Stable Manifolds and Nonuniform Hyperbolicity: Optimal Estimates

Mediterr. J. Math. 10 (2013), 1369–1389DOI 10.1007/s00009-013-0245-51660-5446/13/031369-21, published online February 22, 2013© 2013 Springer Basel

Mediterranean Journalof Mathematics

Stable Manifolds and NonuniformHyperbolicity: Optimal Estimates

Luis Barreira∗ and Claudia Valls

Abstract. We establish the existence of invariant stable manifolds for C1

perturbations of a nonuniform exponential dichotomy with an arbitrarynonuniform part. We consider the general case of sequences of maps,which corresponds to a nonautonomous dynamics with discrete time.We also obtain optimal estimates for the decay of trajectories along thestable manifolds. The optimal C1 smoothness of the invariant manifoldsis obtained using an invariant family of cones.

Mathematics Subject Classification (2010). Primary: 37D10, 37D25.Keywords. Nonuniform hyperbolicity, stable manifolds.

1. Introduction

The concept of nonuniform hyperbolicity originated in the landmark work ofPesin [4, 5], in particular with the study of smooth ergodic theory, also calledPesin theory (see [1] for a detailed exposition). Since then the theory becamean important part of the general theory of dynamical systems and a principaltool in the study of stochastic behavior. Among the important properties dueto nonuniform hyperbolicity is the existence of stable and unstable manifolds,established by Pesin in [4] with an elaboration of the classical work of Perron.In [8], Ruelle obtained a proof of the stable manifold theorem based on thestudy of perturbations of products of matrices in Oseledets’ multiplicativeergodic theorem [3]. Another proof is due to Pugh and Shub in [7] with anelaboration of the classical work of Hadamard using graph transform tech-niques. In all these works the dynamics is assumed to be of class C1+α forsome α > 0. On the other hand, Pugh [6] constructed a C1 diffeomorphismthat is not of class C1+α for any α and for which there exists no invariantmanifold tangent to the stable space such that the trajectories along the in-variant manifold travel with exponential speed. This example shows that the

Partially supported by FCT (grant PTDC/MAT/117106/2010 and through CAMGSD).∗Corresponding author.

1370 L. Barreira and C. Valls Mediterr. J. Math.

hypothesis α > 0 is in general crucial in the stable manifold theorem, but itdoes not forbid the existence of families of C1 dynamics for which there existstable manifolds. Indeed, in this paper we establish the existence of invariantstable manifolds for a large family of maps that in general are at most ofclass C1.

Our main aim is to consider nonuniform exponential dichotomies withan arbitrary nonuniform part. In other words, the exponential dichotomiesmay exhibit a nonuniform exponential stability that is determined by anarbitrary sequence Dn. In this general context, we establish the existence ofstable manifolds and we obtain optimal estimates for the decay of trajectoriesalong the stable manifolds, in terms of Dn. This last aspect amounts todescribe the size of the neighborhood of zero in which we must take the initialconditions so that the speed of decay of solutions along a stable manifold isthe same as that of the linear approximation. More precisely, we establishthe existence of Lipschitz stable manifolds in Banach spaces and we establishtheir optimal C1 smoothness under C1 perturbations in finite-dimensionalspaces. The proofs are inspired in our work [2] for nonuniform exponentialdichotomies (with an exponential nonuniform part), although the argumentsrequire several nontrivial modifications.

2. Existence of Stable Manifolds

Let D = (Dn)n∈N be a sequence of real numbers Dn ≥ 1 such that

ε := lim supn→∞

1n

logDn < +∞. (2.1)

We say that a sequence (An)n∈N of invertible bounded linear operators ina Banach space X admits a D-exponential dichotomy if there exist projec-tions Pm such that

PmA(m,n) = A(m,n)Pn

for each m,n ∈ N with m ≥ n, where

A(m,n) =

{Am−1 · · ·An, m > n,

Id, m = n,

and constantsa < min{0, b} (2.2)

such that for each m,n ∈ N with m ≥ n we have

‖A(m,n)Pn‖ ≤ Dnea(m−n), ‖A(m,n)−1Qm‖ ≤ Dme−b(m−n), (2.3)

where Qm = Id − Pm is the complementary projection. We write

B(m,n) = A(m,n)Pn and C(m,n) = A(m,n)Qn,

and for each n ∈ N we consider the stable and unstable subspaces

En = Pn(X) and Fn = Qn(X).

Vol. 10 (2013) Stable Manifolds and Nonuniform Hyperbolicity 1371

Even though the stable and unstable subspaces usually correspond to takea < 0 < b, for simplicity of the exposition we shall keep these names evenunder condition (2.2).

