Oseledets splitting and invariant manifolds on fields of Banach spaces [original]

Journal of Dynamics and Differential Equations

https://doi.org/10.1007/s10884-021-09969-1

Oseledets Splitting and Invariant Manifolds on Fields of

Banach Spaces

Mazyar Ghani Varzaneh1,2 ·S. Riedel1

Received: 16 June 2020 / Revised: 28 September 2020 / Accepted: 11 February 2021

Abstract

We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of

Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an appli-

cation, we prove an invariant manifold theorem for nonlinear cocycles acting on measurable

fields of Banach spaces.

Keywords Semi-invertible multiplicative ergodic theorem ·Oseledets splitting ·Fields of

Banach spaces ·Invariant manifolds

Mathematics Subject Classification 37H15 ·37L55 ·37B55

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 [2,6,11,15,17,19–23].

In [12], the authors gave a proof for an MET for cocycles acting on measurable fields of

Banach spaces.Letus quicklyrecall thesetting here:If (, F,P)denotesa probabilityspace,

we call a family of Banach spaces {Eω}ω∈a measurable field if there exists a linear subspace

of all sections ω∈Eωand a countable subset 0⊂such that {g(ω) :g∈0}is

dense in Eωfor every ω∈and ω→g(ω)Eωis measurable for every g∈.Notethat

this definition implies that every Banach space Eωis separable. On the other hand, every

separable Banach space defines a field of Banach spaces by simply setting Eω=E.This

structure is similar to a measurable version of a Banach bundle with base and total space

BS. Riedel

[email protected]

Mazyar Ghani Varzaneh

[email protected]

1Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

2Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

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Journal of Dynamics and Differential Equations

ω∈Eωin which every space Eωis a fiber. However, the fundamental difference is that

we do not put any measurable (or topological) structure on the bundle ω∈Eωitself! In

fact, the existence of the set is a substitute for the measurable structure and will help to

prove measurability for functionals defined on ω∈Eωas we will see many times in this

work. If (, F,P,θ) is a measure preserving dynamical systems, a cocycle actingonthe

field {Eω}ω∈consists of a family of maps ϕω:Eω→Eθω. Setting ϕn

ω:= ϕθn−1ω◦···◦ϕω,

we furthermore claim that ω→ϕn

ω(g(ω))Eθnωis measurable for every g∈and every

n∈N.

There are numerous examples in which it is natural to study cocycles on random spaces. In

[12],ourmotivationwastostudydynamicalpropertiesofsingularstochasticdelaydifferential

equations in which the spaces Eωare (essentially) spaces of controlled Brownian paths

known in rough paths theory [8]. In the finite dimensional case, linearizing a C1-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 (, F,P,θ) be an ergodic measurable metric dynamical system and ϕ

be a compact linear cocycle acting on a measurable field of Banach spaces {Eω}ω∈.For

μ∈R∪{−∞}and ω∈, define

Fμ(ω) := x∈Eω:lim sup

n→∞

nlog ϕn

ω(x)≤μ.

Assume that

log+ϕω∈L1().

Then there is a measurable forward invariant set ˜

⊂of full measure and a decreasing

sequence {μi}i≥1,μi∈[−∞,∞)with the properties that limn→∞ μn=−∞and either

μi>μ

i+1or μi=μi+1=−∞such that for every ω∈˜

,

x∈Fμi(ω)\Fμi+1(ω) if and only if lim

n→∞

nlog ϕn

ω(x)=μi.(0.1)

Moreover, there are numbers m1,m2,...such that codim Fμj(ω) =m1+...+mj−1for

every ω∈˜

.

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 {μi}are the Lyapunov exponents, the subspaces Fμ(ω) are sometimes called

slow-growing subspaces and the resulting filtration

Eω=Fμ1(ω) ⊃Fμ2(ω) ⊃···

is called Oseledets filtration. Is is easily seen that the slow-growing spaces are equivariant,

meaning that ϕω(Fμi(ω)) ⊂Fμi(θω). In the proof of this theorem, no invertibility of θor

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Journal of Dynamics and Differential Equations

ϕis 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 θis invertible. In this case, it is

possibletodeduce asplitting of thespaces Eωconsisting of fast-growing subspaces which are

invariantunderϕ. Sucha splittingiscalledOseledets splitting,and thecorrespondingtheorem

is called semi-invertible MET. Let us emphasize that we only need to assume invertibility

of the base θand no invertibility of the cocyle ϕ. In the context of SPDE or stochastic

delay equations, these assumptions are quite natural: θusually 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 θis invertible

with measurable inverse σ:= θ−1and that Assumption 1.1 holds. Then there is a θ-invariant

set ˜

of full measure such that for every i ≥1with μi>μ

i+1and ω∈˜

, there is an mi-

dimensional subspace Hi

ωwith the following properties:

(i) (Invariance) ϕk

ω(Hi

ω)=Hi

θkωfor every k ≥0.

(ii) (Splitting) Hi

ω⊕Fμi+1(ω) =Fμi(ω). In particular,

Eω=H1

ω⊕···⊕Hi

ω⊕Fμi+1(ω).

(iii) (‘Fast-growing’ subspace) For each hω∈Hi

ω\{0},

lim

n→∞

nlog ϕn

ω(hω)=μj

and

lim

n→∞

nlog (ϕn

σnω)−1(hω)=−μj.

