\theta \) is invertible with measurable inverse \(\sigma := \theta ^{-1}\) and that Assumption 1.1 holds. Then there is a \(\theta \)-invariant set \(\tilde{\Omega }\) of full measure such that for every \(i \ge 1\) with \(\mu _i > \mu _{i+1}\) and \(\omega \in \tilde{\Omega }\), there is an \(m_i\)-dimensional subspace \(H^i_\omega \) with the following properties:

  1. (i)

    (Invariance) \(\varphi _{\omega }^k(H^i_{\omega }) = H^i_{\theta ^k \omega }\) for every \(k \ge 0\).

  2. (ii)

    (Splitting) \(H_{\omega }^i \oplus F_{\mu _{i+1}}(\omega ) = F_{\mu _i}(\omega )\). In particular,

    $$\begin{aligned} E_{\omega } = H^1_{\omega } \oplus \cdots \oplus H^i_{\omega } \oplus F_{\mu _{i+1}}(\omega ). \end{aligned}$$
  3. (iii)

    (‘Fast-growing’ subspace) For each \( h_{\omega }\in H^{i}_{\omega }{\setminus } \{0\} \),

    $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \varphi ^{n}_{\omega }(h_{\omega })\Vert = \mu _{j} \end{aligned}$$

    and

    $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert (\varphi ^{n}_{\sigma ^{n}\omega })^{-1}(h_{\omega })\Vert =-\mu _{j}. \end{aligned}$$

Moreover, the spaces are uniquely determined by properties (i), (ii) and (iii).

Clearly, the Oseledets splitting provides much more information about the cocycle than the filtration.

Let us discuss some important preceeding results. In the finite dimensional case, an MET for cocycles acting on measurable bundles can be found in the monograph [1, 4.2.6 Theorem] by L. Arnold. In [17], Mañé proved an MET with Oseledets splitting on a Banach bundle, assuming a topological structure on \(\Omega \) and continuity of the map \(\omega \mapsto \varphi _{\omega }\). He also assumed injectivity of \(\varphi \). Besides these results, we are not aware of any METs for cocycles acting on a bundle-type structure. Lian and Lu [15] proved an MET for cocycles acting on a fixed Banach space, assuming only a measurable structure on \(\Omega \), but injectivity of the cocycle. This assumption was later removed by Doan in [6] without giving an Oseledets splitting, however. In [10], González-Tokman and Quas used this result as a “black-box” and proved that an Oseledets splitting holds in this case, too.

Let us mention that our result is not only the first which provides a splitting on a bundle structure of Banach spaces without using a topological structure on \(\Omega \), it also weakens the measurability assumption on \(\varphi \) significantly in case we are dealing with a single Banach space E only. In fact, the standard measurability assumption, for instance in [11], is strong measurability of \(\varphi \), meaning that for fixed \(x \in E\), the map

$$\begin{aligned} \Omega \ni \omega \mapsto \varphi _{\omega }(x) \in E \end{aligned}$$
(0.2)

should be measurable. In contrast, our assumption means that the maps

$$\begin{aligned} \Omega \ni \omega \mapsto \Vert \varphi ^{k + n}_{\omega }(x) - \varphi ^{k}_{\theta ^n \omega }(\tilde{x}) \Vert _E \in \mathbb {R}\end{aligned}$$

should be measurable for every \(n,k \in \mathbb {N}_0\) and \(x, \tilde{x} \in S\) where S is a countable and dense subset of E. This assumption is clearly implied by (0.2).

The proof of Theorem 0.2 pushes forward the volume growth-approach advocated by Blumenthal [2] and González-Tokman, Quas [11] which provides a clear growth interpretation of the Lyapunov exponents. In a way, our result complements these two works in case of a single Banach space E. In particular, we are not imposing any further assumptions on E like reflexivity or separability of the dual as in [11].

A typical application for an MET is the construction of stable and unstable manifolds, cf. [17, 20, 21]. Here, the existence of the Oseledets splitting is crucial. Our second main contribution is an invariant manifold theorem for nonlinear cocycles acting on fields of Banach spaces. We state an informal version here, the precise statements are formulated in Theorems 2.10 and 2.17.

Theorem 0.3

Let \(\varphi \) be a nonlinear, differentiable cocycle acting on a measurable field of Banach spaces \(\{E_{\omega } \}_{\omega \in \Omega }\). Assume that \(Y_{\omega }\) is a random fixed point of \(\varphi \), in particular \(\varphi _{\omega }(Y_{\omega }) = Y_{\theta \omega }\). Then, under the same measurability and integrability assumptions as in Theorem 0.2, the linearized cocycle \(D_{Y_{\omega }} \varphi _{\omega }\) has a Lyapunov spectrum \(\{\mu _n\}_{n \ge 1}\). Under further assumptions on \(\varphi \) and Y, there is a \(\theta \)-invariant set \(\tilde{\Omega }\) of full measure, closed subspaces \(S_{\omega }\) and \(U_{\omega }\) of \(E_{\omega }\) and immersed submanifolds \(S_{loc}(\omega )\) and \(U_{loc}(\omega )\) of \(E_{\omega }\) such that for every \(\omega \in \tilde{\Omega }\),

$$\begin{aligned} T_{Y(\omega )} S_{loc}(\omega ) = S_{\omega } \qquad \text {and} \qquad T_{Y(\omega )} U_{loc}(\omega ) = U_{\omega } \end{aligned}$$

and the properties that for every \( Z_{\omega }\in S_{loc}(\omega ) \),

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \Vert \varphi ^{n}_{\omega }(Z_{\omega })-Y_{\theta ^{n}\omega }\Vert \leqslant \mu _{j_{0}} < 0 \end{aligned}$$

and for every \( Z_{\omega }\in U_{loc}(\omega ) \) one has \(\varphi ^n_{\sigma ^n \omega }(Z_{\sigma ^n \omega }) = Z_{\omega }\) and

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \Vert Z_{\sigma ^{n}\omega }-Y_{\sigma ^{n}\omega }\Vert \leqslant -\mu _{k_{0}} < 0. \end{aligned}$$

Here we have set \(\mu _{j_0} = \max \{ \mu _j \, :\ \mu _j < 0 \}\) and \(\mu _{k_0} = \min \{ \mu _k \, :\, \mu _k > 0 \}\). In the hyperbolic case, i.e. if all Lyapunov exponents are non-zero, the submanifolds \(S^{\upsilon }_{loc}(\omega ) \) and \(U^{\upsilon }_{loc}(\omega ) \) are transversal, i.e.

$$\begin{aligned} E_{\omega } = T_{Y_{\omega }} U^{\upsilon }_{loc}(\omega ) \oplus T_{Y_{\omega }} S^{\upsilon }_{loc}(\omega ). \end{aligned}$$

The structure of the paper is as follows. In Sect. 1, we prove a semi-invertible MET for cocycles acting on measurable fields of Banach spaces. This result is applied in Sect. 2 to deduce the existence of local stable and unstable manifolds for nonlinear cocycles.

1.1 Notation

  • For Banach spaces \((X, \Vert \cdot \Vert _X)\) and \((Y, \Vert \cdot \Vert _Y)\), L(XY) denotes the space of bounded linear functions from X to Y equipped with usual operator norm. We will often not explicitly write a subindex for Banach space norms and use the symbol \(\Vert \cdot \Vert \) instead. Differentiability of a function \(f :X \rightarrow Y\) will always mean Fréchet-differentiability. A \(C^m\) function denotes an m-times Fréchet-differentiable function. If \(A, B \subseteq X\), we denote by \(d(A,B) := \inf _{a \in A, b \in B} \Vert a - b\Vert \) the distance between two sets A and B. We also set \(d(x,B) := d(B,x) := d(\{x\},B)\) for \(x \in X\), \(B \subseteq X\).

  • Let XY be Banach spaces. For \( x_{1},\ldots ,x_{k}\in X \), set

    $$\begin{aligned} {\text {Vol}}(x_{1},x_{2},\ldots ,x_{k}) := \Vert x_{1}\Vert \prod _{i=2}^{k}d(x_{i},\langle x_{j} \rangle _{1\leqslant j<i}). \end{aligned}$$
    (0.3)

    For a given bounded linear function \( T : X\rightarrow Y \) and \(k \ge 1\), we define

    $$\begin{aligned} D_{k}(T):=\sup _{\Vert x_{i}\Vert =1 ; i=1,\ldots ,k} {\text {Vol}} \big (T(x_{1}),T(x_{2}),\ldots ,T(x_{k})\big ). \end{aligned}$$
  • Let E be a vector space. If we can write E as a direct sum \(E = F \oplus H\) of vector spaces, we have an algebraic splitting. We also say that F is a complement of H and vice versa. The projection operator \(\Pi _{F \Vert H}(e) = f\) with \(e = f + h\), \(f \in F\), \(h \in H\), is called the projection operator onto F parallel to H. If E is a normed space and \(\Pi _{F \Vert H}\) is bounded linear, i.e.

    $$\begin{aligned} \Vert \Pi _{F \Vert H} \Vert = \sup _{f \in F, h \in H, f + h \ne 0} \frac{\Vert f \Vert }{\Vert f + h\Vert } < \infty , \end{aligned}$$

    we call \(E = F \oplus H\) a topological splitting. For normed spaces, a splitting will always mean a topological splitting.

  • Let \((\Omega ,\mathcal {F})\) be a measurable space. We call a family of Banach spaces \(\{E_{\omega }\}_{\omega \in \Omega }\) a measurable field of Banach spaces if there is a set of sections

    $$\begin{aligned} \Delta \subset \prod _{\omega \in \Omega } E_{\omega } \end{aligned}$$

    with the following properties:

    1. (i)

      \(\Delta \) is a linear subspace of \(\prod _{\omega \in \Omega } E_{\omega }\).

    2. (ii)

      There is a countable subset \(\Delta _0 \subset \Delta \) such that for every \(\omega \in \Omega \), the set \(\{g(\omega )\, :\, g \in \Delta _0\}\) is dense in \(E_{\omega }\).

    3. (iii)

      For every \(g \in \Delta \), the map \(\omega \mapsto \Vert g(\omega ) \Vert _{E_{\omega }}\) is measurable.

  • Let \((\Omega ,\mathcal {F})\) be a measurable space. If there exists a measurable map \(\theta :\Omega \rightarrow \Omega \), \(\omega \mapsto \theta \omega \), with a measurable inverse \(\theta ^{-1}\), we call \((\Omega ,\mathcal {F}, \theta )\) a measurable dynamical system. We will use the notation \(\theta ^n \omega \) for n-times applying \(\theta \) to an element \(\omega \in \Omega \). We also set \(\theta ^0 := {\text {Id}}_{\Omega }\) and \(\theta ^{-n} := (\theta ^n)^{-1}\). If \(\mathbb {P}\) is a probability measure on \((\Omega ,\mathcal {F})\) that is invariant under \(\theta \), i.e. \(\mathbb {P}(\theta ^{-1} A) = \mathbb {P}(A) = \mathbb {P}( \theta A)\) for every \(A \in \mathcal {F}\), we call the tuple \(\big (\Omega , \mathcal {F},\mathbb {P},\theta \big )\) a measure-preserving dynamical system. The system is called ergodic if every \(\theta \)-invariant set has probability 0 or 1.

  • Let \((\Omega ,\mathcal {F},\mathbb {P},\theta )\) be a measure-preserving dynamical system and \((\{E_{\omega }\}_{\omega \in \Omega },\Delta )\) a measurable field of Banach spaces. A continuous cocycle on \(\{E_{\omega }\}_{\omega \in \Omega }\) consists of a family of continuous maps

    $$\begin{aligned} \varphi _{\omega } :E_{\omega } \rightarrow E_{\theta \omega }. \end{aligned}$$
    (0.4)

    If \(\varphi \) is a continuous cocycle, we define \(\varphi ^n_{\omega } :E_{\omega } \rightarrow E_{\theta ^n \omega }\) as

    $$\begin{aligned} \varphi ^n_{\omega } := \varphi _{\theta ^{n-1}\omega } \circ \cdots \circ \varphi _{\omega }. \end{aligned}$$

    We also set \(\varphi ^0_{\omega } := {\text {Id}}_{E_{\omega }}\). We say that \(\varphi \) acts on \(\{E_{\omega }\}_{\omega \in \Omega }\) if the maps

    $$\begin{aligned} \omega \mapsto \Vert \varphi (n,\omega ,g(\omega )) \Vert _{E_{\theta ^n \omega }}, \quad n \in \mathbb {N}\end{aligned}$$

    are measurable for every \(g \in \Delta \). In this case, we will speak of a continuous random dynamical system on a field of Banach spaces. If the map (0.4) is bounded linear/compact, we call \(\varphi \) a bounded linear/compact cocycle.

2 Semi-invertible MET on Fields of Banach Spaces

In this section, \((\Omega ,\mathcal {F},\mathbb {P},\theta )\) will denote an ergodic measure-preserving dynamical system and we set \(\sigma := \theta ^{-1}\). Let \((\{E_{\omega })_{\omega \in \Omega },\Delta ,\Delta _0)\) be a measurable field of Banach space and let \(\psi _{\omega } :E_{\omega } \rightarrow E_{\theta \omega }\) be a compact linear cocycle acting on it. In the sequel, we will furthermore assume that the following assumption is satisfied:

Assumption 1.1

For each \( g ,\tilde{g}\in \Delta \) and \(n, k \ge 0\),

$$\begin{aligned} \omega \rightarrow \Vert \psi _{\theta ^n \omega }^k [ \psi ^{n}_{\omega }(g(\omega ))-\tilde{g}(\theta ^{n}\omega ) ]\Vert _{E_{\theta ^{n + k} \omega }} \end{aligned}$$

is measurable.

We will always assume that

$$\begin{aligned} \log ^+ \Vert \psi _{\omega } \Vert \in L^1(\Omega ). \end{aligned}$$

Under this condition, the Multiplicative Ergodic Theorem [12, Theorem 4.17] applies and yields the existence of Lyapunov exponents \(\{\mu _1> \mu _2 > \ldots \} \subset [-\infty ,\infty )\) on a \(\theta \)-invariant set of full measure \(\tilde{\Omega } \subset \Omega \). More precisely, there are numbers \(\Lambda _k \in [-\infty ,\infty )\) such that

$$\begin{aligned} \Lambda _k = \lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big (\psi ^{n}_{\omega }\big ), \quad k \ge 1 \end{aligned}$$

for every \(\omega \in \tilde{\Omega }\). Setting \(\lambda _k = \Lambda _k - \Lambda _{k-1}\), the sequence \((\mu _k)\) is the subsequence of \((\lambda _k)\) defined by removing all multiple elements. For any \(\mu \in [-\infty ,\infty )\), we define the closed subspace

$$\begin{aligned} F_{\mu }(\omega ) = \left\{ \xi \in E_{\omega } \, |\, \limsup _{n \rightarrow \infty } \frac{1}{n} \log \Vert \psi ^n_{\omega }(\xi ) \Vert \le \mu \right\} . \end{aligned}$$

Note that \(\psi \) is invariant on these spaces in the sense that

$$\begin{aligned} \psi ^{n}_{\omega } \vert _{F_{\mu }(\omega )} :{F_{\mu }(\omega )} \rightarrow {F_{\mu }(\theta ^n \omega )}. \end{aligned}$$

We also saw in [12, Theorem 4.17] that there are numbers \(m_i \in \mathbb {N}\) such that \(m_i = {\text {dim}} \left( F_{\mu _i}(\omega ) / F_{\mu _{i+1}}(\omega )\right) \) for every \(\omega \in \tilde{\Omega }\).

If not otherwise stated, \(\tilde{\Omega } \subset \Omega \) will always denote a \(\theta \)-invariant set of full measure. Note that we can always assume w.l.o.g. that a given set of full measure \(\Omega _0 \subset \Omega \) is \(\theta \)-invariant, otherwise we can consider

$$\begin{aligned} \bigcap _{k \in \mathbb {Z}} \theta ^k(\Omega _0) \end{aligned}$$

instead.

Next, we collect some basic Lemmas. Recall the definition of \({\text {Vol}}\) and \(D_k\).

Lemma 1.2

Let XY be Banach spaces and \( T:X\rightarrow Y \) a linear operator. For \(k\in \mathbb {N}\), there exist positive constants \( c_{k}, C_{k} \) depending only on k such that

$$\begin{aligned} c_{k}D_{k}(T)\leqslant D_{k}(T^{*})\leqslant C_{k} D_{k}(T) \end{aligned}$$
(1.1)

where by \( T^{*}:Y^{*}\rightarrow X^{*} \) we mean the dual map of T.

Proof

[11, Lemma 3]. \(\square \)

Lemma 1.3

For a Banach space X and \( k\geqslant 1 \), the map

$$\begin{aligned}&{\text {Vol}} : X^{k} \longrightarrow \mathbb {R}\nonumber \\&(x_{1},x_{2},\ldots ,x_{k}) \mapsto \Vert x_{1}\Vert \prod _{i=2}^{k}d(x_{i},\langle x_{j} \rangle _{1\leqslant j<i}) \end{aligned}$$
(1.2)

is continuous.

Proof

[15, Lemma 4.2].\(\square \)

Lemma 1.4

For every \( g\in \Delta \) and \( j\geqslant 1 \), the map

$$\begin{aligned} \omega \mapsto d\big (g(\omega ) , F_{\mu _{j}}(\omega ))\big ) \end{aligned}$$

is measurable.

Proof

As in the proof to [12, Lemma 4.3]. \(\square \)

For a Banach space X and a closed subspace \(U \subset X\), the quotient space X/U is again a Banach space with norm

$$\begin{aligned} \Vert [x] \Vert _{X/U} = \inf _{u \in U} \Vert x-u \Vert . \end{aligned}$$

For an element \(x \in E_{\omega }\), we denote by \([x]_{\mu }\) its equivalence class in the quotient space \(E_{\omega } / F_{\mu }(\omega )\). From the invariance property of \(\psi \), the map

$$\begin{aligned}{}[\psi ^{n}_{\omega }]_{\mu _{j+1}}: \frac{F_{\mu _{j}}(\omega )}{F_{\mu _{j+1}}(\omega )}&\longrightarrow \frac{F_{\mu _{j}}(\theta ^{n}\omega )}{F_{\mu _{j+1}}(\theta ^{n}\omega )}, \quad [\psi ^{n}_{\omega }]_{\mu _{j+1}}([x]) := [ \psi ^n_{\omega }(x)]_{\mu _{j+1}} \end{aligned}$$

is well-defined for every \(j \ge 1\) and \(n \in \mathbb {N}\). Note also that \([\psi ^{n}_{\omega }]_{\mu _{j+1}}\) is bijective for \(\omega \in \tilde{\Omega }\). Indeed, injectivity is straightforward and surjectivity follows from the fact that \(F_{\mu _{j}}(\omega )/F_{\mu _{j+1}}(\omega )\) and \(F_{\mu _{j}}(\theta ^n\omega )/F_{\mu _{j+1}}(\theta ^n\omega )\) are finite-dimensional with the same dimension \(m_i\).