We also consider maps fn : X → X with fn(0) = 0 for each n ∈ N, andwe assume that there exist constants c > 0 and q > 0 such that

‖fn(u) − fn(v)‖ ≤ c‖u− v‖(‖u‖q + ‖v‖q) (2.4)

for every n ∈ N and u, v ∈ X. Our main aim is to establish the existence ofinvariant stable manifolds for the dynamics in X defined by

vn+1 = Fn(vn), where Fn(v) = Anv + fn(v). (2.5)

We emphasize that the functions fn satisfying (2.4) need not be differentiable(needless to say of class C1+ε). Simple examples can be obtained as follows.

Example. Let p be a polynomial of degree at least 2. Now let f be any functionwhose graph is a polygonal line obtained from connecting points in the graphof p. One can easily verify that if one of the segments contains the origin inits interior and is horizontal, then fn = f satisfies condition (2.4), althoughclearly f is not differentiable.

Now we consider the case of differentiable functions. For simplicity ofthe exposition we assume that X = R. One can easily verify that if fn isdifferentiable, then condition (2.4) holds if and only if there exist constantsc′ > 0 and q > 0 (indeed with the same q) such that

‖dufn‖ ≤ c′‖u‖q (2.6)

for every n ∈ N and u ∈ X. Even when fn is of class C1 it may not be ofclass C1+ε for any ε ∈ (0, 1].

Example. Let fn be a polynomial of degree at least 2. Clearly, fn satisfiescondition (2.6) and so also condition (2.4). On the other hand, fn is not ofclass C1+ε in any unbounded interval.

For each n ∈ N, we denote by Qn(δ) ⊂ En the closed ball of radiusδ > 0 centered at zero, and we let

βn =1

D1+1/qn

(∑j≥n Dj+1eqa(j−n)

)1/q . (2.7)

In view of (2.1) the series in (2.7) converges provided that qa + ε < 0. Thishappens for example when ε = 0 (for every q > 0). Given δ > 0, let X thespace of sequences ϕ = (ϕn)n∈N of functions ϕn : Qn(δβn) → Fn such thatϕn(0) = 0 and

‖ϕn(x) − ϕn(y)‖ ≤ ‖x− y‖ for x, y ∈ Qn(δβn). (2.8)

For each ϕ ∈ X and n ∈ N, we consider the graph

Vn ={(ξ, ϕn(ξ)) : ξ ∈ Qn(δβn)

}.


We also write

F(m,n) =

{Fm−1 ◦ · · · ◦ Fn, m > n,

Id, m = n,

and αn = βn/(2Dn).The following result establishes the existence and uniqueness of invari-

ant stable manifolds Vn. In fact, in view of condition (2.2), we also obtainin this manner strong stable manifolds. These correspond to the invariantsubbundles of the stable spaces. When the constant b is nonnegative, themanifolds Vn are the full stable manifolds corresponding to the whole stablespaces.

Theorem 2.1. If the sequence (An)n∈N admits a D-exponential dichotomy,and the conditions

qa + ε < 0 and a + ε < b (2.9)

hold, then there exist δ > 0 and a unique ϕ ∈ X such that

F(m,n)(ξ, ϕn(ξ)) ⊂ Vm for every m ≥ n and ξ ∈ Qn(δαn). (2.10)

Moreover, for each m ≥ n and ξ, ξ ∈ Qn(δαn) we have

‖F(m,n)(ξ, ϕn(ξ)) − F(m,n)(ξ, ϕn(ξ))‖ ≤ 4Dnea(m−n)‖ξ − ξ‖. (2.11)

We emphasize that the constant δ in Theorem 2.1 only depends ona, b, q, c and D, and not on the specific sequence (An)n∈N.

3. Proof of Theorem 2.1

We separate the proof of Theorem 2.1 into several steps.

3.1. Preliminaries

We write

fm = (gm, hm) ∈ Em × Fm and vm = (xm, ym) ∈ Em × Fm.

Given n ∈ N and vn = (ξ, η) ∈ En × Fn, for each m ≥ n we have

xm = B(m,n)xn +m−1∑l=n

B(m, l + 1)gl(xl, yl),

ym = C(m,n)yn +m−1∑l=n

C(m, l + 1)hl(xl, yl).

(3.1)

In order that the forward invariance property in (2.10) holds, each trajectoryof (2.5) starting in Vn must be in Vm for every m ≥ n, and thus, if vn ∈ Vn,


then the equations in (3.1) take the form

xm = B(m,n)ξ +m−1∑l=n

B(m, l + 1)gl(xl, ϕl(xl)),

ϕm(xm) = C(m,n)ϕn(ξ) +m−1∑l=n

C(m, l + 1)hl(xl, ϕl(xl)).