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 and continuity of the map ω→ ϕω.Healsoassumed

injectivity of ϕ. 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 , 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 , it also weakens the

measurability assumption on ϕsignificantly in case we are dealing with a single Banach

space Eonly. In fact, the standard measurability assumption, for instance in [11], is strong

measurability of ϕ, meaning that for fixed x∈E,themap

ω→ ϕω(x)∈E(0.2)

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Journal of Dynamics and Differential Equations

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

ω→ϕk+n

ω(x)−ϕk

θnω(˜x)E∈R

should be measurable for every n,k∈N0and x,˜x∈Swhere Sis 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 Blu-

menthal [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 Elike

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 ϕbe a nonlinear, differentiable cocycle acting on a measurable field of

Banach spaces {Eω}ω∈. Assume that Yωis a random fixed point of ϕ, in particular ϕω(Yω)=

Yθω. Then, under the same measurability and integrability assumptions as in Theorem 0.2,

the linearized cocycle DYωϕωhas a Lyapunov spectrum {μn}n≥1. Under further assumptions

on ϕand Y , there is a θ-invariant set ˜

of full measure, closed subspaces Sωand Uωof Eω

and immersed submanifolds Sloc(ω) and Uloc(ω) of Eωsuch that for every ω∈˜

,

TY(ω) Sloc(ω) =Sωand TY(ω)Uloc(ω) =Uω

and the properties that for every Zω∈Sloc(ω),

lim sup

n→∞

nlog ϕn

ω(Zω)−Yθnω≤μj0<0

and for every Zω∈Uloc(ω) one has ϕn

σnω(Zσnω)=Zωand

lim sup

n→∞

nlog Zσnω−Yσnω≤−μk0<0.

Here we have set μj0=max{μj:μj<0}and μk0=min{μk:μk>0}. In the hyperbolic

case, i.e. if all Lyapunov exponents are non-zero, the submanifolds Sυ

loc(ω) and Uυ

loc(ω) are

transversal, i.e.

Eω=TYωUυ

loc(ω) ⊕TYωSυ

loc(ω).

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. 2to

deduce the existence of local stable and unstable manifolds for nonlinear cocycles.

Notation

– For Banach spaces (X,·X)and (Y,·

Y),L(X,Y)denotes the space of bounded

linear functions from Xto Yequipped with usual operator norm. We will often not

explicitly write a subindex for Banach space norms and use the symbol ·instead.

Differentiability of a function f:X→Ywill always mean Fréchet-differentiability. A

Cmfunction denotes an m-times Fréchet-differentiable function. If A,B⊆X,wedenote

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Journal of Dynamics and Differential Equations

by d(A,B):= infa∈A,b∈Ba−bthe distance between two sets Aand B.Wealsoset

d(x,B):= d(B,x):= d({x},B)for x∈X,B⊆X.

–LetX,Ybe Banach spaces. For x1,...,xk∈X,set

Vol(x1,x2,...,xk):= x1



i=2

d(xi,xj1≤j<i). (0.3)

For a given bounded linear function T:X→Yand k≥1, we define

Dk(T):= sup

xi=1;i=1,...,k

Vol T(x1), T(x2),...,T(xk).

–LetEbe a vector space. If we can write Eas a direct sum E=F⊕Hof vector spaces,

we have an algebraic splitting. We also say that F is a complement of H and vice versa.

The projection operator FH(e)=fwith e=f+h,f∈F,h∈H, is called the

projection operator onto F parallel to H.IfEis a normed space and FHis bounded

linear, i.e.

FH= sup

f∈F,h∈H,f+h=0f

f+h<∞,

we call E=F⊕Hatopological splitting. For normed spaces, a splitting will always

mean a topological splitting.

–Let(, F)be a measurable space. We call a family of Banach spaces {Eω}ω∈amea-

surable field of Banach spaces if there is a set of sections

⊂

ω∈

Eω

with the following properties:

(i) is a linear subspace of ω∈Eω.

(ii) There is a countable subset 0⊂such that for every ω∈,theset{g(ω) :g∈

0}is dense in Eω.

(iii) For every g∈,themapω→g(ω)Eωis measurable.

–Let(, F)be ameasurable space. Ifthere existsa measurablemap θ:→,ω→ θω,

with a measurable inverse θ−1, we call (, F,θ) ameasurable dynamical system.We

will use the notation θnωfor n-times applying θto an element ω∈.Wealsoset

θ0:= Idand θ−n:= (θn)−1.IfPis a probability measure on (, F)that is invariant

under θ, i.e. P(θ−1A)=P(A)=P(θ A)for every A∈F, we call the tuple ,F,P,θ

ameasure-preserving dynamical system. The system is called ergodic if every θ-invariant

set has probability 0 or 1.

–Let(, F,P,θ)be a measure-preserving dynamical system and ({Eω}ω∈,)a mea-

surable field of Banach spaces. A continuous cocycle on {Eω}ω∈consists of a family

of continuous maps

ϕω:Eω→Eθω.(0.4)

If ϕis a continuous cocycle, we define ϕn

ω:Eω→Eθnωas

ϕn

ω:= ϕθn−1ω◦···◦ϕω.

We also set ϕ0

ω:= IdEω. We say that ϕacts on {Eω}ω∈if the maps

ω→ϕ(n,ω,g(ω))Eθnω,n∈N

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