Lemma 1.5

For \(m, n\in \mathbb {N}\), the maps

$$\begin{aligned} f_{1}(\omega ):= D_{m}(\psi ^{n}_{\omega } \mid _{F_{\mu _{2}}(\omega )}) \ \ \ \text {and} \ \ \ \ f_{2}(\omega ):=D_{m}{(}{[}\psi ^{n}_{\omega }{]}_{\mu _{2}}{)} \end{aligned}$$

are measurable.

Proof

It is not hard to see that

$$\begin{aligned} f_{1}(\omega )=\lim _{l \rightarrow \infty }\liminf _{k \rightarrow \infty }\bigg [\sup _{\lbrace \xi ^{t}_{\omega }\rbrace _{1\leqslant t\leqslant m} \subset B_{\omega }^{l,k}(\mu _{2})} {\text {Vol}}\big (\psi ^{n}_{\omega }(\xi ^{1}_{\omega }),\ldots ,\psi ^{n}_{\omega }(\xi ^{m}_{\omega })\big )\bigg ] \end{aligned}$$
(1.3)

where

$$\begin{aligned} B_{\omega }^{l,k}(\mu _{2})=\big \lbrace \xi \in F_{\mu _{1}}(\omega ):\ \Vert \xi \Vert = 1, \&\Vert \psi ^k_{\omega }(\xi ) \Vert < \exp \big ( k(\mu _{2} + \frac{1}{l}) \big )\ \big \rbrace , \end{aligned}$$

cf. the proof of [12, Lemma 4.3]. Let \(\lbrace g_{t}\rbrace _{1\leqslant t\leqslant m}\subset \Delta _{0} \) and \( C({g_{t}}) := \lbrace \omega \ :\ {g}_{t}(\omega )\in B^{l,k}_{\omega }(\mu _{2})\rbrace \). As a consequence of Lemma 1.4, these sets are measurable and we have

$$\begin{aligned} \sup _{\lbrace \xi _{\omega }^{t}\rbrace _{1\leqslant t\leqslant m} \subset B_{\omega }^{l,k}(\mu _{2})}&{\text {Vol}}\big (\psi ^{n}_{\omega }(\xi ^{1}_{\omega }),\ldots ,\psi ^{n}_{\omega }(\xi ^{m}_{\omega })\big ) = \\&\sup _{\lbrace g_{t}\rbrace _{1\leqslant t\leqslant m}\subset \Delta _0 } {\text {Vol}}\bigg (\psi ^{n}_{\omega }\big (\frac{g_{1}(\omega )}{\Vert g_{1}(\omega )\Vert } \big ),\ldots ,\psi ^{n}_{\omega }\big (\frac{g_{m}(\omega )}{\Vert g_{m}(\omega )\Vert } \big )\bigg )\, \prod _{1\leqslant t\leqslant m}\chi _{C(g_{t})}(\omega ) \end{aligned}$$

which implies measurability of \(f_1\). For \(f_2\), note first that

$$\begin{aligned} f_{2}(\omega )=\lim _{l \rightarrow \infty }\liminf _{k \rightarrow \infty } \bigg [\sup _{\lbrace \xi ^{t}_{\omega }\rbrace _{1\leqslant t\leqslant m} \subset F_{\mu _{1}}(\omega )}&\frac{{\text {Vol}}\big ([\psi ^{n}_{\omega }(\xi ^{1}_{\omega }){]}_{\mu _{2}},\ldots ,{[}\psi ^{n}_{\omega }(\xi ^{m}_{\omega }){]}_{\mu _{2}}\big )}{\prod _{1\leqslant t\leqslant m}\Vert [\xi ^{t}_{\omega }]_{\mu _{2}}\Vert }\bigg ] \end{aligned}$$

where we set \( \frac{0}{0}:=0 \). Again as before

$$\begin{aligned} \sup _{\lbrace \xi ^{t}_{\omega }\rbrace _{1\leqslant t\leqslant m} \subset F_{\mu _{1}}(\omega )}&\frac{{\text {Vol}}\big ([\psi ^{n}_{\omega }(\xi ^{1}_{\omega }){]}_{\mu _{2}},\ldots ,{[}\psi ^{n}_{\omega }(\xi ^{m}_{\omega }){]}_{\mu _{2}}\big )}{\prod _{1\leqslant t\leqslant m}\Vert [\xi ^{t}_{\omega }]_{\mu _{2}}\Vert } = \\&\sup _{\lbrace g_{t}\rbrace _{1\leqslant t\leqslant m}\subset \Delta _{0} }\frac{{\text {Vol}}\big ({[}\psi ^{n}_{\omega }{(}g_{1}(\omega ){)}\big ]_{\mu _{2}},\ldots ,{[}\psi ^{n}_{\omega }(g_{k}(\omega ))]_{\mu _{2}}\big )}{\prod _{1\leqslant t\leqslant m}d\big (g_{t}(\omega ),F_{\mu _{2}}(\omega )\big )} . \end{aligned}$$

It remains to show that for \( g\in \Delta \), \( d\big (\psi ^{n}_{\omega }\big ({g(\omega )}\big ), F_{\mu _{2}}(\theta ^{n}\omega )\big )\) is measurable, which can be achieved using Assumption 1.1 with a proof similar to Lemma 1.4. \(\square \)

Lemma 1.6

For every \(i \ge 0\), there is a constant \(M_i > 0\) such that

$$\begin{aligned} \Vert [\psi ^{1}_{\omega }]_{\mu _{i+1}}\Vert < M_{i}\Vert \psi ^{1}_{\omega }\Vert \end{aligned}$$

for every \(\omega \in \tilde{\Omega }\).

Proof

Since \({\text {dim}}[\frac{F_{\mu _{i}}(\omega )}{F_{\mu _{i+1}}(\omega )}] = m_{i} \), we can choose \( H_{\omega }\subset F_{\mu _{i}}(\omega )\) such that

$$\begin{aligned} H_{\omega }\oplus F_{\mu _{i+1}}(\omega )=F_{\mu _{i}}(\omega )\ \ \ \text {and} \ \ \ \ \Vert \Pi _{H_{\omega }||F_{\mu _{i+1}}(\omega )}\Vert \le \sqrt{m_{i}}+2 =: M_i, \end{aligned}$$
(1.4)

cf. [2, Lemma 2.3]. Let \( \xi _{\omega }\in F_{\mu _{i}}(\omega ){\setminus } F_{\mu _{i+1}}(\omega ) \) with corresponding decomposition \( \xi _{\omega } = h_{\omega } + f_{\omega } \in H_{\omega }\oplus F_{\mu _{i+1}}(\omega )\). From (1.4), we know that \( \frac{\Vert h_{\omega }\Vert }{\Vert [\xi _{\omega }]_{\mu _{i+1}}\Vert }\leqslant M_{i} \) and consequently

$$\begin{aligned} \frac{\Vert [ \psi ^1_{\omega }(\xi _{\omega }) ]_{\mu _{i+1}} \Vert }{ \Vert [ \xi _{\omega }]_{\mu _{i+1}} \Vert } \le M_i \frac{\Vert [ \psi ^1_{\omega }(h_{\omega }) ]_{\mu _{i+1}} \Vert }{ \Vert h_{\omega } \Vert } \le M_i \frac{\Vert \psi ^1_{\omega }(h_{\omega }) \Vert }{ \Vert h_{\omega } \Vert } \le M_i \Vert \psi ^{1}_{\omega }\Vert . \end{aligned}$$

The claim follows. \(\square \)

Lemma 1.7

Assume that \( \lbrace f_{n}(\omega )\rbrace _{n\geqslant 1} \) is a subadditive sequence with respect to \( \theta \) and set \( g_{n}(\omega ):=f_{n}(\sigma ^{n}\omega )\). Assume \( f^+_{1}(\omega ) \in L^{1}(\Omega )\). Then there is a \(\theta \)-invariant set \(\tilde{\Omega } \in \mathcal {F}\) with full measure such that for every \(\omega \in \tilde{\Omega }\),

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}f_{n}(\omega )=\lim _{n\rightarrow \infty }\frac{1}{n}g_{n}(\omega )\in [-\infty ,\infty ) \end{aligned}$$

where the limit does not depend on \(\omega \).

Proof

We can easily check that \( \lbrace g_{n}(\omega )\rbrace _{n\geqslant 1} \) is a subadditive sequence with respect to \( \sigma \). Since \( f_{n}(\omega ) \) and \( g_{n}(\omega ) \) have same law, the result follows from Kingman’s Subadditive Ergodic Theorem. \(\square \)

As a consequence, we obtain the following:

Lemma 1.8

There is a \(\theta \)-invariant set of full measure \(\tilde{\Omega } \in \mathcal {F}\) such that

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big (\psi ^{n}_{\omega }\big )=\lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big (\psi ^{n}_{\sigma ^{n}\omega }\big )=\lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big ((\psi ^{n}_{\sigma ^{n}\omega })^{*}\big )=\Lambda _{k} \end{aligned}$$
(1.5)

and

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big (\psi ^{n}_{\omega }\mid _{F_{\mu _{2}}(\omega )}\big )=\lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big (\psi ^{n}_{\sigma ^{n}\omega }\mid _{F_{\mu _{2}}(\sigma ^{n}\omega )}\big )\nonumber \\&\quad =\lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big ((\psi ^{n}_{\sigma ^{n}\omega })^{*}\mid _{\big (F_{\mu _{2}}(\sigma ^{n}\omega )\big )^{*}}\big ]=\Lambda _{k+m_{1}}-\Lambda _{m_{1}} \end{aligned}$$
(1.6)

Proof

We already noted that \(\lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big (\psi ^{n}_{\omega }\big ) = \Lambda _k\). The equality

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big (\psi ^{n}_{\omega }\mid _{F_{\mu _{2}}(\omega )}\big ) = \Lambda _{k+{m}_{1}}-\Lambda _{{m}_{1}} \end{aligned}$$
(1.7)

was a partial result in the proof of Theorem [12, Theorem4.17]. The remaining inequalities follow by a combination of all Lemmas 1.21.7. \(\square \)

From now on, we will assume that \(\tilde{\Omega }\) is the set provided in Lemma 1.8.

Lemma 1.9

Fix \(\omega \in \tilde{\Omega }\) and let \((\xi _{\sigma ^{n}\omega })_n\) be a sequence such that \( \xi _{\sigma ^{n}\omega }\in F_{\mu _{1}}(\sigma ^{n}\omega ){\setminus } F_{\mu _{2}}(\sigma ^{n}\omega ) \) and \(\Vert [\xi _{\sigma ^{n}\omega }]_{\mu _{2}}\Vert =1\) for every \(n \in \mathbb {N}\). Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert [\psi ^{n}_{\sigma ^{n}\omega }(\xi _{\sigma ^{n}\omega })]_{\mu _{2}}\Vert =\mu _{1}. \end{aligned}$$
(1.8)

Proof

By applying Lemmas 1.5, 1.6 and 1.7, Kingman’s Subadditive Ergodic Theorem shows that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big ( \big [\psi ^{n}_{\omega }\big ]_{\mu _{2}}\big )=\lim _{n\rightarrow \infty }\frac{1}{n}\log D_{k}\big (\big [\psi ^{n}_{\sigma ^{n}\omega }\big ]_{\mu _{2}}\big ) \end{aligned}$$

exist for every \(k \ge 1\). Let \( H_{\omega } \) be a complement subspace for \( F_{\mu _{2}}(\omega ) \) in \( F_{\mu _{1}}(\omega ) \). Using a slight generalization of [12, Lemma 4.4], we have that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \Pi _{\psi ^{n}_{\omega }(H_{\omega })||F_{\mu _{2}}(\theta ^{n}\omega )}\Vert =0. \end{aligned}$$

For \( \xi _{\omega }\in F_{\mu _{1}}(\omega ){\setminus } F_{\mu _{2}}(\omega ) \), since

$$\begin{aligned} \frac{\Vert \psi ^{n}_{\omega }(\Pi _{H_{\omega }||F_{\mu _{2}}(\omega )}(\xi _{\omega }))\Vert }{\Vert [\psi ^{n}_{\omega }(\xi _{\omega })]_{\mu _{2}}\Vert }\leqslant \Vert \Pi _{\psi ^{n}_{\omega }(H_{\omega })||F_{\mu _{2}}(\theta ^{n}\omega )}\Vert \end{aligned}$$

it follows that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert [\psi ^{n}_{\omega }(\xi _{\omega })]_{\mu _{2}}\Vert =\mu _{1}. \end{aligned}$$
(1.9)

Let

$$\begin{aligned} k := \max \big \lbrace m : \lim _{n\rightarrow \infty }\frac{1}{n}\log D_{m}\big ( \big [\psi ^{n}_{\omega }\big ]_{\mu _{2}}\big ) =m \mu _{1} \big \rbrace . \end{aligned}$$

We claim \( k=m_{1} \). Indeed, otherwise from [12, Proposition4.15], there exists a subspace \( F_{\omega }\subset \frac{F_{\mu _{1}}(\omega )}{F_{\mu _{2}}(\omega )} \) with codimension k such that for every \( \xi _{\omega }\in F_{\omega } \)

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \Vert [\psi ^{n}_{\omega }(\xi _{\omega })]_{\mu _{2}} \Vert <\mu _{1}. \end{aligned}$$

Since \({\text {dim}} [\frac{F_{\mu _{1}}(\omega )}{F_{\mu _{2}}(\omega )}]=m_{1}\), we can find a non-zero element in \( F_{\omega }\) which contradicts (1.9). Hence we have shown that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log D_{m_1}\big ( \big [\psi ^{n}_{\omega }\big ]_{\mu _{2}}\big ) =m_1 \mu _{1}. \end{aligned}$$

Therefore, for every \(n\in \mathbb {N}\), we can find \(\lbrace \xi ^{j}_{\sigma ^{n}\omega }\rbrace _{1\leqslant j\leqslant m_{1}}\subset F_{\mu _{1}}(\sigma ^{n}\omega )\) such that \(\Vert [\xi ^{j}_{\omega }]_{\mu _{2}}\Vert =1 \) and

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}{\text {Vol}}\big ([\psi ^{n}_{\sigma ^{n}\omega }(\xi ^{1}_{\sigma ^{n}\omega })]_{\mu _{2}}, \ldots , [\psi ^{n}_{\sigma ^{n}\omega }(\xi ^{m_{1}}_{\sigma ^{n}\omega })]_{\mu _{2}} \big ) \big ]=m_{1} \mu _{1}. \end{aligned}$$
(1.10)

Using the definition of \({\text {Vol}}\), it follows that for every \( 2\leqslant t\leqslant m_{1} \),

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log d\big ([\psi ^{n}_{\sigma ^{n}\omega }(\xi ^{t}_{\omega })]_{\mu _{2}},\langle [\psi ^{n}_{\sigma ^{n}\omega }(\xi ^{j}_{\sigma ^{n}\omega })]_{\mu _{2}}\rangle _{1\leqslant j\leqslant t-1}\big )=\mu _{1}. \end{aligned}$$
(1.11)

We have \( \xi _{\sigma ^{n}\omega }=\sum _{1\leqslant j\leqslant m_{1}}\alpha _{j}\xi ^{j}_{\sigma ^{n}\omega }\) mod \( F_{\mu _{2}}(\sigma ^{n}\omega )\). In the proof of [12, Lemma 4.7], we already saw that the the \( {\text {Vol}} \)-function is symmetric up to a constant. By our assumption on \( \xi _{\sigma ^{n}\omega } \), we can therefore assume that \( \alpha _{m_{1}}\geqslant \frac{1}{ m_{1}}\). Finally from (1.11)

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert [\psi ^{n}_{\sigma ^{n}\omega }(\xi _{\sigma ^{n}\omega })]_{\mu _{2}}\Vert \\&\quad =\lim _{n\rightarrow \infty }\frac{1}{n}\big [d\big ([\psi (\xi ^{m_{i}}_{\sigma ^{n}\omega })]_{\mu _{2}},\langle [\psi ^{n}_{\sigma ^{n}\omega }(\xi ^{j}_{\sigma ^{n}\omega })]_{\mu _{2}}\rangle _{1\leqslant j\leqslant m_{1}-1}\big ) =\mu _{1}. \end{aligned}$$

\(\square \)

Definition 1.10

Let X be a Banach space. We define G(X) to be the Grassmanian of closed subspaces of X equipped with the Hausdorff distance

$$\begin{aligned} d_{H}(A,B):=\max \lbrace \sup _{a\in S_A}d(a,S_{B}),\sup _{b\in S_B}d(b,S_{A})\rbrace \end{aligned}$$

where \( S_{A}=\lbrace a\in A\ : \ \Vert a\Vert =1\rbrace \). Set

$$\begin{aligned} G_{k}(X)=\lbrace A\in G(X)\ :\ {\text {dim}}[A]=k\rbrace \quad \text {and} \quad G^{k}(X)=\lbrace A\in G(X)\ : \ {\text {dim}}[X/A] =k \rbrace . \end{aligned}$$

It can be shown that \((G(X),d_H)\) is a complete metric space and that \(G_k(X)\) and \(G^k(X)\) are closed subsets [13, Chapter IV]. The following lemma will be useful.

Lemma 1.11

For \( A,B\in G(X) \) set

$$\begin{aligned} \delta (A,B):=\sup _{a\in S_{A}}d(a,B). \end{aligned}$$

Then the following holds:

  1. (i)

    \( d_{H}(A,B)\leqslant 2\max \lbrace \delta (A,B),\delta (B,A)\rbrace \).

  2. (ii)

    If \( A , B\in G_{k}(X) \) with \( d(A,B)<\frac{1}{k} \) for some \(k \in \mathbb {N}\), we have

    $$\begin{aligned} \delta (B,A)\leqslant \frac{k\delta (A,B)}{1- k\delta (A,B)}. \end{aligned}$$

Proof

[2, Lemma 2.6]. \(\square \)

Proposition 1.12

Assume \(\mu _1 > -\infty \). Fix \(\omega \in \tilde{\Omega }\). For every \( n\in \mathbb {Z} \), let \( H_{\sigma ^{n}\omega }^{n}\subset F_{\mu _{1}}(\sigma ^{n}\omega )\) be a complementary subspace for \( F_{\mu _{2}}(\omega ) \) satisfying (1.4). Set \( \tilde{H}^{n}_{\omega }:=\psi ^{n}_{\sigma ^{n}\omega }(H^{n}_{\sigma ^{n}\omega }) \). Then the sequence \( \lbrace \tilde{H}^{n}_{\omega }\rbrace _{n\geqslant 1} \) is Cauchy in \(\big (G_{m_{1}}(F_{\mu _{1}}(\omega )),d_{H}\big ) \).