(3.2)

Now we equip the space X with the norm

‖ϕ‖ = sup{‖ϕn(x)‖/‖x‖ : n ∈ N and x ∈ Qn(δβn) \ {0}}

for each ϕ = (ϕn)n∈N ∈ X. Since Dn ≥ 1, it follows from (2.7) that

βn ≤(∑

j≥n

eqa(j−n)

)−1/q

= (1 − eqa)1/q.

Clearly, ‖ϕ‖ ≤ 1, and given n ∈ N and x = 0, we have

‖ϕn(x)‖ ≤ δβn‖ϕn(x)‖/‖x‖≤ δ(1 − eqa)1/q‖ϕ‖ ≤ δ(1 − eqa)1/q

for every ϕ ∈ X. This readily implies that X is a complete metric space withthe distance induced by ‖·‖. Now let

N ={(n, ξ) : n ∈ N and ξ ∈ Qn(δβn)

}.

For technical reasons, we also consider the space X∗ of all sequences ϕ =(ϕn)n∈N with ϕn : En → Fn for each n ∈ N such that ϕ|N ∈ X and

ϕn(ξ) = ϕn(δβnξ/‖ξ‖) whenever ξ ∈ Qn(δβn).

Clearly, X∗ is a complete metric space with the norm X∗ � ϕ �→ ‖ϕ|N‖.Lemma 3.1. For each ϕ ∈ X∗ and n ∈ N, we have

‖ϕn(x) − ϕn(y)‖ ≤ 2‖x− y‖ for x, y ∈ En.

Proof. In view of (2.8), we may assume that x ∈ Qn(δβn). Take also y outsideQn(δβn). Letting c = δβn, we have

‖ϕn(x) − ϕn(y)‖ =∥∥∥∥ϕn

(c

x

‖x‖)− ϕn

(c

y

‖y‖)∥∥∥∥ ≤ c

∥∥∥∥ x

‖x‖ − y

‖y‖∥∥∥∥.

Since ∥∥∥∥ x

‖x‖ − y

‖y‖∥∥∥∥ =

‖(x− y)‖y‖ + y(‖y‖ − ‖x‖)‖‖x‖ · ‖y‖ ≤ 2‖x− y‖

‖x‖ ,

we obtain

‖ϕn(x) − ϕn(y)‖ ≤ 2‖x− y‖.


Now take y ∈ Qn(δβn) and κ ∈ (0, 1) such that z = κx + (1 − κ)y has norm‖z‖ = c. Then

‖ϕn(x) − ϕn(y)‖ ≤ ‖ϕn(x) − ϕn(z)‖ + ‖ϕn(z) − ϕn(y)‖≤ ‖x− z‖ + 2‖z − y‖= ‖x− y‖ + ‖z − y‖ ≤ 2‖x− y‖.

This completes the proof of the lemma. �

3.2. Solution on the Stable Direction

We first consider the stable component of the solutions.

Lemma 3.2. For any sufficiently small δ > 0 and each ϕ ∈ X∗, given (n, ξ) ∈N there exists a unique sequence (xm)m≥n ⊂ X with xn = ξ and xm ∈ Em

for each m > n satisfying (3.2) for m ≥ n. In addition, we have

‖xm‖ ≤ 2Dnea(m−n)‖ξ‖ for m ≥ n. (3.3)

Proof. Given δ > 0, we consider the space D of all sequences (xm)m≥n withxm ∈ Em for each m ≥ n such that ‖x‖′ ≤ δβn, with the norm

‖x‖′ = (2Dn)−1 sup{‖xm‖e−a(m−n) : m ≥ n

}.

One can easily verify that D is a complete metric space. Now we define anoperator J in D by

(Jx)m =m−1∑l=n

B(m, l + 1)gl(xl, ϕl(xl))

for each x ∈ D and m ≥ n. Since ϕ ∈ X, one can always compute ϕl(xl).Given x, y ∈ D and l ≥ n, we have

‖(xl, ϕl(xl))‖ = ‖(xl, ϕl(xl) − ϕl(0))‖ ≤ 3‖xl‖,

and

‖(xl, ϕl(xl)) − (yl, ϕl(yl))‖ ≤ 3‖xl − yl‖.

Therefore, by (2.4),

A : = ‖gl(xl, ϕl(xl)) − gl(yl, ϕl(yl))‖≤ 3q+1c‖xl − yl‖(‖xl‖q + ‖yl‖q)≤ KD1+q

n δqea(q+1)(l−n)βqn‖x− y‖′

(3.4)


for some constant K = K(c, q) > 0. By the first inequality in (2.3), we obtain

‖(Jx)m − (Jy)m‖ ≤m−1∑l=n

‖B(m, l + 1)‖A

≤ KD1+qn δqβq

n‖x− y‖′m−1∑l=n

Dl+1ea(m−l−1)ea(q+1)(l−n)

≤ KD1+qn δq e−aβq

n‖x− y‖′ea(m−n)m−1∑l=n

Dl+1eqa(l−n)

≤ Kδqe−a‖x− y‖′ea(m−n),

and thus,‖Jx− Jy‖′ ≤ θ‖x− y‖′, (3.5)

for some θ < 1/2, provided that δ > 0 is sufficiently small (depending on a, qand c). Now we consider the operator J in D defined by

(Jx)m = B(m,n)ξ + (Jx)m.

For y = 0 ∈ D we have Jy = 0 (because ϕm(0) = 0 for every m ∈ N),and thus, by (3.5), ‖Jx‖′ ≤ θ‖x‖′. By the first inequality in (2.3), we obtain‖B(·, n)ξ‖′ ≤ ‖ξ‖/2 and hence,

‖Jx‖′ ≤ ‖B(·, n)ξ‖′ + ‖Jx‖′ ≤ 12‖ξ‖ + θ‖x‖′

≤ 12δβn +

12δβn = δβn.

(3.6)

Therefore, J : D → D is well defined. In view of (3.5), we have

‖Jx− Jy‖′ = ‖Jx− Jy‖′ ≤ θ‖x− y‖′,and hence, J is a contraction. By (3.6), the unique x ∈ D such that Jx = xsatisfies

‖x‖′ ≤ 12‖ξ‖ + θ‖x‖′

and hence,

‖x‖′ ≤ ‖ξ‖2(1 − θ)

.


3.3. Auxiliary Results

Given ϕ ∈ X∗ and (n, ξ), (n, ξ) ∈ N , we denote by x and x the uniquesequences given by Lemma 3.2 with xn = ξ and xn = ξ.

Lemma 3.3. Given δ > 0 sufficiently small, for every ϕ ∈ X∗ and (n, ξ),(n, ξ) ∈ N we have

‖xm − xm‖ ≤ 2Dnea(m−n)‖ξ − ξ‖ for m ≥ n. (3.7)


Proof. Take l ≥ n. By (3.4), we have

‖gl(xl, ϕl(xl)) − gl(xl, ϕl(xl))‖ ≤ KδqD1+qn eqa(l−n)βq

n‖xl − xl‖. (3.8)

Now let

ρm = ‖xm − xm‖ and Tm = e−a(m−n)ρm.

By first inequality in (2.3), it follows from (3.2) that

ρm ≤ ‖B(m,n)‖ · ‖ξ − ξ‖ +m−1∑l=n

‖B(m, l + 1)‖KδqD1+qn βq

neqa(l−n)ρl

≤ Dnea(m−n)‖ξ − ξ‖ + KδqD1+q

n βqne

−am−1∑l=n

Dl+1ea(m−l)+qa(l−n)ρl.

Therefore,

Tm ≤ Dn‖ξ − ξ‖ + KδqD1+qn βq

ne−a

m−1∑l=n

Dl+1eqa(l−n)Tl.

Letting T = supm∈N Tm, for any sufficiently small δ > 0 (depending on a, qand c) we obtain

T ≤ Dn‖ξ − ξ‖ + Kδqe−aT ≤ Dn‖ξ − ξ‖ +T

2.

This yields the desired result. �

Given ϕ, ψ ∈ X∗ and (n, ξ) ∈ N , we denote by x and y the uniquesequences given by Lemma 3.2 such that xn = yn = ξ.

Lemma 3.4. Given δ > 0 sufficiently small, for every ϕ, ψ ∈ X∗ and (n, ξ) ∈N we have

‖xm − ym‖ ≤ Dnea(m−n)‖ξ‖ · ‖ϕ− ψ‖ for m ≥ n. (3.9)

Proof. Take l ≥ n. Proceeding as in (3.4), and using (3.3) in Lemma 3.2, weobtain

‖gl(xl, ϕl(xl)) − gl(yl, ψl(yl))‖≤ 3qc‖(xl − yl, ϕl(xl) − ψl(yl))‖(‖xl‖q + ‖yl‖q)≤ K1D

qnδ

qβqne

qa(l−n)(‖xl‖ · ‖ϕ− ψ‖ + 2‖xl − yl‖

) (3.10)

for some constant K1 = K1(q, c) > 0. Now let

ρm = ‖xm − ym‖ and Tm = e−a(m−n)ρm.