Proof

From (1.4), we can deduce that for every \( n\in \mathbb {N}\) and \( \xi _{\sigma ^{n}\omega }\in S_{H^{n}_{\sigma ^n\omega }}\),

$$\begin{aligned} \frac{1}{M_{1}} < \Vert [\xi _{\sigma ^{n}\omega }]_{\mu _{2}}\Vert \le 1. \end{aligned}$$
(1.12)

Note that \(\psi ^{k}_{\sigma ^{n}\omega } \vert _{H_{\sigma ^{n}\omega }^{n}}\) is injective for any \(k \ge 1\), therefore \({\text {dim}}( \tilde{H}^{n}_{\omega }) = {\text {dim}}( H_{\sigma ^{n}\omega }^{n})= m_1\). Since \( \mu _{2} < \mu _{1} \), we know that \(\tilde{H}^{n}_{\omega } \cap F_{\mu _{2}}(\omega ) = \{0\}\) and since \( {\text {dim}}[\frac{F_{\mu _{1}}(\omega )}{F_{\mu _{2}}(\omega )}] =m_{1} \), we obtain that

$$\begin{aligned} \tilde{H}^{n}_{\omega }\oplus F_{\mu _{2}}(\omega )=F_{\mu _{1}}(\omega ) \end{aligned}$$

for any \(n \in \mathbb {N}\). Let \(\lbrace \xi ^{j}_{\sigma ^{n}\omega }\rbrace _{1\leqslant j\leqslant m_{1}} \subset S_{F_{\mu _{1}}(\sigma ^{n}\omega )} \) be a base for \( H^{n}_{\sigma ^{n}\omega }\). Then for \( \xi _{\sigma ^{n+1}\omega }\in S_{F_{\mu _{1}}(\sigma ^{n+1}\omega )} \cap H^{n+1}_{\sigma ^{n+1}\omega }\), there exist \( \lbrace \beta _{j}\rbrace _{1\leqslant j\leqslant m_{1}}\subset \mathbb {R} \) such that

$$\begin{aligned} Z^{n}_{\omega } := \frac{\psi ^{n+1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })}{\Vert \psi ^{n+1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })\Vert }-\sum _{1\leqslant j\leqslant m_{1}}\beta _{j}\frac{\psi ^{n}_{\sigma ^{n}\omega }(\xi _{\sigma ^{n}\omega }^{j})}{\Vert \psi ^{n}_{\sigma ^{n}\omega }(\xi _{\sigma ^{n}\omega }^{j})\Vert }\in F_{\mu _{2}}(\omega ). \end{aligned}$$

It follows that

$$\begin{aligned} Y^{n}_{\sigma ^{n}\omega } := \frac{\psi ^{1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })}{\Vert \psi ^{n+1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })\Vert }-\sum _{1\leqslant j\leqslant m_{1}}\beta _{j}\frac{\xi _{\sigma ^{n}\omega }^{j}}{\Vert \psi ^{n}_{\sigma ^{n}\omega }(\xi _{\sigma ^{n}\omega }^{j})\Vert }\in F_{\mu _{2}}(\sigma ^{n}\omega ), \end{aligned}$$

thus

$$\begin{aligned} \big \Vert \sum _{1\leqslant j\leqslant m_{1}}\beta _{j}\frac{\xi _{\sigma ^{n}\omega }^{j}}{\Vert \psi ^{n}_{\sigma ^{n}\omega }(\xi _{\sigma ^{n}\omega }^{j})\Vert }\big \Vert&\leqslant \Vert \Pi _{H^{n}_{\sigma ^{n}\omega }||F_{\mu _{2}}(\sigma ^{n}\omega )}\Vert \frac{\Vert \psi ^{1}_{\sigma ^{n+1}\omega }\Vert }{\Vert \psi ^{n+1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })\Vert }\\&\leqslant M_{1} \frac{ \Vert \psi ^{1}_{\sigma ^{n+1}}\Vert }{\Vert \psi ^{n+1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })\Vert } \end{aligned}$$

and so

$$\begin{aligned} d\bigg (\frac{\psi ^{n+1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })}{\Vert \psi ^{n+1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })\Vert },\tilde{H}^{n}_{\omega }\bigg )&\leqslant \Vert Z^{n}_{\omega } \Vert = \Vert \psi ^{n}_{\sigma ^{n}\omega }(Y^{n}_{\sigma ^{n}\omega })\Vert \nonumber \\&\leqslant \ (M_{1}+1)\frac{\Vert \psi ^{n}_{\sigma ^{n}\omega }|_{F_{\mu _{2}}(\sigma ^n \omega )}\Vert \Vert \psi ^{1}_{\sigma ^{n+1}\omega }\Vert }{\Vert \psi ^{n+1}_{\sigma ^{n+1}\omega }(\xi _{\sigma ^{n+1}\omega })\Vert }. \end{aligned}$$
(1.13)

Note that \( \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \psi ^{1}_{\sigma ^{n}\omega }\Vert =0 \) from Birkhoff’s Ergodic Theorem. Using Lemma 1.7 and (1.7) for \(k = 1\), we have

$$\begin{aligned} \limsup _{n \rightarrow \infty } \frac{1}{n} \log \Vert \psi ^{n}_{\sigma ^{n}\omega }|_{F_{\mu _{2}}(\sigma ^n \omega )}\Vert \le \mu _{2}. \end{aligned}$$

From Lemma 1.9 the estimate 1.12 and Lemma 1.11, (1.13) implies that for \( \epsilon >0 \) small and large n,

$$\begin{aligned} d_{H}\big (\tilde{H}^{n}_\omega ,\tilde{H}^{n+1}_{\omega }\big )< M \exp \big (n(\mu _{2}-\mu _{1}+\epsilon )\big ) \end{aligned}$$

for a constant \(M > 0\). The claim is proved. \(\square \)

Next, we collect some facts about the limit of the sequence above.

Lemma 1.13

Assume \( \tilde{H}^{n}_{\omega }\xrightarrow {d_{H}}\tilde{H}_{\omega } \). Then the following holds:

  1. (i)

    \( \tilde{H}_{\omega } \) is invariant, i.e. \(\psi _{\omega }^k (\tilde{H}_{\omega }) = \tilde{H}_{\theta ^k \omega }\) for any \(k \ge 0\).

  2. (ii)

    \(\tilde{H}_{\omega }\cap F_{\mu _{2}}(\omega ) = \lbrace 0\rbrace \).

  3. (iii)

    \( \tilde{H}_{\omega } \) only depends on \( \omega \). In particular, it does not depend on the choice of the sequence \(\{ \tilde{H}^{n}_{\omega } \}_{n \ge 1}\).

Proof

By construction, \( \tilde{H}_{\omega } \) is invariant. We proceed with (ii). Consider the dual map

$$\begin{aligned} \big (\psi ^{n}_{\sigma ^{n}\omega }\big )^{*}_{\mu _{1}} :\big (F_{\mu _{1}}(\omega )\big )^{*}\rightarrow \big (F_{\mu _{1}}(\sigma ^{n}\omega )\big )^{*}. \end{aligned}$$

It is straightforward to see that \( \big (\psi ^{n}_{\sigma ^{n}\omega }\big )^{*}_{\mu _{1}} \) enjoys the cocycle property. From (1.5) and [12, Proposition 4.15], we can find a closed subspace \( G^{*}_{\mu _{2}}(\omega )\subset \big (F_{\mu _{1}}(\omega )\big )^{*} \) such that \( {\text {dim}} [ (F_{\mu _{1}}(\omega ))^{*} / G^{*}_{\mu _{2}}(\omega ) ] = m_{1} \) and for \( \xi ^{*}_{\omega }\in G^{*}_{\mu _{2}}(\omega ) \), \( \limsup _{n\rightarrow \infty }\frac{1}{n}\log \big \Vert \big (\psi ^{n}_{\sigma ^{n}\omega }\big )^{*}_{\mu _{1}}( \xi ^{*}_{\omega }) \big \Vert \leqslant \mu _{2}\). Set

$$\begin{aligned} \big (F_{\mu _{2}}(\omega )\big )_{\mu _{1}}^{\perp }=\big \lbrace \xi ^{*}_{\omega }\in \big (F_{\mu _{1}}(\omega )\big )^{*}\ : \ \xi ^{*}_{\omega }|_{F_{\mu _{2}}(\omega )} =0\big \rbrace . \end{aligned}$$

By Hahn–Banach separation theorem,

$$\begin{aligned} {\text {dim}} \left[ \big (F_{\mu _{2}}(\omega )\big )_{\mu _{1}}^{\perp } \right] = {\text {dim}} \left[ F_{\mu _1}(\omega ) / F_{\mu _{2}}(\omega ) \right] = m_{1}. \end{aligned}$$

Let \( \xi ^{*}_{\omega }\in \big (F_{\mu _{2}}(\omega )\big )_{\mu _{1}}^{\perp }\cap G^{*}_{\mu _{2}}(\omega )\) and assume that \( \xi ^{*}_{\omega }\ne 0 \). Then for some \( \xi _{\omega }\notin F_{\mu _{1}}(\omega ){\setminus } F_{\mu _{2}}(\omega ) \), \(\langle \xi ^{*}_{\omega },\xi _{\omega } \rangle = 1 \). Using surjectivity of \([\psi ^n_{\sigma ^n \omega }]_{\mu _{2}}\), for every \( n\in \mathbb {N} \), we can find \( \xi _{\sigma ^{n}\omega }\in H^{n}_{\sigma ^{n}\omega } \) such that

$$\begin{aligned} \psi ^{n}_{\sigma ^{n}\omega }(\xi _{\sigma ^{n}\omega })=\xi _{\omega }\ \ \ \text {mod } F_{\mu _{2}}(\omega ). \end{aligned}$$

Consequently, \(\langle (\psi ^{n}_{\sigma ^{n}\omega })_{\mu _{1}}^{*}(\xi ^{*}_{\omega }),\xi _{\sigma ^{n}\omega } \rangle =1 \). From Lemma 1.9 ,

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \big \Vert \big [\psi ^{n}_{\sigma ^{n}\omega }(\frac{\xi _{\sigma ^{n}\omega }}{\Vert [\xi _{\sigma ^{n}\omega }]_{\mu _{2}}\Vert })\big ]_{\mu _{2}}\big \Vert =\lim _{n\rightarrow \infty }\frac{1}{n}\log \big \Vert \frac{\Vert [\xi _{\omega }]_{\mu _{2}}\Vert }{\Vert [\xi _{\sigma ^{n}\omega }]_{\mu _{2}}\Vert }\big \Vert =\mu _{1}. \end{aligned}$$
(1.14)

Hence for \( \epsilon >0 \) and large n,

$$\begin{aligned} \Vert [\xi _{\sigma ^{n}\omega }]_{\mu _{2}}\Vert < \exp (-n\big (\mu _{1}-\epsilon )\big ) \end{aligned}$$

which is a contradiction since \(\Vert (\psi ^{n}_{\sigma ^{n}\omega })_{\mu _{1}}^{*}(\xi ^{*}_{\omega })\Vert \leqslant \exp \big (n(\mu _{2}+\epsilon )\big )\). Thus we have shown that

$$\begin{aligned} \big (F_{\mu _{1}}(\omega )\big )^{*}=\big (F_{\mu _{2}}(\omega )\big )_{\mu _{1}}^{\perp }\oplus G^{*}_{\mu _{2}}(\omega ). \end{aligned}$$
(1.15)

Now let \( \xi _{\omega }\in \tilde{H}_{\omega }\cap F_{\mu _{2}}(\omega )\) and assume that \( \Vert \xi _{\omega }\Vert =1 \). From 1.15, we can find \( \xi _{\omega }^{*}\in G^{*}_{\mu _{2}}(\omega ) \) such that \( \langle \xi ^{*}_{\omega } ,\xi _{\omega } \rangle =1 \). By definition of \(\tilde{H}_{\omega }\), there exist \( \xi ^{n}_{\sigma ^{n}\omega }\in S_{H^{n}_{\sigma ^{n}\omega }} \) such that \( \frac{\psi ^{n}_{\sigma ^{n}\omega }(\xi ^{n}_{\sigma ^{n}\omega })}{\Vert \psi ^{n}_{\sigma ^{n}\omega }(\xi ^{n}_{\sigma ^{n}\omega })\Vert }\rightarrow \xi _{\omega } \) as \(n \rightarrow \infty \), and consequently

$$\begin{aligned} \langle \xi ^{*}_{\omega },\frac{\psi ^{n}_{\sigma ^{n}\omega }(\xi ^{n}_{\sigma ^{n}\omega })}{\Vert \psi ^{n}_{\sigma ^{n}\omega }(\xi ^{n}_{\sigma ^{n}\omega })\Vert } \rangle = \langle (\psi ^{n}_{\sigma ^{n}\omega })^{*}(\xi ^{*}_{\omega }),\frac{\xi ^{n}_{\sigma ^{n}\omega }}{\Vert \psi ^{n}_{\sigma ^{n}\omega }(\xi ^{n}_{\sigma ^{n}\omega })\Vert } \rangle \rightarrow 1 \end{aligned}$$

as \(n \rightarrow \infty \). With Lemma 1.9 and a similar argument as above, this is again a contradiction and we have shown (ii). It remains to prove (iii). For \( \xi _{\omega }\in \tilde{H}_{\omega }\subset (F_{\mu _{1}}(\omega ))^{**} \), \( \xi ^{*}_{\omega }\in G_{\mu _{2}}^{*}(\omega ) \) and a sequence \( \xi ^{n}_{\sigma ^{n}\omega }\) chosen as above,

$$\begin{aligned} \langle \frac{\psi ^{n}_{\sigma ^{n}\omega }(\xi ^{n}_{\sigma ^{n}\omega })}{\Vert \psi ^{n}_{\sigma ^{n}\omega }(\xi ^{n}_{\sigma ^{n}\omega })\Vert },\xi ^{*}_{\omega } \rangle \rightarrow 0 \end{aligned}$$

as \(n \rightarrow \infty \). Therefore, \(\tilde{H}_{\omega }\subset \big (G_{\mu _{2}}^{*}(\omega )\big )_{\mu _{1}}^{\perp }=\big \lbrace \xi ^{**}_{\omega }\in \big (F_{\mu _{1}}(\omega )\big )^{**}\ : \ \xi ^{**}_{\omega }|_{G_{\mu _{2}}^{*}(\omega )} =0\big \rbrace \) and since \({\text {dim}} \big [ \big (G_{\mu _{2}}^{*}(\omega )\big )_{\mu _{1}}^{\perp }\big ] =m_{1} \), we obtain \(\tilde{H}_{\omega }=\big (G_{\mu _{2}}^{*}(\omega )\big )_{\mu _{1}}^{\perp }\) which proves (iii). \(\square \)

Combining Proposition 1.12 and Lemma 1.13, we see that if \(\mu _1 > -\infty \), there is a \(\theta \)-invariant set \(\tilde{\Omega } \subset \Omega \) of full measure such that for every \(\omega \in \tilde{\Omega }\), there is an \(m_1\)-dimensional subspace \(H^1_\omega \) with the properties

  • \(\psi ^k_{\omega }(H^1_{\omega }) = H^1_{\theta ^k \omega }\) for every \(k \ge 0\) and

  • \(H_{\omega }^1 \oplus F_{\mu _{2}}(\omega ) = F_{\mu _1}(\omega )\).

Thanks to the following lemma, we can invoke an induction argument to deduce the existence of a sequence of invariant spaces \(H^i_\omega \), \(i \ge 1\).

Lemma 1.14

The family of Banach spaces \(\lbrace F_{\mu _{2}}(\omega )\rbrace _{\omega \in \tilde{\Omega }} \) is a measurable field of Banach spaces with

$$\begin{aligned} \tilde{\Delta } = \lbrace \tilde{g} := \Pi _{F_{\mu _{2}} || H^{1}} \circ g , \ g\in \Delta \rbrace \quad \text {and} \quad \tilde{\Delta }_0 = \lbrace \tilde{g} := \Pi _{F_{\mu _{2}} || H^{1}} \circ g , \ g\in \Delta _0 \rbrace . \end{aligned}$$

In addition, \( \psi _{\omega }|_{F_{\mu _{2}}(\omega )}: F_{\mu _{2}}(\omega )\rightarrow F_{\mu _{2}}(\theta \omega ) \) is a linear compact cocycle satisfying Assumption 1.1 with \(\Delta \) replaced by \(\tilde{\Delta }\). Moreover, the maps

$$\begin{aligned} f_{1}(\omega ) := \Vert \Pi _{{H}^{1}_{\omega }||F_{\mu _{2}}(\omega )}\Vert \ \ \text {and} \ \ f_{2}(\omega ) := \Vert \Pi _{F_{\mu _{2}}(\omega )||{H}^{1}_{\omega }}\Vert \end{aligned}$$

are measurable.