By the first inequality in (2.3) and (3.3), it follows from (3.2) that

Tm ≤ K1e−a(m−n)e−aDq

nβqnδ

q

×m−1∑l=n

Dl+1ea(m−l)+qa(l−n)

(‖xl‖ · ‖ϕ− ψ‖ + 2ρl)

≤ K1e−aDq

nβqnδ

q

×(

2Dn‖ξ‖ · ‖ϕ− ψ‖m−1∑l=n

Dl+1eqa(l−n) + 2

m−1∑l=n

Dl+1eqa(l−n)Tl

).

Letting T = supm∈N Tm, we have

T ≤ K1e−aδq

(2Dn‖ξ‖ · ‖ϕ− ψ‖ + 2T

),

and for any sufficiently small δ > 0 (depending on a, q and c), we obtain

T ≤ Dn

2‖ξ‖ · ‖ϕ− ψ‖ +

T

2.


3.4. Fixed Point Problem

Lemma 3.5. Given δ > 0 sufficiently small, there exists a unique ϕ ∈ X∗ suchthat for every (n, ξ) ∈ N we have

ϕn(ξ) = −∞∑l=n

C(l + 1, n)−1hl(xl, ϕl(xl)), (3.11)

where x = (xl)l≥n is the unique sequence given by Lemma 3.2 with xn = ξ.

Proof. Let h∗l (ξ) = hl(xl, ϕl(xl)). We look for a fixed point of the operator Φ

in X∗ defined by

(Φϕ)n(ξ) = −∞∑l=n

C(l + 1, n)−1h∗l (ξ) (3.12)

for (n, ξ) ∈ N , and by

(Φϕ)n(ξ) = (Φϕ)n(δβnξ/‖ξ‖) for (n, ξ) ∈ N.

We first show that for each fixed n ∈ N the series in (3.12) converges.By (2.4) and (3.3), we have

‖h∗l (ξ)‖ ≤ K2D

q+2n βq+1

n δq+1e(q+1)a(l−n)


for some constant K2 > 0. Since T := a − b < 0, it follows from the secondinequality in (2.3) that

∞∑l=n

‖C(l + 1, n)−1‖ · ‖h∗l (ξ)‖ ≤ K2e

−bDq+2n δq+1βq+1

n

∞∑l=n

Dl+1e(T+qa)(l−n)

≤ K2e−bDnβnδ

q+1Dq+1n βq

n

∞∑l=n

Dl+1eqa(l−n)

= K2e−bDnβnδ

q+1.

When ξ = 0 we have xm = 0 for every ϕ ∈ X∗ and m ≥ n. Thus,(Φϕ)m(0) = 0. Given ϕ ∈ X∗, and (n, ξ), (n, ξ) ∈ N , consider the sequencesx and x given by Lemma 3.2 with xn = ξ and xn = ξ. Proceeding as in (3.8),and using (3.3) and (3.7) we obtain

c(l) := ‖h∗l (ξ) − h∗

l (ξ)‖ ≤ K3Dq+1n δqβq

ne(q+1)a(l−n)‖ξ − ξ‖,

for some constant K3 = K3(c, q) > 0. Using the second inequality in (2.3) weconclude that

‖(Φϕ)n(ξ) − (Φϕ)n(ξ)‖ ≤∞∑l=n

‖C(l + 1, n)−1‖c(l)

≤ K3e−bDq+1

n δqβqn

∞∑l=n

Dl+1e(T+qa)(l−n)

= K3e−bδq.

(3.13)

For any sufficiently small δ > 0 (depending on b, q and c), we have

‖(Φϕ)n(ξ) − (Φϕ)n(ξ)‖ ≤ ‖ξ − ξ‖for every ξ, ξ ∈ Qn(δβn). This shows that Φ(X∗) ⊂ X∗.

Now we show that Φ: X∗ → X∗ is a contraction. Proceeding as in (3.10),and using (3.3) and (3.9), we obtain

d(l) : = ‖hl(xl, ϕl(xl)) − hl(yl, ψl(yl))‖≤ K4D

q+1n δqβq

ne(q+1)a(l−n)‖ξ‖ · ‖ϕ− ψ‖

for some constant K4 = K4(c, q) > 0. Proceeding as in (3.13), we concludethat

‖(Φϕ)n(ξ) − (Φψ)n(ξ)‖ ≤∞∑l=n

‖C(l + 1, n)−1‖d(l)

≤ K4e−bδq‖ξ‖ · ‖ϕ− ψ‖.

For any sufficiently small δ > 0 (depending on b, q and c), we have

‖Φϕ− Φψ‖ ≤ θ‖ϕ− ψ‖for some θ < 1, and Φ is a contraction in the complete metric space X∗.Hence, there exists a unique ϕ ∈ X∗ satisfying Φϕ = ϕ. In particular, in view


of (3.12), identity (3.11) holds for every (n, ξ) ∈ N . This completes the proofof the lemma. �

We can now establish Theorem 2.1.

Proof of Theorem 2.1. By Lemma 3.2, for each ϕ ∈ X∗ there exists a uniquesequence x = xϕ satisfying the first equation in (3.2), and thus it remains tosolve the second equation in (3.2) with x = xϕ. One can easily verify that ifϕ satisfies this equation for (n, ξ) ∈ N , then identity (3.11) holds for every(n, ξ) ∈ N . Indeed, it follows from (3.2) that

ϕn(ξ) = C(n,m)ϕm(xm) −m−1∑l=n

C(n, l + 1)hl(xl, ϕl(xl)), (3.14)

and

‖C(n,m)ϕm(xm)‖ ≤ Dme−b(m−n)‖xm‖ ≤ 2DmDne(a−b)(m−n)‖ξ‖.

By (2.9), we obtain

C(n,m)ϕm(xm) → 0 when m → ∞,

and identity (3.11) follows from (3.14).On the other hand, by Lemma 3.5, there exists a unique ϕ ∈ X∗ such

that (3.11) holds for every (n, ξ) ∈ N . Since C(m,n)C(l+1, n)−1 = C(m, l+1),for each m ≥ n we obtain

C(m,n)ϕn(ξ) +m−1∑l=n

C(m, l + 1)hl(xl, ϕl(xl))

= −∞∑

l=m

C(l + 1,m)−1hl(xl, ϕl(xl)).

(3.15)

Furthermore,

‖xm‖ ≤ 2Dnea(m−n)‖ξ‖ ≤ 2Dne

a(m−n) δβn

2Dn≤ δβn

for every ξ ∈ Qn(δαn), and hence, (m,xm) ∈ N . Thus, it follows from (3.11)that the series in the right-hand side of (3.15) is ϕm(xm). This shows thatthe sequence x satisfies the second identity in (3.2) for every n ∈ N andξ ∈ Qn(δαn).

It remains to establish inequality (2.11). It follows from Lemma 3.3 that

‖(xm, ϕm(xm)) − (xm, ϕm(xm))‖ ≤ 2‖xm − xm‖≤ 4Dne

a(m−n)‖ξ − ξ‖for every m ≥ n. This completes the proof of the theorem. �


4. Smoothness of the Stable Manifolds

We consider in this section the finite-dimensional case and we establish the C1

regularity of the stable manifolds when the functions fn are of class C1. Thereason for this separation between the finite and infinite-dimensional cases isthat only in the finite-dimensional case we are able to establish the optimalC1 regularity of the stable manifolds (without requiring further regularity forthe maps fn).

More precisely, we take X = Rk and we consider the sets

V′n =

{(ξ, ϕn(ξ)) : ξ ∈ intQn(δαn)

} ⊂ Vn. (4.1)

Theorem 4.1. If the sequence (An)n∈N admits a D-exponential dichotomy,(fn)n∈N is a sequence of C1 maps satisfying (2.4) for some q > 1, and theconditions in (2.9) hold, then there exists δ > 0 such that for each n ∈ N theset V′

n is a smooth manifold of class C1 such that T0V′n = En.

The proof of Theorem 4.1 is given in several steps. Because of thenonuniform hyperbolicity in (2.3), we introduce Lyapunov norms. Let

‖u‖′m = supk≥m

(‖B(k,m)u‖e−a(k−m))

for u ∈ Em,

‖v‖′m = supk≤m

(‖C(m, k)−1v‖eb(m−k))

for v ∈ Fm.(4.2)

By (2.3), we have

‖u‖ ≤ ‖u‖′m ≤ Dm‖u‖, ‖v‖ ≤ ‖v‖′m ≤ Dm‖v‖. (4.3)

Furthermore,

‖Bmu‖′m+1 ≤ ea supk≥m

(‖B(k,m)u‖e−a(k−m)) ≤ ea‖u‖′m (4.4)

and

‖C−1m v‖′m ≤ e−b sup

k≤m+1

(‖C(m + 1, k)−1v‖eb(m+1−k)) ≤ e−b‖v‖′m+1. (4.5)

4.1. Existence of Invariant Families of Cones

The first step in the proof of Theorem 4.1 is to establish the existence of aninvariant family of cones along each trajectory of F(m,n).

We first give a bound for the derivatives of the perturbation.

Lemma 4.2. We have

max{∥∥∥∥∂gm∂x

∥∥∥∥ , ∥∥∥∥∂gm∂y∥∥∥∥ , ∥∥∥∥∂hm

∂x

∥∥∥∥ , ∥∥∥∥∂hm

∂y

∥∥∥∥} ≤ 2c‖(x, y)‖q.