Proof

The only non-trivial part in proving that \(\lbrace F_{\mu _{2}}(\omega )\rbrace _{\omega \in \tilde{\Omega }} \) is a measurable field of Banach spaces is to show that

$$\begin{aligned} \omega \mapsto \Vert \Pi _{F_{\mu _{2}}(\omega ) || H_{\omega }^{1}}(g(\omega )) \Vert \end{aligned}$$
(1.16)

is measurable for every \(g \in \Delta \). Let

$$\begin{aligned} \{g_i\, :\, i \in \mathbb {N}\} = \Delta _0 \quad \text {and} \quad \{ (g_{k_1}, \ldots , g_{k_{m_1}})\, :\, k \in \mathbb {N}\} = \Delta _0^{m_1}. \end{aligned}$$

Fix \(n \in \mathbb {N}\) and \(\omega \in \tilde{\Omega }\). We define \( \lbrace U^{k}_{\sigma ^{n}\omega }\rbrace _{k\geqslant 1} \) to be the family of subspaces of \( E_{\sigma ^{n}\omega } \) given by \( U^{k}_{\sigma ^{n}\omega } = \langle g_{k_{i}}(\sigma ^{n}\omega ) \rangle _{1\leqslant i\leqslant m_{1}, g_{k_{i}}\in \Delta _{0}} \). Using the same technique as in Lemma 1.5, one can show that the map

$$\begin{aligned} \omega \mapsto G_{k}(\sigma ^{n}\omega )= {\left\{ \begin{array}{ll} \Vert \Pi _{U^{k}_{\sigma ^{n}\omega }||F_{\mu _{2}}(\sigma ^{n}\omega )}\Vert &{}\quad \ U^{k}_{\sigma ^{n}\omega }\oplus F_{\mu _{2}}(\sigma ^{n}\omega ) = F_{\mu _{1}}(\sigma ^{n}\omega )\\ \infty &{}\quad \ \text {otherwise} \end{array}\right. } \end{aligned}$$

is measurable. Set \( \psi _{n}(\omega ):=\inf \lbrace k: G_{k}(\sigma ^{n}\omega )<M_{1}\rbrace \) with \(M_1\) as in Lemma 1.6. This map is clearly measurable. By Proposition 1.12, \( \tilde{H}^{n}_{\omega } := \psi ^{n}_{\sigma ^{n}\omega }\big (U^{\psi _{n}(\omega )}_{\sigma ^{n}\omega }\big ) \xrightarrow {d_{H}}{H}_{\omega }^{1}\) and consequently

$$\begin{aligned} \Pi _{\tilde{H}^{n}_{\omega }||F_{\mu _{2}}(\omega )}\rightarrow \Pi _{H^{1}_{\omega }||F_{\mu _{2}}(\omega )} \quad \text { as } n \rightarrow \infty . \end{aligned}$$
(1.17)

Let \( g\in \Delta \). Then we have a decomposition of the form

$$\begin{aligned} \Pi _{\tilde{H}^{n}_{\omega }||F_{\mu _2}(\omega )}g({\omega }) = \sum _{1\leqslant t\leqslant m_{1}}\alpha _{t}(\omega ) \psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _t(\omega )}(\sigma ^n \omega )) \end{aligned}$$

where \(\iota _1, \ldots , \iota _{m_1} :\Omega \rightarrow \mathbb {N}\) are measurable. We assume \( m_{1}=1 \) first. To ease notation, set \(\iota := \iota _1\). Since \(g(\omega ) - \alpha _1(\omega ) \psi ^n_{\sigma ^n \omega }(g_{\iota (\omega )}(\sigma ^n \omega )) \in F_{\mu _2}(\omega )\), we have \(\Vert [g(\omega )]_{\mu _2} \Vert = |\alpha _1(\omega )| \Vert [\psi ^n_{\sigma ^n \omega }(g_{\iota (\omega )}(\sigma ^n \omega ))]\Vert \) and therefore

$$\begin{aligned} |\alpha _1(\omega )| = \frac{d\big (g(\omega ),F_{\mu _{2}}(\omega )\big )}{d\big (\psi _{\sigma ^{n}\omega }(g_{\iota (\omega )}(\sigma ^{n}\omega )),F_{\mu _{2}}(\omega )\big )}. \end{aligned}$$

Set

$$\begin{aligned} d_{0}(\omega ) := d\big (g(\omega ),F_{\mu _{2}}(\omega )\big ) \quad \text {and} \quad d_{1}(\omega ) := d\big (\psi _{\sigma ^{n}\omega }(g_{\iota (\omega )}(\sigma ^{n}\omega )),F_{\mu _{2}}(\omega )\big ). \end{aligned}$$

From Lemma 1.4, we know that \(d_0\) is measurable. Furthermore, a slight adaptation of the proof yields the measurability of \(\omega \mapsto d\big (\psi _{\sigma ^{n}\omega }(g_{k}(\sigma ^{n}\omega )),F_{\mu _{2}}(\omega )\big )\) for any fixed \(k \in \mathbb {N}\). Since \(\iota \) is measurable, this implies the measurability of \(d_1\), too. We have

$$\begin{aligned} \Pi _{\tilde{H}^{n}_{\omega }||F_{\mu _2}(\omega )}g({\omega }) = G(\omega ) \frac{d_0(\omega )}{d_1(\omega )} \psi ^{n}_{\sigma ^{n}\omega }(g_{\iota (\omega )}(\sigma ^n \omega )) \end{aligned}$$

where \(G(\omega )\) takes values in \(\{-1,0,1\}\). Set \( h_{0}(\omega ) := g(\omega )-\frac{d_{0}(\omega )}{d_{1}(\omega )}\psi ^{n}_{\sigma ^{n}\omega }(g_{\iota (\omega )}(\sigma ^{n}\omega )) \) and \(h_{1}(\omega ) := g(\omega )+\frac{d_{0}(\omega )}{d_{1}(\omega )}\psi ^{n}_{\sigma ^{n}\omega }(g_{\iota (\omega )}(\sigma ^{n}\omega ))\) and define

$$\begin{aligned} J_{0}(\omega ):=\lim _{m\rightarrow \infty }\frac{1}{m}\log \big \Vert \psi ^{m}_{\omega }\big (h_{0}(\omega )\big )\big \Vert , \ \ \ \ \ J_{1}(\omega ):=\lim _{m\rightarrow \infty }\frac{1}{m}\log \big \Vert \psi ^{m}_{\omega }\big (h_{1}(\omega )\big )\big \Vert . \end{aligned}$$

It follows that \( J_{0} \) and \( J_{1}\) are measurable and that

$$\begin{aligned} \Pi _{\tilde{H}^{n}_{\omega }||F_{\mu _2}(\omega )}g({\omega }) = (1 - \chi _{\{ g(\omega ) \in F_{\mu _2}(\omega )\}}) \left[ g(\omega ) - \chi _{\mu _{2}} \big (J_0(\omega )\big )h_0(\omega ) - \chi _{\mu _{2}}\big (J_{1}(\omega )\big ) h_{1}(\omega ) \right] . \end{aligned}$$
(1.18)

Then (1.18) and (1.17) prove the measurability of (1.16) for every \(g \in \Delta \) in the case \(m_1 = 1\). Furthermore, measurability of \(f_1\) and \(f_2\) and Assumption 1.1 for \(\tilde{\Delta }\) can also be deduced. It remains to consider the case \( m_{1}>1 \) for which we invoke the same technique: Let

$$\begin{aligned}&d_{0}(\omega )=d\big (g(\omega ),F_{\mu _{2}}(\omega )\oplus \langle \psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _t(\omega )}(\sigma ^{n}\omega ))\rangle _{{2\leqslant t\leqslant m_{1}}}\big ) , \\&d_{1}(\omega ) = d\big (\psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _1(\omega )}(\sigma ^{n}\omega )),F_{\mu _{2}}(\omega )\oplus \langle \psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _t(\omega )}(\sigma ^{n}\omega ))\rangle _{{2\leqslant t\leqslant m_{1}}}\big ). \end{aligned}$$

For \( h_{0}(\omega )=g(\omega )-\frac{d_{0}(\omega )}{d_{1}(\omega )}\psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _1(\omega )}(\sigma ^{n}\omega )) \) and \( h_{1}(\omega ) = g(\omega ) + \frac{d_{0}(\omega )}{d_{1}(\omega )}\psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _1(\omega )}(\sigma ^{n}\omega )) \) let

$$\begin{aligned}&d_{i0}(\omega ) := d\big (h_{i}(\omega ),F_{\mu _{2}}(\omega )\oplus \langle \psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _t(\omega )}(\sigma ^{n}\omega ))\rangle _{{3\leqslant t\leqslant m_{1}}}\big ), \ i\in \lbrace 0,1\rbrace \\&d_{01}(\omega ) = d_{11}(\omega ) = d\big (\psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _2(\omega )}(\sigma ^{n}\omega )),F_{\mu _{2}}(\omega )\oplus \langle \psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _t(\omega )}(\sigma ^{n}\omega ))\rangle _{{3\leqslant t\leqslant m_{1}}}\big ). \end{aligned}$$

For \( i\in \lbrace 0,1\rbrace \) define

$$\begin{aligned} h_{0,i}&=h_{0}(\omega )+(-1)^{i+1}\frac{d_{00}(\omega )}{d_{01}(\omega )}\psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _2(\omega )}(\sigma ^{n}\omega ))\\ h_{1,i}&=h_{1}(\omega )+(-1)^{i+1}\frac{d_{10}(\omega )}{d_{11}(\omega )}\psi ^{n}_{\sigma ^{n}\omega }(g_{\iota _2(\omega )}(\sigma ^{n}\omega )). \end{aligned}$$

We repeat the same procedure with our four new functions. Iterating this, we end up with \( 2^{m_1} \) functions \( \lbrace I_{t}(\omega )\rbrace _{1\leqslant t\leqslant 2^{m_{1}}} \) for which we define \( J_{t}(\omega ) := \lim _{m\rightarrow \infty }\frac{1}{m}\log \big \Vert \psi ^{m}_{\omega }(I_{t}(\omega ))\big \Vert \). Since

$$\begin{aligned} \Pi _{\tilde{H}^{n}_{\omega }||F_{\mu _2}(\omega )}g({\omega }) = (1 - \chi _{\{ g(\omega ) \in F_{\mu _2}(\omega )\}}) \left[ g(\omega )-\sum _{0\leqslant t\leqslant 2^{m_{1}}}\chi _{\mu _{2}}\big (J_{t}(\omega )\big )I_{t}(\omega ) \right] , \end{aligned}$$

our claim follows for arbitrary \(m_1\). \(\square \)

Proposition 1.15

Let \(i \in \mathbb {N}\) and assume \(\mu _i > \infty \). Then there is a \(\theta \)-invariant set of full measure \(\tilde{\Omega }\) such that for every \(\omega \in \tilde{\Omega }\), there is an \(m_i\)-dimensional space \(H^i_{\omega }\) with the properties

  1. (1)

    \(\psi ^k_{\omega }(H^i_{\omega }) = H^i_{\theta ^k \omega }\) for every \(k \ge 0\) and

  2. (2)

    \(H_{\omega }^i \oplus F_{\mu _{i+1}}(\omega ) = F_{\mu _{i}}(\omega )\).

Proof

For \(i = 1\), the statement follows from Proposition 1.12 and Lemma 1.13. For \(i = 2\), we consider the restricted cocycle \(\psi ^k_{\omega }|_{F_{\mu _{2}}(\omega )}\). From Lemma 1.14, we know that this cocycle acts on the measurable field of Banach spaces \(\{F_{\mu _{2}}(\omega )\}_{\omega \in \Omega }\) and we can thus apply Proposition 1.12 and Lemma 1.13 to this cocycle again. It remains to make sure that the top Lyapunov exponent of the restricted cocycle coincides with \(\mu _2\). This, however, was deduced in Lemma 1.8. We can now repeat the argument until we reach i. \(\square \)

From now on, \(H_{\omega }^i\) will always denote the spaces deduced in Proposition 1.15.

Remark 1.16

Using identities of the form

$$\begin{aligned} \Pi _{F_{\mu _{j}}(\omega )||\oplus _{l \leqslant i < j}H^{i}_{\omega }}=\Pi _{F_{\mu _{j}}(\omega )||H^{j-1}_{\omega }}\circ \Pi _{F_{\mu _{j-1}}(\omega )||H^{j-2}_{\omega }}\circ \cdots \circ \Pi _{F_{\mu _{l+1}}(\omega )||H^{l}_{\omega }}, \end{aligned}$$

we can use the same strategy as in Lemma 1.14 to see that for each \( 1 \le l \le j \) and \(k \ge 0\),

$$\begin{aligned} f_{1}(\omega )&:= \big \Vert \Pi _{\oplus _{l \leqslant i< j}H_{\omega }^{i}\oplus F_{\mu _{j}}(\omega )}\big \Vert , \ f_{2}(\omega ) := \big \Vert \Pi _{F_{\mu _{j}}(\omega )||\oplus _{l \leqslant i< j}H^{i}_{\omega }}\big \Vert \ \text {and} \ f_{3}(\omega )\\&:= \Vert \psi ^{k}_{\omega }|_{\oplus _{l \leqslant i < j}H^{i}_{\omega }}\Vert \end{aligned}$$

are measurable.

Lemma 1.17

For a measurable and non-negative function \(f :\Omega \rightarrow \mathbb {R} \)

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}f(\theta ^{n}\omega )=0 \text { a.s.}\ \ \ \text {if and only if} \ \ \ \lim _{n\rightarrow \infty }\frac{1}{n}f(\sigma ^{n}\omega )=0 \text { a.s.} \end{aligned}$$

Proof

The main idea is due to Jack Feldman, cf. [16, Lemma 7.2]. Assume that \( \lim _{n\rightarrow \infty }\frac{1}{n}f(\theta ^{n}\omega )=0 \) on a set of full measure \(\Omega ^0\). Let \(\epsilon > 0\) and set

$$\begin{aligned} \Omega _{n}:=\lbrace \omega \in \Omega ^0 \, :\, \forall i\geqslant n \ \ \frac{f(\theta ^{i}\omega )}{i}\leqslant \epsilon \rbrace . \end{aligned}$$

Fom our assumptions, for some \( n_{0} \in \mathbb {N}\),

$$\begin{aligned} \mathbb {P}(\Omega _{n_{0}})>\frac{9}{10}. \end{aligned}$$

From Birkhoff’s ergodic theorem, there is a set of full measure \(\Omega ^1\) such that for every \( \omega \in \Omega ^1\), we can find \( m_{0}=m_{\omega } \) such that for \( m\geqslant m_{0} \),

$$\begin{aligned} \frac{1}{m}\sum _{0\leqslant j\leqslant m}\chi _{\Omega _{n_{0}}}(\sigma ^{j}\omega ) > \frac{9}{10}. \end{aligned}$$
(1.19)

W.l.o.g., we may assume that \(\Omega ^0 = \Omega ^1\). Now for \( k\geqslant \max \lbrace 3n_{0}, m_{0}\rbrace \), set \( m=\lfloor \frac{5}{3}k\rfloor +1\). Then from (1.19)

$$\begin{aligned} \frac{1}{m}\big [\sum _{0\leqslant j\leqslant \frac{4m}{5}}\chi _{\Omega _{n_{0}}}(\sigma ^{j}\omega )+\sum _{\frac{4m}{5}<j\leqslant m}\chi _{\Omega _{n_{0}}}(\sigma ^{j}\omega )\big ]>\frac{9}{10}. \end{aligned}$$

Consequently, there exists \( \frac{4m}{5}<j\leqslant m \) such that \( \sigma ^{j}\omega \in \Omega _{n_{0}} \). Set \( i := j-k >n_{0}\). Then by the definition of \( \Omega _{n_{0}} \),

$$\begin{aligned} \frac{f(\theta ^{i}\sigma ^{j}\omega )}{i}=\frac{f(\sigma ^{k}\omega )}{j-k}\leqslant \epsilon . \end{aligned}$$

Since \( j-k \le \frac{2}{3}k + 1 \) and \( \epsilon \) is arbitrary, our claim is shown. The other direction can be proved similarly. \(\square \)

As a consequence, we obtain the following:

Lemma 1.18

For each \( 1 \le l \le j \) and \(\omega \in \tilde{\Omega }\),

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \Pi _{\oplus _{l\leqslant i<j}H^{i}_{\theta ^{n}\omega }||F_{\mu _{j}}(\theta ^{n}\omega )}\Vert =\lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \Pi _{\oplus _{l\leqslant i<j}H^{i}_{\sigma ^{n}\omega }||F_{\mu _{j}}(\sigma ^{n}\omega )}\Vert =0. \end{aligned}$$
(1.20)

Proof

Follows from a straightforward generalization of [12, Lemma 4.4] and Lemma 1.17. \(\square \)

The following lemma characterizes the spaces \(H^{i}_{\omega }\) as ‘fast-growing’ subspaces.

Proposition 1.19

For \(\omega \in \tilde{\Omega }\), every \( i\geqslant N \) and \( \xi _{\omega }\in H^{i}_{\omega }{\setminus }\lbrace 0\rbrace \),

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \psi ^{n}_{\omega }(\xi _{\omega }) \Vert = \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \psi ^{n}_{\omega } |_{H^{i}_{\omega }}\Vert = \mu _{i} \end{aligned}$$
(1.21)

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi _{\omega })\Vert = \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert (\psi ^{n}_{\sigma ^{n}\omega }|_{H^{i}_{\omega }} )^{-1} \Vert = -\mu _{i}. \end{aligned}$$
(1.22)

Proof

The equalities (1.21) follow by applying the Multiplicative Ergodic Theorem [12, Theorem 4.17] to the map \(\psi ^n_\omega \vert _{H^i_{\omega }} :H^i_{\omega } \rightarrow H^i_{\theta ^n \omega }\). It remains to prove (1.22). By definition, for every \( \xi _{\omega }\in H^{i}_{\omega } \),

$$\begin{aligned} \frac{\Vert (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi _{\omega })\Vert }{\Vert [\xi _{\omega }]_{\mu _{i+1}}\Vert }&\times \frac{\big \Vert \big [\psi ^{n}_{\sigma ^{n}\omega }\big ((\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi _{\omega }) \big ) \big ]_{\mu _{i+1}}\big \Vert }{\Vert [(\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi _{\omega })]_{\mu _{i+1}}\Vert } = \frac{\Vert (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi _{\omega })\Vert }{\Vert [(\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi _{\omega })]_{\mu _{i+1}}\Vert }\\&\leqslant \Vert \Pi _{H^{i}_{\sigma ^{n}\omega }||F_{\mu _{i+1}}(\sigma ^{n}\omega )}\Vert . \end{aligned}$$

From Lemma 1.9,

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\bar{\xi }_{\sigma ^{n}\omega }\in H^{i}_{\sigma ^{n}\omega }}\frac{\Vert [\psi ^{n}_{\sigma ^{n}\omega }(\bar{\xi }_{\sigma ^{n}\omega })]_{\mu _{i+1}}\Vert }{\Vert [\bar{\xi }_{\sigma ^{n}\omega }]_{\mu _{i+1}}\Vert } = \lim _{n\rightarrow \infty }\frac{1}{n} \frac{\Vert [\psi ^{n}_{\sigma ^{n}\omega }(\hat{\xi }_{\sigma ^{n}\omega })]_{\mu _{i+1}}\Vert }{\Vert [\hat{\xi }_{\sigma ^{n}\omega }]_{\mu _{i+1}}\Vert } = \mu _{i} \end{aligned}$$

where \(\hat{\xi }_{\sigma ^{n}\omega } \in H^i_{\sigma ^n \omega }\) is chosen such that

$$\begin{aligned} \frac{\Vert [\psi ^{n}_{\sigma ^{n}\omega }(\hat{\xi }_{\sigma ^{n}\omega })]_{\mu _{i+1}}\Vert }{\Vert [\hat{\xi }_{\sigma ^{n}\omega }]_{\mu _{i+1}}\Vert } = \min _{\bar{\xi }_{\sigma ^{n}\omega }\in H^{i}_{\sigma ^{n}\omega }} \frac{\Vert [\psi ^{n}_{\sigma ^{n}\omega }(\bar{\xi }_{\sigma ^{n}\omega })]_{\mu _{i+1}}\Vert }{\Vert [\bar{\xi }_{\sigma ^{n}\omega }]_{\mu _{i+1}}\Vert }. \end{aligned}$$

Consequently, from (1.20),

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \Vert (\psi ^{n}_{\sigma ^{n}\omega } |_{H^{i}_{\omega }} )^{-1}\Vert \leqslant -\mu _{i} \end{aligned}$$

Finally, from inequality \( \Vert \xi _{\omega }\Vert \leqslant \Vert \psi ^{n}_{\sigma ^{n}\omega }|_{H^{i}_{\sigma ^{n}\omega }}\Vert \Vert (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi _{\omega })\Vert \), Lemma 1.7 and (1.21), the equalities (1.22) can be deduced. \(\square \)