Proof. Since gm is differentiable, it follows from (2.4) that for every v ∈ Em,∥∥∥∥∂gm∂xv

∥∥∥∥ = limh→0

∥∥gm(t, x + vh, y) − gm(t, x, y)∥∥

|h|

≤ c limh→0

‖x + vh− x‖(‖(x + vh, y)‖q + ‖(x, y)‖q)|h|

≤ 2c‖(x, y)‖q‖v‖.


Therefore, ‖∂gm/∂x‖ ≤ 2c‖(x, y)‖q. Proceeding in a similar manner with thederivatives ∂gm/∂y, ∂hm/∂x and ∂hm/∂y we obtain the desired result. �

Fix γ > 0. For each m ∈ N, we consider the cone

Cm ={(v, w) ∈ Em × Fm : ‖w‖′m < γ‖v‖′m

} ∪ {(0, 0)}.The following lemma shows that this family is invariant under the differentialof F(m,n). Let

Nα ={(n, ξ) : n ∈ N and ξ ∈ Qn(δαn)

},

andpm = (xm, ϕm(xm)) = F(m,n)(ξ, ϕn(ξ)) for each m ≥ n. (4.6)

Lemma 4.3. Given δ > 0 sufficiently small, for each (n, ξ) ∈ Nα and m ≥r ≥ n we have (

dpmF(m, r)−1)Cm ⊂ Cr. (4.7)

Proof. Take (v, w) ∈ Cm+1 and consider the vector (v′, w′) satisfying

v = Bmv′ + dpmgm(v′, w′), w′ = C−1

m w − C−1m dpm

hm(v′, w′). (4.8)

By Lemma 4.2, (4.3), and Lemma 3.2 we have

‖dpmgm‖′m+1 ≤ 2q+1cDm+1‖xm‖q

≤ 22q+1cDqne

qa(m−n)δqαqnDm+1.

By the definition of ε in (2.1) and since qa + ε < 0 we have

κ1 := supm≥n

(eqa(m−n)Dm+1

)< +∞

and also

κ2 := αqnD

qn ≤ βq

nDq+1n ≤ 1∑

j≥n Dj+1eqa(j−n)< +∞.

Hence,‖dpmgm‖′m+1 ≤ 22q+1cδqκ1κ2 =: ζ. (4.9)

We have a similar inequality with gm replaced by hm.By (4.4), (4.8) and (4.9), we obtain

‖v‖′m+1 ≤ ea‖v′‖′m + ζ(‖v′‖′m + ‖w′‖′m). (4.10)

Similarly, in view of (4.5), (4.8) and (4.9) (with gm replaced by hm), we have

‖w′‖′m ≤ e−b‖w‖′m+1 + e−bζ(‖v′‖′m + ‖w′‖′m). (4.11)

Since ‖w‖′m+1 ≤ γ‖v‖′m+1, it follows from (4.10) that

‖w′‖′m ≤ γea−b‖v′‖′m + e−b(1 + γ)ζ(‖v′‖′m + ‖w′‖′m).

Thus, we obtain

‖w′‖′m ≤ γea−b + e−b(1 + γ)ζ1 − e−b(1 + γ)ζ

‖v′‖′m.


Since a − b < 0, for any sufficiently small ζ (and thus for any sufficientlysmall δ, depending on a, b, q, c and D), we conclude that ‖w′‖′m < γ‖v′‖′mwhenever v′ = 0. This shows that(

dpm+1F(m + 1,m)−1)Cm+1 ⊂ Cm (4.12)

for every m ≥ n. The desired result follows now from a repeated applicationof (4.12). �

For each m ∈ N, we also consider the complementary cone

C ′m =

{(v, w) ∈ Em × Fm : ‖v‖′m < γ−1‖w‖′m

} ∪ {(0, 0)}.Lemma 4.4. Given δ > 0 sufficiently small, for each (n, ξ) ∈ Nα and m ≥r ≥ n we have

(dprF(m, r))C ′

r ⊂ C ′m. (4.13)

Proof. We note that

C ′r = (Rk \ Cr) ∪ {(0, 0)} and C ′

r = (Rk \ Cr) ∪ {(0, 0)}.Therefore, by Lemma 4.3,

(dprF(m, r))C ′

r =(R

k \ (dprF(m, r))Cr

) ∪ {(0, 0)}⊂ (Rk \ Cm) ∪ {(0, 0)} = C ′

m,

which yields the desired result. �

4.2. Construction of the Stable Spaces

Given (n, ξ) ∈ Nα, let (see (4.6))

En(ξ) =⋂

m≥n

(dpm

F(m,n)−1)Cm. (4.14)

It is shown in Lemma 4.7 that En(ξ) is a vector space (independent of γ),with the same dimension as the space En. We call it a stable space.