Lemma 1.20

Let \(\omega \in \tilde{\Omega }\) and \(i < k\). For every \(i \le j < k\), let \(\{\xi ^t_{\omega } \}_{t \in I_j}\) be a basis of \(H^j_{\omega }\). Set \(I := \cup _{i \le j < k} I_j\) and assume \(\xi ^t_{\omega } \in H^j_{\omega }\). Then

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{1}{n} \log d (\psi ^n_{\omega }(\xi ^t_{\omega }), \langle \psi ^n_{\omega }(\xi ^{t'}_{\omega }) \rangle _{t' \in I{\setminus } \{t\}} ) = \mu _j \end{aligned}$$
(1.23)

and

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{1}{n} \log d ((\psi ^n_{\sigma ^n \omega })^{-1} (\xi ^t_{\omega }), \langle (\psi ^n_{\sigma ^n \omega })^{-1} (\xi ^{t'}_{\omega }) \rangle _{t' \in I{\setminus } \{t\}} ) = - \mu _j. \end{aligned}$$
(1.24)

Proof

We will prove (1.24) only, the proof for (1.23) is completely analogous. First, we claim that the statement is true for \(j = i\) and \(k = i+1\). Indeed, in this case we have the inequalities

$$\begin{aligned} \frac{1}{\Vert \psi ^{n}_{\sigma ^{n}\omega }|_{H^{i}_{\sigma ^{n}\omega }}\Vert }\leqslant \frac{d\big ((\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{t}_{\omega }),\langle (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{t^{\prime }}_{\omega })\rangle _{t^{\prime }\in I{\setminus }\lbrace t\rbrace }\big )}{d\big (\xi ^{t}_{\omega },\langle \xi ^{t^{\prime }}_{\omega }\rangle _{t^{\prime }\in I{\setminus }\lbrace t\rbrace }\big )}\leqslant \Vert (\psi ^{n}_{\sigma ^{n}\omega })^{-1}|_{H^{i}_{\omega }}\Vert \end{aligned}$$

and we can conclude with Proposition 1.19. For arbitrary k and \(j = i\), we can use the inequalities

$$\begin{aligned} 1\leqslant \frac{d\big ((\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{t}_{\omega }),\langle (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{t^{\prime }}_{\omega })\rangle _{t^{\prime }\in I_{i}{\setminus }\lbrace t\rbrace }\big )}{d\big ((\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{t}_{\omega }),\langle (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{t^{\prime }}_{\omega })\rangle _{t^{\prime }\in I{\setminus }\lbrace t\rbrace }\big )}\leqslant \Vert \Pi _{H^{i}_{\sigma ^{n}\omega }||F_{\mu _{i+1}}(\sigma ^{n}\omega )}\Vert , \end{aligned}$$

Lemma 1.18 and our previous result above. The definition of \({\text {Vol}}\) allows to deduce that

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{n} \log {\text {Vol}}\big (\big ((\psi ^{n}_{\sigma ^n \omega })^{-1}(\xi ^{t}_{\omega })\big )_{t\in I_{k-1}},\ldots ,\big ((\psi ^{n}_{\sigma ^n \omega })^{-1}(\xi ^{t}_{\omega })\big )_{t\in I_{i}}\big )=\sum _{i\leqslant j<k}-\mu _{j}\vert I_{j}\vert . \end{aligned}$$
(1.25)

Since \({\text {Vol}}\) is symmetric up to a constant, the claim (1.24) follows for arbitrary j. \(\square \)

The following theorem is the announced semi-invertible Oseledets theorem on fields of Banach spaces. It summarizes the main result of this section.

Theorem 1.21

There is a \(\theta \)-invariant set of full measure \(\tilde{\Omega }\) such that for every \(i \ge 1\) with \(\mu _i > \mu _{i+1}\) and \(\omega \in \tilde{\Omega }\), there is an \(m_i\)-dimensional subspace \(H^i_\omega \) with the following properties:

  1. (i)

    (Invariance) \(\psi _{\omega }^k(H^i_{\omega }) = H^i_{\theta ^k \omega }\) for every \(k \ge 0\).

  2. (ii)

    (Splitting) \(H_{\omega }^i \oplus F_{\mu _{i+1}}(\omega ) = F_{\mu _i}(\omega )\). In particular,

    $$\begin{aligned} E_{\omega } = H^1_{\omega } \oplus \cdots \oplus H^i_{\omega } \oplus F_{\mu _{i+1}}(\omega ). \end{aligned}$$
  3. (iii)

    (‘Fast-growing’ subspace I) For each \( h_{\omega }\in H^{i}_{\omega }{\setminus }\lbrace 0\rbrace \),

    $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert \psi ^{n}_{\omega }(h_{\omega })\Vert = \mu _{i}. \end{aligned}$$
  4. (iv)

    (‘Fast-growing’ subspace II) For each \( h_{\omega }\in H^{i}_{\omega } {\setminus }\lbrace 0\rbrace \),

    $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(h_{\omega })\Vert =-\mu _{i} . \end{aligned}$$
  5. (v)

    If \(\{\xi ^{t}_{\omega } \}_{1\leqslant t\leqslant m}\) is a basis of \( \oplus _{1\leqslant i\leqslant j}H^{i}_{\omega }\), then

    $$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n}\log {\text {Vol}}\big (\psi ^{n}_{\omega }(\xi ^{1}_{\omega }),\ldots ,\psi ^{n}_{\omega }(\xi ^{m}_{\omega })\big ) = \sum _{1\leqslant i\leqslant j}m_{i}\mu _{i} \quad \text {and} \nonumber \\&\lim _{n\rightarrow \infty }\frac{1}{n}\log {\text {Vol}}\big ((\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{1}_{\omega }),\ldots ,(\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{m}_{\omega })\big )=\sum _{1\leqslant i\leqslant j}-m_{i}\mu _{i}. \end{aligned}$$
    (1.26)

Moreover, the properties (i)–(iv) uniquely determine the spaces \(H^i_\omega \).

Proof

Properties (i) and (ii) are proven in Proposition 1.15. (iii) and (iv) are shown in Proposition 1.19 and (v) can be deduced from Lemma 1.20, using the definition of \({\text {Vol}}\) and symmetry modulo a constant of this function. It remains to prove the uniqueness statement. Fix \(i \ge 1\) and assume \(\mu _i > \mu _{i+1}\). We define \(G^*_{\mu _{i+1}}(\omega )\) and \(\big (G^{*}_{\mu _{i+1}}(\omega )\big )_{\mu _{i}}^{\perp }\) as in Lemma 1.13 and claim that

$$\begin{aligned} H^{i}_{\omega } = \big (G^{*}_{\mu _{i+1}}(\omega )\big )_{\mu _{i}}^{\perp }. \end{aligned}$$
(1.27)

Let \(h_{\omega } \in H^i_{\omega }\), \( h^{*}_{\omega }\in G^{*}_{\mu _{i+1}}(\omega ) \) and set \( h_{\sigma ^{n}\omega } := (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(h_{\omega }) \). Property (iv) implies that there is an \(\epsilon > 0\) sufficiently small such that

$$\begin{aligned} \langle h_{\omega }, h^{*}_{\omega } \rangle = \langle \psi ^{n}_{\sigma ^{n}\omega }(h_{\sigma ^{n}\omega }),h^{*}_{\omega }\rangle =\langle h_{\sigma ^{n}\omega },(\psi ^{n}_{\sigma ^{n}\omega })^{*}(h^{*}_{\omega }) \rangle \leqslant \exp \big (-n(\mu _{i}-\mu _{i+1}-\epsilon )\big )\rightarrow 0 \end{aligned}$$

as \(n \rightarrow \infty \) which reveals \( H^{i}_{\omega }\subset \big (G^{*}_{\mu _{i+1}}(\omega )\big )_{\mu _{i}}^{\perp } \). Finally, since these spaces have the same dimension, (1.27) follows. \(\square \)

Remark 1.22

Property (iv) seems to be new in the context of Banach spaces. As seen in the proof, it is crucial for the uniqueness statement

3 Invariant Manifolds

Let \( \lbrace E_{\omega }\rbrace _{\omega \in \Omega } \) be a measurable field of Banach spaces and \( \varphi ^{n}_{\omega } \) a nonlinear cocycle on acting on it, i.e.

$$\begin{aligned}&\varphi ^{n}_{\omega } :E_{\omega } \rightarrow E_{\theta ^{n}\omega }\\&\varphi ^{n+m}_{\omega }(.) = \varphi ^{n}_{\theta ^{m}\omega }\big (\varphi ^{m}_{\omega }(.)\big ). \end{aligned}$$

Definition 2.1

We say that \( \varphi ^{n}_{\omega } \) admits a stationary solution if there exists a map \(Y:\Omega \longrightarrow \prod _{\omega \in \Omega }E_{\omega }\) such that

  1. (i)

    \(Y_{\omega }\in E_{\omega }\),

  2. (ii)

    \(\varphi ^n_\omega (Y_{\omega })=Y_{\theta ^{n}\omega }\) and

  3. (iii)

    \(\omega \rightarrow \Vert Y_{\omega }\Vert \) is measurable.

Stationary solutions should be thought of random analogues to fixed points in (deterministic) dynamical systems. If \( \varphi ^{n}_{\omega }\) is Fréchet differentiable, one can easily check that the derivative around a stationary solution also enjoys the cocycle property, i.e for \( \psi ^n_\omega (.) = D_{Y_{\omega }}\varphi ^{n}_{\omega }(.) \), one has

$$\begin{aligned} \psi ^{n+m}_{\omega }(.)=\psi ^{n}_{\theta ^{m}\omega }\big (\psi ^m_\omega (.)\big ). \end{aligned}$$

In the following, we will assume that \(\varphi \) is Fréchet differentiable, that there exists a stationary solution Y and that the linearized cocycle \(\psi \) around Y is compact and satisfies Assumption 1.1. Furthermore, we will assume that

$$\begin{aligned} \log ^{+}\Vert \psi _{\omega } \Vert \in L^{1}(\Omega ). \end{aligned}$$

Therefore, we can apply the MET to \(\psi \). In the following, we will use the same notation as in the previous section.

3.1 Stable Manifolds

Definition 2.2

Let Y be a stationary solution, let \(\lbrace \cdots<\mu _{j}< \mu _{j-1}<\cdots <\mu _{1} \rbrace \in [-\infty ,\infty )\) be the corresponding Lyapunov spectrum and \(\tilde{\Omega }\) the \(\theta \)-invariant set on which the MET holds. Set \( \mu _{j_{0}} = \max \lbrace \mu _{j} : \mu _{j} < 0 \rbrace \) and \( \mu _{j_{0}} = -\infty \) if all finite \( \mu _{j} \) are nonnegative. We define the stable subspace

$$\begin{aligned} S_{\omega } := F_{\mu _{j_{0}}}(\omega ). \end{aligned}$$

By the unstable subspace we mean

$$\begin{aligned} U_{\omega } := \oplus _{1\leqslant i < j_{0}} H^{i}_{\omega }. \end{aligned}$$

Note that \(\dim [E_{\omega }/S_{\omega }] = \dim [ U_{\omega }] =: k<\infty \) for every \(\omega \in \tilde{\Omega }\).

Lemma 2.3

For \(\omega \in \tilde{\Omega }\) and \(\epsilon \in (0,-\mu _{j_0})\), set

$$\begin{aligned} F(\omega ) := \sup _{p\geqslant 0}\exp [-p(\mu _{j_{0}} + \epsilon )] \Vert \psi ^p_\omega |_{S_{\omega }}\Vert . \end{aligned}$$

Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\log ^{+}\big [F(\theta ^{n}\omega ){]} = 0. \end{aligned}$$
(2.1)

Proof

Follows from (1.7). \(\square \)

Lemma 2.4

Let \( \omega \in \tilde{\Omega } \), \( U_{\omega } = \langle \xi ^{t}_{\omega }\rangle _{1\leqslant t\leqslant k} \) and \(n, p \ge 0\). Then

$$\begin{aligned} \Vert [\psi ^{n}_{\theta ^{p}\omega }]^{-1}\Vert _{L[{U_{\theta ^{n+p}\omega }},U_{\theta ^{p}\omega }]}\leqslant \sum _{1\leqslant t\leqslant k}\frac{\Vert \psi ^{p}_{\omega }(\xi ^{t}_{\omega })\Vert }{\Vert \psi ^{n+p}_{\omega }(\xi ^{t}_{\omega })\Vert }\times \frac{\Vert \psi _{\omega }^{n+p}(\xi ^{t}_{\omega })\Vert }{d\big (\psi _{\omega }^{n+p}(\xi ^{t}_{\omega }), \langle \psi _{\omega }^{n+p}(\xi ^{t^{\prime }}_{\omega }) \rangle _{t^{\prime }\ne t}\big )} \end{aligned}$$
(2.2)

and

$$\begin{aligned} \Vert [\psi ^{p}_{\sigma ^{n}\omega }]^{-1}\Vert _{L[{U_{\sigma ^{n-p}\omega }},U_{\sigma ^{n}\omega }]}&\leqslant \sum _{1\leqslant t\leqslant k}\frac{\Vert (\psi ^{n}_{\sigma ^{n}\omega })^{-1}(\xi ^{t}_{\omega })\Vert }{ \Vert (\psi ^{n-p}_{\sigma ^{n-p}\omega })^{-1}(\xi ^{t}_{\omega })\Vert }\nonumber \\&\times \frac{ \Vert (\psi ^{n-p}_{\sigma ^{n-p}\omega })^{-1}(\xi ^{t}_{\omega })\Vert }{d\big ((\psi ^{n-p}_{\sigma ^{n-p}\omega })^{-1}(\xi ^{t}_{\omega }),\langle (\psi ^{n-p}_{\sigma ^{n-p}(\omega )})^{-1}(\xi ^{t^{\prime }}_{\omega })\rangle _{t^{\prime }\ne t}\big )}. \end{aligned}$$
(2.3)

Proof

Choose \( u\in U_{\theta ^{p}\omega } \) and assume that \(u = \sum _{1\leqslant t\leqslant k}u^{t}\frac{\psi ^{p}_{\omega }(\xi ^{t}_{\omega })}{\Vert \psi ^{p}_{\omega }(\xi ^{t}_{\omega })\Vert } \). Then

$$\begin{aligned} \frac{\vert u^{t}\vert }{\Vert u\Vert }\leqslant \frac{\Vert \psi _{\omega }^{p}(\xi ^{t}_{\omega })\Vert }{d\big (\psi _{\omega }^{p}(\xi ^{t}_{\omega }),\langle \psi _{\omega }^{p}(\xi ^{t^{\prime }}_{\omega })\rangle _{t^{\prime }\ne t}\big )}. \end{aligned}$$
(2.4)

From \( \psi ^{n}_{\theta ^{p}\omega }u = \sum _{1\leqslant t\leqslant k}u^{t} \frac{\Vert \psi ^{n+p}_{\omega }(\xi ^{t}_{\omega })\Vert }{\Vert \psi ^{p}_{\omega }(\xi ^{t}_{\omega })\Vert }\frac{\psi ^{n+p}_{\omega }(\xi ^{t}_{\omega })}{\Vert \psi ^{n+p}_{\omega }(\xi ^{t}_{\omega })\Vert }\) and (2.4),

$$\begin{aligned} \frac{\vert u^{t}\vert }{\Vert \psi ^{n}_{\theta ^{p}\omega }u\Vert }\leqslant \frac{\Vert \psi ^{p}_{\omega }(\xi ^{t}_{\omega })\Vert }{\Vert \psi ^{n+p}_{\omega }(\xi ^{t}_{\omega })\Vert } \times \frac{\Vert \psi _{\omega }^{n+p}(\xi ^{t}_{\omega })\Vert }{d\big (\psi _{\omega }^{n+p}(\xi ^{t}_{\omega }) ,\langle \psi _{\omega }^{n+p}(\xi ^{t^{\prime }}_{\omega })\rangle _{t^{\prime }\ne t}\big )} \end{aligned}$$

and (2.2) follows. The estimate (2.3) is proven similarly. \(\square \)

Definition 2.5

For \(\omega \in \Omega \) set \( \Sigma _\omega :=\prod _{j\geqslant 0}E_{\theta ^{j}\omega } \). For \( \upsilon >0 \) we define

$$\begin{aligned} \Sigma ^{\upsilon }_{\omega }:=\bigg \lbrace \Gamma \in \Sigma _{\omega } :\Vert \Gamma \Vert =\sup _{j\geqslant 0}\big [\Vert \Pi _{\omega }^{j}\Gamma \Vert \exp (\upsilon j)\big ]<\infty \bigg \rbrace \end{aligned}$$

where \( \Pi _{\omega }^{j}:\prod _{i\geqslant 0} E_{\theta ^{i}\omega }\rightarrow E_{\theta ^{j}\omega } \) denotes the projection map.

One can check that \( \Sigma _{\omega }^{\upsilon } \) is a Banach space.

Lemma 2.6

Let \(\omega \in {\Omega }\) and \(0< \upsilon < - \mu _{j_0}\). Define

$$\begin{aligned} P_{\omega } : E_{\omega }&\rightarrow E_{\theta \omega }\\ \xi _{\omega }&\mapsto \varphi ^{1}_{\omega } (Y_{\omega }+\xi _{\omega })-\varphi ^{1}_{\omega }(Y_{\omega })-\psi ^{1}_{\omega }(\xi _{\omega }). \end{aligned}$$

Let \(\rho :\Omega \rightarrow \mathbb {R}^+\) be a random variable with the property that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{1}{n}\log \rho (\theta ^{n}\omega ) \ge 0 \end{aligned}$$

almost surely. Assume that for \( \Vert \xi _{\omega }\Vert , \Vert \tilde{\xi }_{\omega }\Vert <\rho (\omega ) \),

$$\begin{aligned} \Vert P_{\omega } (\xi _{\omega }) - P_{\omega } (\tilde{\xi }_{\omega })\Vert \leqslant \Vert \xi _{\omega }-\tilde{\xi }_{\omega }\Vert f(\omega ) h{(}\Vert \xi _{\omega }\Vert +\Vert \tilde{\xi }_{\omega }\Vert {)} \end{aligned}$$
(2.5)

almost surely where \( f:\Omega \rightarrow \mathbb {R}^{+} \) is a measurable function such that \(\lim _{n\rightarrow \infty }\frac{1}{n}\log ^{+}f(\theta ^{n}\omega ) = 0\) almost surely and \( h(x)=x^{r}g(x)\) for some \(r > 0\) where \( g:\mathbb {R}\rightarrow \mathbb {R}^{+} \) is an increasing \( C^{1} \) function. Set

$$\begin{aligned} \tilde{\rho }(\omega ) := \inf _{n\geqslant 0}\exp (n\upsilon )\rho (\theta ^{n}\omega ). \end{aligned}$$
(2.6)

Then the map

$$\begin{aligned}&I_{_{\omega }} :S_{\omega }\times \Sigma _{\omega }^{\upsilon }\cap B(0,\tilde{\rho }(\omega ))\rightarrow \Sigma _{\omega }^{\upsilon }, \\&{\Pi }^{n}_{\omega }\big [I_{\omega }(v_{\omega } ,\Gamma )\big ] \\&\quad = {\left\{ \begin{array}{ll} \psi ^{n}_{\omega }(v_{\omega }) + \sum _{0\leqslant j\leqslant n-1}\big [\psi ^{n-1-j}_{\theta ^{1+j}\omega }\circ \Pi _{ S_{\theta ^{1+j}\omega }\parallel U_{\theta ^{1+j}\omega }}\big ]P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big ) &{}\text {}\\ \qquad - \sum _{j\geqslant n}\big [[\psi ^{j-n+1}_{\theta ^{n}\omega }]^{-1}\circ \Pi _{U_{\theta ^{1+j}\omega }\parallel S_{\theta ^{1+j}\omega }}\big ] P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big ) &{}\text {for } n \ge 1,\\ v_{\omega }- \sum _{j\geqslant 0}\big [[\psi ^{j+1}_{\omega }]^{-1}\circ \Pi _{U_{\theta ^{1+j}\omega }\parallel S_{\theta ^{1+j}\omega }}\big ] P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big ) &{}\text {for } n = 0. \end{array}\right. } \end{aligned}$$

is well-defined on a \(\theta \)-invariant set of full measure \(\tilde{\Omega }\).

Proof

We collect some estimates first. Let \(\epsilon \in (0, -\mu _{j_0})\). From (1.20), we can find a random variable \( R(\omega )>1 \) such that for \( j\geqslant 0 \),

$$\begin{aligned} \Vert \Pi _{U_{\theta ^{j}\omega }\parallel S_{\theta ^{j}\omega }}\Vert \leqslant R(\omega )\exp (\epsilon j) \ ,\ \ \ \ \ \Vert \Pi _{ S_{\theta ^{j}\omega }\parallel U_{\theta ^{j}\omega }}\Vert \leqslant R(\omega )\exp (\epsilon j). \end{aligned}$$
(2.7)

Also from (2.1), for \( n,p\geqslant 0 \),

$$\begin{aligned} \Vert \psi ^{p}_{\theta ^{n}\omega }|_{S_{\theta ^{n}\omega }}\Vert \leqslant R(\omega )\exp \big (p\mu _{j_{0}}+\epsilon (n+p)\big ). \end{aligned}$$
(2.8)

In addition, from (1.23) and (2.2) for \( n, p \geqslant 0 \),

$$\begin{aligned} \Vert [\psi ^{n}_{\theta ^{p}\omega }]^{-1}\Vert _{L[{U_{\theta ^{n+p} \omega }},U_{\theta ^{p}\omega }]}\leqslant R(\omega ) \exp \big (\epsilon (n+p)\big ) \exp (-n\mu _{j_{0}-1}). \end{aligned}$$
(2.9)

From our assumptions,

$$\begin{aligned} \big \Vert P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big )\big \Vert \leqslant \big \Vert \Pi ^{j}_{\omega }[\Gamma ]\big \Vert ^{1+r}\big [f(\theta ^{j}\omega ) g(\Vert \Pi ^{j}_{\omega }[\Gamma ]\Vert )\big ]. \end{aligned}$$

So for \( j\geqslant 0 \) and a random variable \( \tilde{R}({\omega })>1 \),

$$\begin{aligned} \big \Vert P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big )\big \Vert \leqslant \tilde{R}({\omega })\big \Vert \Pi ^{j}_{\omega }[\Gamma ]\big \Vert ^{1+r}g(\Vert \Pi ^{j}_{\omega }[\Gamma ]\Vert ) \exp (\epsilon j). \end{aligned}$$
(2.10)

Now from (2.7), (2.8), (2.9) and (2.10), we obtain

$$\begin{aligned}&\big \Vert {\Pi }^{n}_{\omega }\big [I_{\omega }(v_{\omega } ,\Gamma )\big ]\big \Vert \leqslant R(\omega )\bigg [\exp ((\mu _{j_{0}}+\epsilon )n)\Vert v_{\omega }\Vert + \\&\sum _{0\leqslant j\leqslant n-1}R(\omega ) \tilde{R}(\omega )\exp \big (\epsilon n+2\epsilon (1+j)+(n-1-j)\mu _{j_{0}} \big )\Vert \Pi ^{j}_{\omega }(\Gamma )\Vert ^{1+r}g({\Vert } \Pi ^{j}_{\omega }[\Gamma ]\Vert ) + \\&\sum _{j\geqslant n}R(\omega )\tilde{R}(\omega )\exp \big (3\epsilon (1+j)-(j-n+1)\mu _{j_{0}-1}\big )\Vert \Pi ^{j}_{\omega }(\Gamma )\Vert ^{1+r}g({\Vert } \Pi ^{j}_{\omega }[\Gamma ]\Vert )\bigg ]. \end{aligned}$$

Since g is increasing,

$$\begin{aligned}&\big \Vert {\Pi }^{n}_{\omega }\big [I_{\omega }(v_{\omega } ,\Gamma )\big ]\big \Vert \leqslant R(\omega )\bigg [\exp \big ((\mu _{j_{0}}+\epsilon )n\big ).\Vert v_{\omega }\Vert + \\&R(\omega )\tilde{R}(\omega )\Vert \Gamma \Vert ^{1+r}_{\Sigma ^{\upsilon }_{\omega }}g(\Vert \Gamma \Vert _{\Sigma ^{\upsilon }_{\omega }})\exp \big (\epsilon n+2\epsilon +(n-1)\mu _{j_{0}}\big )\sum _{0\leqslant j\leqslant n-1}\exp \big (j\big (2\epsilon -\mu _{j_{0}}-(1+r)\upsilon \big )\big )+\\&R(\omega )\tilde{R}(\omega )\Vert \Gamma \Vert ^{1+r}_{\Sigma ^{\upsilon }_{\omega }}g(\Vert \Gamma \Vert _{\Sigma ^{\upsilon }_{\omega }})\exp \big (3\epsilon +(n-1)\mu _{j_{0}-1}\big )\sum _{j\geqslant n}\exp \big (j\big (3\epsilon -\mu _{j_{0}-1}-(1+r)\upsilon \big )\big )\bigg ]. \end{aligned}$$

Since \(\mu _{j_{0}-1} \geqslant 0 \) and \( 0<\upsilon <-\mu _{j_{0}} \), we can choose \(\epsilon > 0\) smaller if necessary to see that

$$\begin{aligned} \sup _{n\geqslant 0}\bigg [\big \Vert {\Pi }^{n}_{\omega }\big [I_{\omega }(v_{\omega } ,\Gamma )\big ]\big \Vert \exp (\upsilon n)\bigg ]<\infty . \end{aligned}$$

As a result, \( I_{{\omega }} \) is well-defined . \(\square \)

Lemma 2.7

With the same setting as in Lemma 2.6, for \(\Gamma \in \Sigma ^{\upsilon }_{\omega } \cap B(0,\tilde{\rho }(\omega ))\),

$$\begin{aligned} I_{{\omega }}[v_{\omega },\Gamma ]=\Gamma \ \ \ \ {\Longleftrightarrow }\ \ \ \ \forall j\geqslant 0: \Pi ^{j}_{\omega }[\Gamma ]=\varphi ^{j}_{\omega }(Y_{\omega }+\xi _{\omega })-\varphi ^{j}_{\omega }(Y_{\omega }) \end{aligned}$$
(2.11)

where

$$\begin{aligned} \xi _{\omega }=v_{\omega }-\sum _{j\geqslant 0}\big [[\psi ^{j+1}_{\omega }]^{-1}\circ \Pi _{U_{\theta ^{1+j}\omega }\parallel S_{\theta ^{1+j}\omega }}\big ] P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big ). \end{aligned}$$
(2.12)

Proof

The strategy of the proof is similar to [17, Lemma VI.5]. Let \( I_{\omega }[v_{\omega },\Gamma ]=\Gamma \). Then \( \xi _{\omega }=\Pi ^{0}_{\omega }[\Gamma ] \) and the claim is shown for \(j = 0\). We proceed by induction. Assume that \( \Pi ^{n}_{\omega }[\Gamma ]=\varphi ^{n}_{\omega }(Y_{\omega }+\xi _{\omega })-\varphi ^{n}_{\omega }(Y_{\omega }) \). By definition,

$$\begin{aligned}&\varphi ^{n+1}_{\omega }(Y_{\omega }+\xi _{\omega })-\varphi ^{n+1}_{\omega }(Y_{\omega })=\varphi ^{1}_{\theta ^{n}\omega }\big (\varphi ^{n}_{\omega }(Y_{\omega }+\xi _{\omega })\big )-\varphi ^{1}_{\theta ^{n}\omega }(Y_{\theta ^{n}\omega })=\\&P_{\theta ^{n}\omega }\big (\varphi ^{n}_{\omega }(Y_{\omega }+\xi _{\omega })-Y_{\theta ^{n}\omega }\big )\\&+ \psi ^{1}_{\theta ^{n}\omega }\big (\varphi ^{n}_{\omega }(Y_{\omega }+\xi _{\omega })-Y_{\theta ^{n}\omega }\big ) = P_{\theta ^{n}\omega }(\Pi ^{n}_{\omega }[\Gamma ]) + \psi ^{1}_{\theta ^{n}\omega }\big (\Pi ^{n}_{\omega }\big [I_{\omega }(v_{\omega } ,\Gamma )\big ]\big ). \end{aligned}$$

Note that for \( j\geqslant n\),

$$\begin{aligned} \psi ^{1}_{\theta ^{n}\omega }\circ [\psi ^{j-n+1}_{\theta ^{n}\omega }]^{-1}=[\psi ^{j-n}_{\theta ^{n+1}\omega }]^{-1}:U_{\theta ^{1+j}\omega }\rightarrow U_{\theta ^{1+n}\omega }. \end{aligned}$$

By definition

$$\begin{aligned}&\psi ^{1}_{\theta ^{n}\omega }\big ( \Pi ^n_{\omega } [I_{\omega }(v_{\omega } ,\Gamma )] \big )=\psi ^{n+1}_{\omega }(v_{\omega })+\sum _{0\leqslant j\leqslant n-1}\big [\psi ^{n-j}_{\theta ^{1+j}\omega }\circ \Pi _{ S_{\theta ^{1+j}\omega }\parallel U_{\theta ^{1+j}\omega }}\big ]P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big )-\\&\sum _{j\geqslant n}\big [[\psi ^{j-n}_{\theta ^{n}\omega }]^{-1}\circ \Pi _{U_{\theta ^{1+j}\omega }\parallel S_{\theta ^{1+j}\omega }}\big ] P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big ). \end{aligned}$$

Consequently, \(\Pi ^{n+1}_{\omega }[\Gamma ]=\varphi ^{n+1}_{\omega }(Y_{\omega }+\xi _{\omega })-\varphi ^{n+1}_{\omega }(Y_{\omega }) \) which finishes the induction step.

Conversely, for \(\xi _{\omega }\in E_{\omega }\) and \(\Gamma \in \Sigma ^{\nu }_{\omega } \cap B(0,\tilde{\rho }(\omega ))\), assume that for every \( j\geqslant 0 \), \( \Pi ^{j}_{\omega }[\Gamma ]=\varphi ^{j}_{\omega }(Y_{\omega }+\xi _{\omega })-\varphi ^{j}_{\omega }(Y_{\omega }) \). Set

$$\begin{aligned} v_{\omega }:=\xi _{\omega }+\sum _{j\geqslant 0}\big [[\psi ^{j+1}_{\omega }]^{-1}\circ \Pi _{U_{\theta ^{1+j}\omega }\parallel S_{\theta ^{1+j}\omega }}\big ] P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big ). \end{aligned}$$

Similar to Lemma 2.6, we can see that \( v_{\omega } \) is well-defined. Morever,

$$\begin{aligned} {\Pi }^{n}_{\omega }\big [I_{\omega }(v_{\omega } ,\Gamma )\big ]&=\psi ^{n}_{\omega }(\xi _{\omega })+\sum _{0\leqslant j\leqslant n-1}\psi ^{n-1-j}_{\theta ^{1+j}\omega }P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\Gamma ]\big )\\&=\varphi ^{j}_{\omega }(Y_{\omega }+\xi _{\omega })-\varphi ^{j}_{\omega }(Y_{\omega })=\Pi ^{j}_{\omega }[\Gamma ] \end{aligned}$$

which proves the claim. \(\square \)

Lemma 2.8

Under the same assumptions as in Lemma 2.7, set

$$\begin{aligned} h_{1}^{\upsilon }(\omega )&:= \sup _{n\geqslant 0}\big [\exp (n\upsilon )\Vert \psi ^{n}_{\omega }|_{S_{\omega }}\Vert \big ] \quad \text {and} \\ h_{2}^{\upsilon }(\omega )&:= \sup _{n\geqslant 0}\big [\exp (n\upsilon )\sum _{0\leqslant j\leqslant n-1}\exp (-j\upsilon (1+r))f(\theta ^{j}\omega )\Vert \psi ^{n-j}_{\theta ^{j+1}\omega }|_{S_{\theta ^{j+1}\omega }}\Vert \Vert \Pi _{S_{\theta ^{j+1}\omega }||U_{\theta ^{j+1}\omega }}\Vert \\&\quad + \exp (n\upsilon )\sum _{j\geqslant n}\exp (-j\upsilon (1+r))f(\theta ^{j}\omega )\Vert (\psi ^{j-n+1}_{\theta ^{n}\omega } |_{U_{\theta ^{j+1}}} )^{-1}\Vert \Vert \Pi _{U_{\theta ^{j+1}\omega }||S_{\theta ^{j+1}\omega }}\Vert \big ]. \end{aligned}$$

Then \( h_{1}^{\upsilon } \) and \( h_{2}^{\upsilon }\) are measurable and finite on a \(\theta \)-invariant set of full measure \(\tilde{\Omega }\). In addition,

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{1}{n} \log ^+ h^{\upsilon }_1(\theta ^n \omega ) = \lim _{n \rightarrow \infty } \frac{1}{n} \log ^+ h^{\upsilon }_2(\theta ^n \omega ) = 0 \end{aligned}$$

for every \(\omega \in \tilde{\Omega }\). Furthermore, the estimates

$$\begin{aligned} \Vert I_{\omega }(v_{\omega },\Gamma )\Vert&\leqslant h_{1}^{\upsilon }(\omega )\Vert v_{\omega }\Vert + h_{2}^{\upsilon }(\omega )\Vert \Gamma \Vert ^{1+r} g(\Vert \Gamma \Vert ) \quad \text {and} \\ \Vert I_{\omega }(v_{\omega },\Gamma )-I_{\omega }(v_{\omega },\tilde{\Gamma })\Vert&\leqslant h_{2}^{\upsilon }(\omega )h(\Vert \Gamma \Vert +\Vert \tilde{\Gamma }\Vert )\ \Vert \Gamma -\tilde{\Gamma }\Vert \end{aligned}$$

hold for every \(\omega \in \tilde{\Omega }\), \(\Gamma , \tilde{\Gamma } \in \Sigma ^{\upsilon }_{\omega } \cap B(0,\tilde{\rho }(\omega ))\) and \(v_{\omega } \in S_{\omega }\).

Proof

The statements about \(h_{1}^{\upsilon }\) and \(h_{2}^{\upsilon }\) follow from our assumption on f, (1.7), Lemma 1.8 and Proposition 1.19. The claimed estimates follow by definition of \(I_{\omega }\). \(\square \)

Recall that \( h(x)=x^{r}g(x) \). In particular, h is invertible and h and \(h^{-1}\) are strictly increasing.

Lemma 2.9

Assume that for \(v_{\omega } \in S_{\omega }\),

$$\begin{aligned} \Vert v_{\omega }\Vert \leqslant \frac{1}{2h_{1}^{\upsilon }(\omega )}\min \big \lbrace \frac{1}{2} h^{-1}(\frac{1}{2h_{2}^{\upsilon }(\omega )}),\tilde{\rho }(\omega )\big \rbrace . \end{aligned}$$

Then the equation

$$\begin{aligned} I_{\omega }(v_{\omega },\Gamma )=\Gamma \end{aligned}$$

admits a uniques solution \(\Gamma = \Gamma (v_{\omega })\) and the bound

$$\begin{aligned} \Vert \Gamma (v_{\omega })\Vert \leqslant \min \big \lbrace \frac{1}{2}h^{-1}(\frac{1}{2h_{2}^{\upsilon }(\omega )}), \tilde{\rho }(\omega )\big \rbrace =: H^{\upsilon }_{1}(\omega ) \end{aligned}$$
(2.13)

holds true.

Proof

We can use the estimates provided in Lemma 2.8 to conclude that \(I(v_{\omega },\cdot )\) is a contraction on the closed ball with radius \(\min \big \lbrace \frac{1}{2}h^{-1}(\frac{1}{2h_{2}^{\upsilon }(\omega )}), \tilde{\rho }(\omega )\big \rbrace \). \(\square \)

Now we can formulate the main theorem about the existence of local stable manifolds.

Theorem 2.10

Let \((\Omega ,\mathcal {F},\mathbb {P},\theta )\) be an ergodic measure-preserving dynamical systems and \(\varphi \) a Fréchet-differentiable cocycle acting on a measurable field of Banach spaces \(\{E_{\omega }\}_{\omega \in \Omega }\). Assume that \(\varphi \) admits a stationary solution Y and that the linearized cocycle \(\psi \) around Y is compact, satisfies Assumption 1.1 and the integrability condition

$$\begin{aligned} \log ^+ \Vert \psi _{\omega } \Vert \in L^1(\omega ). \end{aligned}$$

Moreover, assume that (2.5) holds for \(\varphi \) and \(\psi \). Let \(\mu _{j_0} < 0\) and \(S_{\omega }\) be defined as in Definition 2.2. For \( 0< \upsilon < -\mu _{j_{0}} \), \(\omega \in {\Omega }\) and \( R^{\upsilon }(\omega ) :=\frac{1}{2h_{1}^{\upsilon }(\omega )}\min \big \lbrace \frac{1}{2}h^{-1}(\frac{1}{2h_{2}^{\upsilon }(\omega )}),\tilde{\rho }(\omega )\big \rbrace \) with \(\tilde{\rho }\) defined as in (2.6), let

$$\begin{aligned} S^{\upsilon }_{loc}(\omega ) := \big \lbrace Y_{\omega }+\Pi ^{0}_{\omega }[\Gamma (v_{\omega })], \ \ \Vert v_{\omega }\Vert < R^{\upsilon }(\omega )\big \rbrace . \end{aligned}$$
(2.14)

Then there is a \(\theta \)-invariant set of full measure \(\tilde{\Omega }\) on which the following properties are satisfied for every \(\omega \in \tilde{\Omega }\):

  1. (i)

    There are random variables \( \rho _{1}^{\upsilon }(\omega ), \rho _{2}^{\upsilon }(\omega )\), positive and finite on \(\tilde{\Omega }\), for which

    $$\begin{aligned} \liminf _{p \rightarrow \infty } \frac{1}{p} \log \rho _i^{\upsilon }(\theta ^p \omega ) \ge 0, \quad i = 1,2 \end{aligned}$$
    (2.15)

    and such that

    $$\begin{aligned}&\big \lbrace Z_{\omega } \in E_{\omega }\, :\, \sup _{n\geqslant 0}\exp (n\upsilon )\Vert \varphi ^{n}_{\omega }(Z_{\omega })-Y_{\theta ^{n}\omega }\Vert<\rho _{1}^{\upsilon }(\omega )\big \rbrace \subseteq S^{\upsilon }_{loc}(\omega )\\&\subseteq \big \lbrace Z_{\omega } \in E_{\omega }\, :\, \sup _{n\geqslant 0}\exp (n\upsilon )\Vert \varphi ^{n}_{\omega }(Z_{\omega })-Y_{\theta ^{n}\omega }\Vert <\rho _{2}^{\upsilon }({\omega })\big \rbrace . \end{aligned}$$
  2. (ii)

    \( S^{\upsilon }_{loc}(\omega ) \) of \( E_{\omega } \) and

    $$\begin{aligned} T_{Y_{\omega }}S^{\upsilon }_{loc}(\omega ) = S_{\omega }. \end{aligned}$$
  3. (iii)

    For \( n\geqslant N(\omega ) \),

    $$\begin{aligned} \varphi ^{n}_{\omega }(S^{\upsilon }_{loc}(\omega ))\subseteq S^{\upsilon }_{loc}(\theta ^{n}\omega ). \end{aligned}$$
  4. (iv)

    For \( 0<\upsilon _{1}\leqslant \upsilon _{2}< - \mu _{j_{0}} \),

    $$\begin{aligned} S^{\upsilon _{2}}_{loc}(\omega )\subseteq S^{\upsilon _{1}}_{loc}(\omega ). \end{aligned}$$

    Also for \(n\geqslant N(\omega ) \),

    $$\begin{aligned} \varphi ^{n}_{\omega }(S^{\upsilon _{1}}_{loc}(\omega ))\subseteq S^{\upsilon _{2}}_{loc}(\theta ^{n}(\omega )) \end{aligned}$$

    and consequently for \( Z_{\omega }\in S^{\upsilon }_{loc}(\omega ) \),

    $$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \Vert \varphi ^{n}_{\omega }(Z_{\omega })-Y_{\theta ^{n}\omega }\Vert \leqslant \mu _{j_{0}}. \end{aligned}$$
    (2.16)
  5. (v)
    $$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \bigg [\sup \bigg \lbrace \frac{\Vert \varphi ^{n}_{\omega }(Z_{\omega })-\varphi ^{n}_{\omega }(\tilde{Z}_{\omega })\Vert }{\Vert Z_{\omega }-\tilde{Z}_{\omega }\Vert },\ \ Z_{\omega }\ne \tilde{Z}_{\omega },\ Z_{\omega },\tilde{Z}_{\omega }\in S^{\upsilon }_{loc}(\omega ) \bigg \rbrace \bigg ]\leqslant \mu _{j_{0}}. \end{aligned}$$

Proof

We start with (i). For the first inclusion, note that we can find a random variable \( \rho _1^{\upsilon }(\omega ) \) satisfying

$$\begin{aligned} \liminf _{p\rightarrow \infty }\frac{1}{p}\log \rho _{1}^{\upsilon }(\theta ^{p}\omega ) \geqslant 0 \end{aligned}$$
(2.17)

and such that whenever \(\Vert \Gamma \Vert \leqslant \rho ^{\upsilon }_{1}(\omega )\),

$$\begin{aligned} \Vert \Gamma \Vert +h_{2}^{\upsilon }(\omega )\Vert \Gamma \Vert ^{r+1} g(\Vert \Gamma \Vert )\leqslant \frac{1}{2h_{1}^{\upsilon }(\omega )}\min \big \lbrace \frac{1}{2} h^{-1}(\frac{1}{2h_{2}^{\upsilon }(\omega )}),\tilde{\rho }(\omega )\big \rbrace =: H^{\upsilon }_{2}(\omega ). \end{aligned}$$

For example, we can define

$$\begin{aligned} \rho _1^{\upsilon }(\omega ) :=\min \big \lbrace h^{-1}(\frac{1}{h_{2}^{\upsilon }(\omega )}), H^{\upsilon }_{2}(\omega ) / 2, H^{\upsilon }_{1}(\omega )\big \rbrace \end{aligned}$$

with \(H^{\upsilon }_{1}\) defined as in (2.13). Assume that \(Z_{\omega } \in E_{\omega }\) has the property that

$$\begin{aligned} \sup _{n\geqslant 0}\exp (n\upsilon )\Vert \varphi ^{n}_{\omega }(Z_{\omega }) - Y_{\theta ^{n}\omega } \Vert < \rho _1^{\upsilon }(\omega ). \end{aligned}$$

Setting

$$\begin{aligned} \tilde{v}_{\omega } := Z_{\omega } - Y_{\omega } + \sum _{j\geqslant 0}\big [[\psi ^{j+1}_{\omega }]^{-1}\circ \Pi _{U_{\theta ^{1+j}\omega }\parallel S_{\theta ^{1+j}\omega }}\big ] P_{\theta ^{j}\omega }\big (\Pi ^{j}_{\omega }[\tilde{\Gamma }]\big ), \end{aligned}$$

it follows that \(\Vert \tilde{v}_{\omega } \Vert < R^{\upsilon }(\omega )\). From Lemma 2.7, we conclude that \(I_{\omega }[\tilde{v}_{\omega },\tilde{\Gamma }] = \tilde{\Gamma }\). By uniqueness of the fixed point map, we have \(\tilde{\Gamma } = \Gamma (\tilde{v}_{\omega })\), therefore \(Z_{\omega } = Y_{\omega } + \Pi ^0_{\omega }(\Gamma (\tilde{v}_{\omega })) \in S^{\upsilon }_{loc}(\omega )\). Next, let \(Z_{\omega } \in S^{\upsilon }_{loc}(\omega )\), i.e. \(Z_{\omega } = Y_{\omega } + \Pi ^0_{\omega }(\Gamma ({v}_{\omega }))\) for some \(\Vert v_{\omega } \Vert < R^{\upsilon }(\omega )\). From Lemmas 2.7 and 2.9,

$$\begin{aligned} \Vert \Gamma (v_{\omega })\Vert = \sup _{n\geqslant 0}\exp (n\upsilon )\Vert \varphi ^{n}_{\omega }(Z_{\omega })-Y_{\theta ^{n}\omega } \Vert \leqslant R^{\upsilon }(\omega ). \end{aligned}$$

We can therefore choose \(\rho _{2}^{\upsilon }(\omega ) = R^{\upsilon }(\omega )\) and the second inclusion is shown.

The second item immediately follows from our definition for \( S^{\upsilon }_{loc}(\omega ) \).

For item (iii), by (2.15), we can find \( N(\omega ) \) such that for \( n\geqslant N(\omega ) \),

$$\begin{aligned} \exp (-n\upsilon )\rho _{2}^{\upsilon }(\omega )\leqslant \rho _{1}^{\upsilon }(\theta ^{n}\omega ). \end{aligned}$$

Now the claim follows from item (i).

For item (iv), note first that \( R^{\upsilon _{2}}(\omega )\leqslant R^{\upsilon _{1}}(\omega )\). By definition of \( \Gamma ^{\upsilon }_{\omega }(v_{\omega }) \), it immediately follows that

$$\begin{aligned} S^{\upsilon _{2}}_{loc}(\omega )\subseteq S^{\upsilon _{1}}_{loc}(\omega ). \end{aligned}$$

Now take \( Z_{\omega }\in S^{\upsilon _{1}}_{loc}(\omega ) \). From Lemma 1.18 and (i), we can find \( N(\omega ) \) such that for \( n\geqslant N(\omega ) \),

$$\begin{aligned} \Vert \Pi _{ S_{\theta ^{n}\omega }\parallel U_{\theta ^{n}\omega }}\big (\varphi ^{n}_{\omega }(Z_{\omega })-Y_{\theta ^{n}\omega }\big )\Vert < R^{\upsilon _{2}}(\theta ^{n}\omega ). \end{aligned}$$

We may also assume that \( \exp (-n\upsilon _{1})\rho _{2}^{\upsilon _{1}}(\omega )\leqslant \rho _{1}^{\upsilon _{1}}(\theta ^{n}\omega ) \) for \(n \ge N(\omega )\). For

$$\begin{aligned} v_{\theta ^{n}\omega } := \Pi _{ S_{\theta ^{n}\omega }\parallel U_{\theta ^{n}\omega }}\big (\varphi ^{n}_{\omega }(Z_{\omega })-Y_{\theta ^{n}\omega }\big ) \end{aligned}$$

let

$$\begin{aligned} Z_{\theta ^{n}\omega } := \Pi ^{0}_{\theta ^{n}\omega }(\Gamma (v_{\theta ^{n}\omega }))+Y_{\theta ^{n}\omega }\in S^{\upsilon _{2}}_{loc}(\theta ^{n}\omega )\subset S^{\upsilon _{1}}_{loc}(\theta ^{n}\omega ). \end{aligned}$$

We claim that \( Z_{\theta ^{n}\omega }=\varphi ^{n}_{\omega }(Z_{\omega }) \). Since \( Z_{\omega }\in S^{\upsilon _{1}}_{loc}(\omega ) \),

$$\begin{aligned} \sup _{j\geqslant 0}\exp (j\upsilon _{1})\Vert \varphi ^{j}_{\theta ^{n}\omega }(\varphi ^{n}_{\omega }(Z_{\omega }))-Y_{\theta ^{j}\theta ^{n}\omega }\Vert \leqslant \exp (-n\upsilon _{1})\rho _{2}^{\upsilon _{1}}(\omega )\leqslant \rho _{1}^{\upsilon _{1}}(\theta ^{n}\omega ). \end{aligned}$$

So from item (i), \( \varphi ^{n}_{\omega }(Z_{\omega })\in S^{\upsilon _{1}}_{loc}(\theta ^{n}\omega ) \). Remember \( Z_{\theta ^{n}\omega }\in S^{\upsilon _{1}}_{loc}(\theta ^{n}\omega )\cap S^{\upsilon _{2}}_{loc}(\theta ^{n}\omega ) \) and

$$\begin{aligned} \Pi _{S^{\theta ^{n}\omega }||U^{\theta ^{n}\omega }}(Z_{\theta ^{n}\omega }-Y_{\theta ^{n}\omega })=\Pi _{S^{\theta ^{n}\omega }||U^{\theta ^{n}\omega }}(\varphi ^{n}_{\omega }(Z_{\omega })-Y_{\theta ^{n}\omega }). \end{aligned}$$

So by uniqueness of the fixed point, we indeed have

$$\begin{aligned} \varphi ^{n}_{\omega }(Z_{\omega }) = Z_{\theta ^{n}\omega } \in S^{\upsilon _{2}}_{loc}(\theta ^{n}\omega ). \end{aligned}$$

To prove (2.16), let \(\upsilon \le \upsilon _2 < - \mu _0\) and take \(Z_{\omega } \in S^{\upsilon }_{loc}(\omega )\). Then we know that for large enough N, \(\varphi ^N_{\omega } (Z_{\omega }) \in S^{\upsilon _2}_{loc}(\theta ^N \omega )\), therefore

$$\begin{aligned} \sup _{j \ge 0} \exp (j \upsilon _2) \Vert \varphi ^{j+N}_{\omega }(Z_{\omega }) - Y_{\theta ^{j+N} \omega } \Vert < \infty \end{aligned}$$

and it follows that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \Vert \varphi ^{n}_{\omega }(Z_{\omega }) - Y_{\theta ^{n}\omega }\Vert \leqslant - \upsilon _2. \end{aligned}$$

We can choose \(\upsilon _2\) arbitrarily close to \(- \mu _0\), therefore the claim follows and item (iv) is proved.

For item (v), first by definition,

$$\begin{aligned} \Vert \Gamma (v_{\omega })-\Gamma (\tilde{v}_{\omega })\Vert&=\Vert I_{\omega }(v_{\omega },\Gamma (v_{\omega }))-I_{\omega }(\tilde{v}_{\omega },\Gamma (\tilde{v}_{\omega }))\Vert \\&\leqslant \Vert I_{\omega }(v_{\omega },\Gamma (v_{\omega }))-I_{\omega }(\tilde{v}_{\omega },\Gamma ({v}_{\omega }))\Vert +\Vert I_{\omega }(\tilde{v}_{\omega },\Gamma (v_{\omega }))-I_{\omega }(\tilde{v}_{\omega },\Gamma (\tilde{v}_{\omega }))\Vert \\&\leqslant h_{1}^{\upsilon }(\omega )\Vert v_{\omega }-\tilde{v}_{\omega }\Vert +\frac{1}{2}\Vert \Gamma (v_{\omega })-\Gamma (\tilde{v}_{\omega })\Vert \end{aligned}$$

for every \(v_{\omega }, \tilde{v}_{\omega } \in S_{\omega }\) with \(\Vert v_{\omega }\Vert , \Vert \tilde{v}_{\omega } \Vert \le R^{\upsilon }(\omega )\). Consequently,

$$\begin{aligned} \Vert \Gamma (v_{\omega })-\Gamma (\tilde{v}_{\omega })\Vert \leqslant 2h_{1}^{\upsilon }(\omega )\Vert v_{\omega }-\tilde{v}_{\omega }\Vert . \end{aligned}$$
(2.18)

Also by definition, cf. (2.12),

$$\begin{aligned}&\Vert \Pi ^{0}_{\omega }(\Gamma (v_{\omega }))-\Pi ^{0}_{\omega }(\Gamma (\tilde{v}_{\omega }))\Vert \\&\quad \geqslant \Vert v_{\omega } - \tilde{v}_{\omega }\Vert - h_{2}^{\upsilon }(\omega )\, \Vert \Gamma (v_{\omega })-{\Gamma }_{\omega }(\tilde{v}_{\omega })\Vert \, h(\Vert \Gamma (v_{\omega })\Vert +\Vert {\Gamma }_{\omega }(\tilde{v}_{\omega })\Vert ). \end{aligned}$$

So from (2.18)

$$\begin{aligned} \Vert \Pi ^{0}_{\omega }(\Gamma (v_{\omega }))-\Pi ^{0}_{\omega }(\Gamma (\tilde{v}_{\omega }))\Vert \geqslant \Vert v_{\omega }-\tilde{v}_{\omega }\Vert \big [1 -2h_{1}^{\upsilon }(\omega )h_{2}^{\upsilon }(\omega ) h(\Vert \Gamma (v_{\omega })\Vert +\Vert {\Gamma }_{\omega }(\tilde{v}_{\omega })\Vert )\big ]. \end{aligned}$$
(2.19)

First assume that

$$\begin{aligned} \max \lbrace \Vert \Gamma (v_{\omega }), \Gamma (\tilde{v}_{\omega })\Vert \rbrace \leqslant \frac{1}{2}h^{-1}(\frac{1}{4h_{1}^{\upsilon }(\omega )h_{2}^{\upsilon }(\omega )}). \end{aligned}$$

Then from (2.18) and (2.19),

$$\begin{aligned} \frac{\Vert \Gamma (v_{\omega })-\Gamma (\tilde{v}_{\omega })\Vert }{\Vert \Pi ^{0}_{\omega }(\Gamma (v_{\omega }))-\Pi ^{0}_{\omega }(\Gamma (\tilde{v}_{\omega }))\Vert }\leqslant 4h_{1}^{\upsilon }(\omega ). \end{aligned}$$
(2.20)

Thus if \(Z_{\omega } = Y_{\omega } + \Pi ^0_{\omega }[\Gamma (v_{\omega })]\) and \(\tilde{Z}_{\omega } = Y_{\omega } + \Pi ^0_{\omega }[\Gamma (v_{\omega })]\), it follows that

$$\begin{aligned} \frac{\Vert \varphi ^{n}_{\omega }(Z_{\omega })-\varphi ^{n}_{\omega }(\tilde{Z}_{\omega })\Vert }{\Vert Z_{\omega }-\tilde{Z}_{\omega }\Vert }\leqslant 4\exp (-n\upsilon ) h_{1}^{\upsilon }(\omega ) \end{aligned}$$

for every \(n \ge 1\). In the general case, we can use item (i) and that \( h^{-1}(\frac{1}{4h_{1}^{\upsilon }(\omega )h_{2}^{\upsilon }(\omega )}) \) satisfies (2.15) to see that for some \( N=N(\omega ) \),

$$\begin{aligned}&\sup _{j\geqslant 0}\exp (j\upsilon )\Vert \varphi ^{j}_{\theta ^{N}\omega }(\varphi ^{N}_{\omega }(Z_{\omega }))-Y_{\theta ^{j}\theta ^{N}\omega }\Vert \leqslant \exp (-N\upsilon )\rho ^{\upsilon }_{2}(\omega )\\&\quad \leqslant \frac{1}{2}h^{-1}(\frac{1}{4h_{1}^{\upsilon }(\theta ^{N}\omega )h_{2}^{\upsilon }(\theta ^{N}\omega )}). \end{aligned}$$

Consequently, from (2.20),

$$\begin{aligned} \sup _{j\geqslant 0} \frac{\exp (j\upsilon )\Vert \varphi ^{j+N}_{\omega }(Z_{\omega })-\varphi ^{j+N}_{\omega }(\tilde{Z}_{\omega })\Vert }{\Vert \varphi ^{N}_{\omega }(Z_{\omega })-\varphi ^{N}_{\omega }(\tilde{Z}_{\omega })\Vert }\leqslant 4h_{1}^{\upsilon }(\theta ^{N}\omega ) \end{aligned}$$

and hence for every \( n\geqslant N \),

$$\begin{aligned} \frac{\Vert \varphi ^{n}_{\omega }(Z_{\omega })-\varphi ^{n}_{\omega }(\tilde{Z}_{\omega })\Vert }{\Vert Z_{\omega }-\tilde{Z}_{\omega }\Vert }\leqslant 4\exp ((-n-N)\upsilon ) h_{1}^{\upsilon }(\theta ^{N}\omega )H^{\upsilon }_{N}(\omega ) \end{aligned}$$
(2.21)

where

$$\begin{aligned} H^{\upsilon }_{N}(\omega )=\sup \bigg \lbrace \frac{\Vert \varphi ^{N}_{\omega }(Z_{\omega })-\varphi ^{N}_{\omega }(\tilde{Z}_{\omega })\Vert }{\Vert Z_{\omega }-\tilde{Z}_{\omega }\Vert },\ \ Z_{\omega }\ne \tilde{Z}_{\omega },\ Z_{\omega },\tilde{Z}_{\omega }\in S^{\upsilon }_{loc}(\omega ) \bigg \rbrace . \end{aligned}$$

We claim that \( H^{\upsilon }_{N}(\omega ) \) is finite. Indeed, by assumption (2.5),

$$\begin{aligned}&\Vert \varphi ^{N}_{\omega }(Z_{\omega })-\varphi ^{N}_{\omega }(\tilde{Z}_{\omega })\Vert \leqslant \Vert \psi ^{1}_{\theta ^{N-1}\omega }\Vert \ \Vert \varphi ^{N-1}_{\omega }(Z_{\omega })-\varphi ^{N-1}_{\omega }(\tilde{Z}_{\omega })\Vert \\&\qquad + f(\theta ^{N}\omega ) \ \Vert \varphi ^{N-1}_{\omega }(Z_{\omega })-\varphi ^{N-1}_{\omega }(\tilde{Z}_{\omega })\Vert h\\&\quad \times \big (\Vert \varphi ^{N-1}_{\omega }(Z_{\omega })-Y_{\theta ^{N-1}\omega }\Vert +\Vert \varphi ^{N-1}_{\omega }(\tilde{Z}_{\omega })-Y_{\theta ^{N-1}\omega }\Vert \big ) \end{aligned}$$

and we can proceed by induction to conclude. Finally, from (2.21) and item (iv), our claim is proved. \(\square \)

Remark 2.11

Assume that for \( \omega \in \tilde{\Omega } \) the function \( \varphi _{\omega } \) is \( C^{m} \). Then, since

$$\begin{aligned} I_{\omega }(0,0)=\frac{\partial }{\partial \Gamma }I_{\omega }(0,0)=0, \end{aligned}$$

we can deduce from the Implicit function theorem that \(S^{\upsilon }_{loc}(\omega ) \) is locally \( C^{m-1} \).

3.2 Unstable Manifolds

We invoke same strategy for proving the existence of unstable manifolds. Since the arguments are very similar, we will only sketch them briefly. In this section, we will assume that the largest Lyapunov exponent is strictly positive, i.e. that \(\mu _1 > 0\).

Definition 2.12

Set \(k_{0} := \min \lbrace k :\mu _{k} > 0\rbrace \), \(\tilde{S}_{\omega } := F_{\mu _{k_{0}+1}}(\omega ) \) and \( \tilde{U}_{\omega }=\oplus _{1\leqslant i\leqslant k_{0}}H^{i}_{\omega } \) for \(\omega \in \tilde{\Omega }\). For \( \tilde{\Sigma }_\omega := \prod _{j\geqslant 0}E_{\sigma ^{j}\omega } \) and \( \upsilon >0 \), we define the Banach space

$$\begin{aligned} \tilde{\Sigma }^{\upsilon }_{\omega } := \bigg \lbrace {\Gamma }\in \tilde{\Sigma }_{\omega }\ :\ \Vert {\Gamma }\Vert = \sup _{k\geqslant 0}\big [\Vert \tilde{\Pi }_{\omega }^{k}{\Gamma }\Vert \exp ( k\upsilon )\big ]<\infty \bigg \rbrace \end{aligned}$$

where \( \tilde{\Pi }_{\omega }^{k}:\prod _{i\geqslant 0} E_{\sigma ^{i}\omega }\rightarrow E_{\sigma ^{k}\omega } \) is the projection map. Similar to last section, we also set

$$\begin{aligned} \tilde{h}_{1}^{\upsilon }(\omega )&:= \sup _{n\geqslant 0}\big [\exp (n\upsilon )\Vert (\psi ^{n}_{\sigma ^{n}\omega } |_{\tilde{U}_{\omega }} )^{-1} \Vert \big ] \quad \text {and} \\ \tilde{h}_{2}^{\upsilon }(\omega )&:= \sup _{n\geqslant 0}\big [\exp (n\upsilon ) \sum _{0\leqslant k\leqslant n-1} \exp \big (-\upsilon (n-k) (1+r) \big ) f(\sigma ^{n-k}\omega ) \Vert (\psi ^{k+1}_{\sigma ^{n}\omega } |_{\tilde{U}_{\sigma ^{n-1-k}\omega }} )^{-1}\Vert \\&\qquad \times \Vert \Pi _{ \tilde{U}_{\sigma ^{n-1-k}\omega }\parallel \tilde{S}_{\sigma ^{n-1-k}\omega }}\Vert \\&\quad + \exp (n\upsilon )\sum _{k\geqslant n}\exp (-\upsilon (k+1)(1+r))f(\sigma ^{k+1}\omega )\Vert \psi ^{k-n}_{\sigma ^{k}\omega }|_{\tilde{S}_{\sigma ^{k}\omega }}\Vert \Vert \Pi _{\tilde{S}_{\sigma ^{k}\omega }||\tilde{U}_{\sigma ^{k}\omega }}\Vert \big ]. \end{aligned}$$

Lemma 2.13

Let \(\omega \in \Omega \), \( 0< \upsilon < \mu _{k_{0}}\) and assume that \(\rho :\Omega \rightarrow \mathbb {R}^+\) satisfies

$$\begin{aligned} \liminf _{n \rightarrow \infty } \frac{1}{n} \log \rho (\sigma ^n \omega ) \ge 0 \end{aligned}$$
(2.22)

almost surely. Define P as in Lemma 2.6 and assume that (2.5) holds for a random variable \(f :\Omega \rightarrow \mathbb {R}^+\) which satisfies \(\lim _{n \rightarrow \infty } f(\sigma ^n \omega ) = 0\) almost surely. Set

$$\begin{aligned} \tilde{\rho }(\omega ) := \inf _{n \ge 0} \exp (n \upsilon ) \rho (\sigma ^n \omega ). \end{aligned}$$
(2.23)

Then the map

$$\begin{aligned}&\tilde{I}_{_{\omega }} :\tilde{U}_{\omega }\times \tilde{\Sigma }_{\omega }^{\upsilon }\cap B(0,\tilde{\rho }(\omega ))\rightarrow \tilde{\Sigma }_{\omega }^{\upsilon }, \\&{\tilde{\Pi }}^{n}_{\omega }\big [ \tilde{I} _{\omega }(u_{\omega } ,\Gamma )\big ] \\&\quad = {\left\{ \begin{array}{ll} [\psi ^{n}_{\sigma ^{n}\omega }]^{-1}(u_{\omega }) &{}\text {}\\ \qquad -\sum _{0\leqslant k\leqslant n-1}\big [[\psi ^{k+1}_{\sigma ^{n}\omega }]^{-1}\circ \Pi _{ \tilde{U}_{\sigma ^{n-1-k}\omega }\parallel \tilde{S}_{\sigma ^{n-1-k}\omega }}\big ]P_{\sigma ^{n-k}\omega }\big (\tilde{\Pi }^{n-k}_{\omega }[\Gamma ]\big ) &{}\text {}\\ \qquad + \sum _{k\geqslant n}\big [\psi ^{k-n}_{\sigma ^{k}\omega }\circ \Pi _{\tilde{S}_{\sigma ^{k}\omega }\parallel \tilde{U}_{\sigma ^{k}\omega }}\big ] P_{\sigma ^{k+1}\omega }\big (\tilde{\Pi }^{k+1}_{\omega }[\Gamma ]\big ) &{}\text {for } n \ge 1,\\ u_{\omega }+ \sum _{k\geqslant 0}\big [\psi ^{k}_{\sigma ^{k}\omega }\circ \Pi _{\tilde{S}_{\sigma ^{k}\omega }\parallel \tilde{U}_{\sigma ^{k}\omega }}\big ] P_{\sigma ^{k+1}\omega }\big (\tilde{\Pi }^{k+1}_{\omega }[\Gamma ]\big ) &{}\text {for } n = 0. \end{array}\right. } \end{aligned}$$

is well-defined on a \(\theta \)-invariant set of full measure \(\tilde{\Omega }\).

Proof

We can use Lemma 1.17 to obtain a version of Lemma 2.3 where we replace \(\theta \) by \(\sigma \). The rest of the proof is similar to Lemma 2.6. \(\square \)

Lemma 2.14

For \( 0< \upsilon < \mu _{k_{0}}\), \(\omega \in \tilde{\Omega }\) and \(\Gamma \in \Sigma ^{\upsilon }_{\omega } \cap B(0,\tilde{\rho }(\omega ))\),

$$\begin{aligned} \tilde{I}_{\omega }(u_{\omega },{\Gamma })={\Gamma } \ \ \ \ {\Longleftrightarrow }\ \ \ \ \forall \ 0 \le k\leqslant n: \ \ \ \tilde{\Pi }^{n-k}_{\omega }{\Gamma } = \varphi ^{k}_{\sigma ^{n}\omega }( \tilde{\Pi }^{n}_{\omega }{\Gamma }+Y_{\sigma ^{n}\omega })-Y_{\sigma ^{n-k}\omega } . \end{aligned}$$
(2.24)

Proof

Similar to Lemma 2.7. \(\square \)

Lemma 2.15

For \( 0< \upsilon < \mu _{k_{0}}\), \(\tilde{h}_{1}^{\upsilon } \) and \(\tilde{h}_{2}^{\upsilon }\) are measurable and finite on a \(\theta \)-invariant set of full measure \(\tilde{\Omega }\). Moreover,

$$\begin{aligned} \lim _{p\rightarrow \infty }\frac{1}{p}\log ^{+}\tilde{h}^{\upsilon }_{1}(\sigma ^{p}\omega )=\lim _{p\rightarrow \infty }\frac{1}{p}\log ^{+}\tilde{h}^{\upsilon }_{2}(\sigma ^{p}\omega )=0 \end{aligned}$$
(2.25)

and

$$\begin{aligned}&\Vert \tilde{I}_{\omega }(u_{\omega },{\Gamma })\Vert \leqslant \tilde{h}_{1}^{\upsilon }(\omega )\Vert u_{\omega }\Vert +\tilde{h}_{2}^{\upsilon }(\omega )\Vert {\Gamma }\Vert ^{r+1} g(\Vert {\Gamma }\Vert )\\&\Vert \tilde{I}_{\omega }(u_{\omega },\Gamma ) - \tilde{I}_{\omega }(u_{\omega },\tilde{\Gamma })\Vert \leqslant \tilde{h}_{2}^{\upsilon }(\omega )h(\Vert \Gamma \Vert +\Vert \tilde{\Gamma }\Vert )\ \Vert \Gamma -\tilde{\Gamma }\Vert \end{aligned}$$

hold for every \(\omega \in \tilde{\Omega }\), \(\Gamma , \tilde{\Gamma } \in \tilde{\Sigma }^{\upsilon }_{\omega } \cap B(0,\tilde{\rho }(\omega ))\) and \(u_{\omega } \in \tilde{U}_{\omega }\).

Proof

As in Lemma 2.8. \(\square \)

Lemma 2.16

Assume that for \(u_{\omega } \in \tilde{U}_{\omega }\),

$$\begin{aligned} \Vert u_{\omega }\Vert \leqslant \frac{1}{2\tilde{h}_{1}^{\upsilon }(\omega )}\min \big \lbrace \frac{1}{2}h^{-1}(\frac{1}{2\tilde{h}_{2}^{\upsilon }(\omega )}),\tilde{\rho }(\omega )\big \rbrace . \end{aligned}$$

Then the equation

$$\begin{aligned} \tilde{I}_{\omega }(u_{\omega },\Gamma ) = \Gamma \end{aligned}$$

admits a uniques solution \(\Gamma = \Gamma (u_{\omega })\) and the bound

$$\begin{aligned} \Vert \Gamma (u_{\omega })\Vert \leqslant \min \big \lbrace \frac{1}{2}h^{-1}(\frac{1}{2\tilde{h}_{2}^{\upsilon }(\omega )}), \tilde{\rho }(\omega )\big \rbrace \end{aligned}$$

holds true.

Proof

We can show that \(\tilde{I}(u_{\omega },\cdot )\) is a contraction using Lemma 2.15. \(\square \)

Finally we can formulate our main results about the existence of local unstable manifolds.

Theorem 2.17

Let \((\Omega ,\mathcal {F},\mathbb {P},\theta )\) be an ergodic measure-preserving dynamical systems, \(\sigma := \theta ^{-1}\) and \(\varphi \) a Fréchet-differentiable cocycle acting on a measurable field of Banach spaces \(\{E_{\omega }\}_{\omega \in \Omega }\). Assume that \(\varphi \) admits a stationary solution Y and that the linearized cocycle \(\psi \) around Y is compact, satisfies Assumption 1.1 and the integrability condition

$$\begin{aligned} \log ^+ \Vert \psi _{\omega } \Vert \in L^1(\omega ). \end{aligned}$$

Moreover, assume that (2.5) holds for \(\varphi \) and \(\psi \) and a random variable \(\rho :\Omega \rightarrow \mathbb {R}^+\) satisfying (2.22). Assume that \(\mu _1 > 0 \) and let \(\mu _{k_0} > 0\) and \(\tilde{U}_{\omega }\) be defined as in Definition 2.12. For \( 0< \upsilon < \mu _{k_{0}} \), \(\omega \in {\Omega }\) and \( R^{\upsilon }(\omega ) :=\frac{1}{2 \tilde{h}_{1}^{\upsilon }(\omega )} \min \big \lbrace \frac{1}{2}h^{-1}(\frac{1}{2\tilde{h}_{2}^{\upsilon }(\omega )}),\tilde{\rho }(\omega )\big \rbrace \) with \(\tilde{\rho }\) defined as in (2.23), let

$$\begin{aligned} U^{\upsilon }_{loc}(\omega ) := \big \lbrace Y_{\omega }+\tilde{\Pi }^{0}_{\omega }[\Gamma (u_{\omega })], \ \ \Vert u_{\omega }\Vert < \tilde{R}^{\upsilon }(\omega )\big \rbrace . \end{aligned}$$
(2.26)

Then there is a \(\theta \)-invariant set of full measure \(\tilde{\Omega }\) on which the following properties are satisfied for every \(\omega \in \tilde{\Omega }\):

  1. (i)

    There are random variables \( \tilde{\rho }_{1}^{\upsilon }(\omega ), \tilde{\rho }_{2}^{\upsilon }(\omega )\), positive and finite on \(\tilde{\Omega }\), for which

    $$\begin{aligned} \liminf _{p \rightarrow \infty } \frac{1}{p} \log \tilde{\rho }_i^{\upsilon }(\sigma ^p \omega ) \ge 0, \quad i = 1,2 \end{aligned}$$

    and such that

    $$\begin{aligned}&\bigg \lbrace Z_{\omega } \in E_{\omega }\, :\, \exists \lbrace Z_{\sigma ^{n}\omega }\rbrace _{n\geqslant 1} \text { s.t. } \varphi ^{m}_{\sigma ^{n}\omega }(Z_{\sigma ^{n}\omega }) = Z_{\sigma ^{n-m}\omega } \text { for all } 0 \le m \le n \text { and }\\&\quad \sup _{n\geqslant 0}\exp (n\upsilon )\Vert Z_{\sigma ^{n}\omega } - Y_{\sigma ^{n}\omega } \Vert<\tilde{\rho }_{1}^{\upsilon }(\omega )\bigg \rbrace \subseteq U^{\upsilon }_{loc}(\omega ) \subseteq \bigg \lbrace Z_{\omega } \in E_{\omega }\, :\, \exists \lbrace Z_{\sigma ^{n}\omega }\rbrace _{n\geqslant 1} \text { s.t. } \\&\qquad \varphi ^{m}_{\sigma ^{n}\omega }(Z_{\sigma ^{n}\omega } ) = Z_{\sigma ^{n-m}\omega } \text { for all } 0 \le m \le n \text { and } \sup _{n\geqslant 0}\exp (n\upsilon )\Vert Z_{\sigma ^{n}\omega } - Y_{\sigma ^{n}\omega }\Vert <\tilde{\rho }_{2}^{\upsilon }(\omega )\bigg \rbrace . \end{aligned}$$
  2. (ii)

    \( U^{\upsilon }_{loc}(\omega ) \) is an immersed submanifold of \(E_{\omega } \) and

    $$\begin{aligned} T_{Y_{\omega }}U^{\upsilon }_{loc}(\omega ) = \tilde{U}_{\omega }. \end{aligned}$$
  3. (iii)

    For \( n\geqslant N(\omega ) \),

    $$\begin{aligned} U^{\upsilon }_{loc}(\omega )\subseteq \varphi ^{n}_{\sigma ^{n}\omega }(U^{\upsilon }_{loc}(\sigma ^{n}\omega )). \end{aligned}$$
  4. (iv)

    For \( 0<\upsilon _{1} \leqslant \upsilon _{2} < \mu _{k_{0}} \),

    $$\begin{aligned} U^{\upsilon _{2}}_{loc}(\omega )\subseteq U^{\upsilon _{1}}_{loc}(\omega ). \end{aligned}$$

    Also for \(n\geqslant N(\omega ) \),

    $$\begin{aligned} U^{\upsilon _{1}}_{loc}(\omega )\subseteq \varphi ^{n}_{\sigma ^{n}\omega }(U^{\upsilon _{2}}_{loc}(\sigma ^{n}(\omega )) \end{aligned}$$

    and consequently for \( Z_{\omega }\in U^{\upsilon }_{loc}(\omega ) \),

    $$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \Vert Z_{\sigma ^{n}\omega } - Y_{\sigma ^{n}\omega }\Vert \leqslant -\mu _{k_{0}}. \end{aligned}$$
  5. (v)
    $$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log \bigg [\sup \bigg \lbrace \frac{\Vert Z_{\sigma ^{n}\omega }- \tilde{Z}_{\sigma ^{n}\omega }\Vert }{\Vert Z_{\omega }-\tilde{Z}_{\omega }\Vert },\ \ Z_{\omega }\ne \tilde{Z}_{\omega },\ Z_{\omega },\tilde{Z}_{\omega }\in U^{\upsilon }_{loc}(\omega ) \bigg \rbrace \bigg ]\leqslant -\mu _{k_{0}}. \end{aligned}$$

Proof

One uses the same arguments as in the proof of Theorem 2.10. \(\square \)

Remark 2.18

  1. (i)

    As in the stable case, if \( \varphi _{\omega } \) is \( C^{m} \) for every \( \omega \in \tilde{\Omega } \), one can deduce that \({U}^{\upsilon }_{loc}(\omega ) \) is locally \( C^{m-1} \).

  2. (ii)

    In the hyperbolic case, i.e. if all Lyapunov exponents are non-zero, if the assumptions of Theorem 2.10 and 2.17 are satisfied, we have \(S_{\omega } = \tilde{S}_{\omega }\) and \(U_{\omega } = \tilde{U}_{\omega }\). In particular, the submanifolds \(S^{\upsilon }_{loc}(\omega ) \) and \(U^{\upsilon }_{loc}(\omega ) \) are transversal, i.e.

    $$\begin{aligned} E_{\omega } = T_{Y_{\omega }} U^{\upsilon }_{loc}(\omega ) \oplus T_{Y_{\omega }} S^{\upsilon }_{loc}(\omega ). \end{aligned}$$