We first establish some auxiliary results concerning the speed at whichthe norms of vectors inside and outside the cones vary along a given trajec-tory.

Lemma 4.5. Given δ > 0 sufficiently small, for each (n, ξ) ∈ Nα, j ≥ 0 and(v, w) ∈ Cn+j, we have∥∥(dpn+jF(n + j, n)−1

)(v, w)

∥∥′n≥ e−aj

2(1 + γ)‖(v, w)‖′n+j .

Proof. Take l ∈ N with l ≤ n + j and consider the vector

(vl, wl) =(dpn+j

F(n + j, l)−1)(v, w).

By Lemma 4.3 (see (4.7)), we have (vl, wl) ∈ Cl and thus ‖wl‖′l ≤ γ‖vl‖′l forevery l. We note that (

vlwl

)=

(∂xl

∂xn

∂xl

∂yn∂yl

∂xn

∂yl

∂yn

)(vnwn

),


and thus,

v = vn+j = B(n + j, n)vn +n+j−1∑l=n

B(n + j, l + 1)dplgl(vl, wl).

Proceeding as in (4.9), we obtain

‖dplgl‖ ≤ 22q+1cDq

nαqnδ

qeaq(l−n) =: ζβqne

aq(l−n).

By (4.4), we have‖B(n + j, n)vn‖′n+j ≤ eaj‖vn‖′n,

and hence, using (2.3) and (4.3) we conclude that

‖v‖′n+j ≤ eaj‖vn‖′n + ζβqn

n+j−1∑l=n

Dl+1ea(n+j−l−1)eaq(l−n)‖(vl, wl)‖′l

≤ eaj‖vn‖′n + eaj ζβqn(1 + γ)e−a

n+j−1∑l=n

Dl+1ea(q−1)(l−n)‖vl‖′l.

Letting T (j) = e−aj‖vn+j‖′n+j , we obtain

T (j) ≤ ‖vn‖′n + ζβqn(1 + γ)e−a

n+j−1∑l=n

Dl+1eaq(l−n)T (l − n).

Now we set T = supl≥0 T (l). Since aq < 0, we obtain

T ≤ ‖vn‖′n + ζ(1 + γ)e−aT,

and taking δ sufficiently small (depending on a, q and c, for each given γ) sothat

ζ(1 + γ)e−a ≤ 12,

yields the inequality T ≤ 2‖vn‖′n. This implies that

‖vn‖′n ≥ e−aj

2‖v‖′n+j . (4.15)

Since (v, w) ∈ Cn+j , we have ‖(v, w)‖′n+j ≤ (1 + γ)‖v‖′n+j , and it followsfrom (4.15) that

‖(vn, wn)‖′n ≥ ‖vn‖′n ≥ e−aj

2‖v‖′n+j ≥

e−aj

2(1 + γ)‖(v, w)‖′n+j .


Now we establish an analogous result for vectors outside the cones.

Lemma 4.6. Given δ > 0 sufficiently small, for each (n, ξ) ∈ Nα, j ≥ 0 andz ∈ F we have

‖(d(ξ,ϕn(ξ))F(n + j, n))(0, z)‖′n+j ≥ e(b+κ)j‖(0, z)‖′n, (4.16)

where κ = log(1 − e−bζ(1 + γ−1)).


Proof. Let(vn+j , wn+j) = (d(ξ,ϕn(ξ))F(n + j, n))(0, z).

Since‖(vn+j , wn+j)‖′n+j ≥ ‖wn+j‖′n+j ,

it is sufficient to find a lower bound for ‖wn+j‖′n+j . Since (0, z) ∈ Dn, itfollows from (4.13) that (vn+j , wn+j) ∈ Dn+j for every j ≥ 0, and thus,

‖vn+j‖n+j ≤ γ−1‖wn+j‖′n+j .

We prove (4.16) by induction on j. For j = 0 there is nothing to prove.Assume now that (4.16) holds for j = k. Proceeding as in (4.11), we obtain

‖wn+k‖′n+k ≤ e−b‖wn+k+1‖′n+k+1 + e−bζ(‖vn+k‖′n+k + ‖wn+k‖′n+k)

≤ e−b‖wn+k+1‖′n+k+1 + e−bζ(1 + γ−1)‖wn+k‖′n+k,

and by the induction hypothesis,

‖wn+k+1‖′n+k+1 ≥ eb(1 − e−bζ(1 + γ−1))‖wn+k‖′n+k

≥ e(b+κ)k‖(0, z)‖′n.This establishes (4.16). �

We note that κ can be made as small as desired by taking δ sufficientlysmall.

4.3. Continuity of the Stable Spaces

We use the lemmas above to show that the set En(ξ) in (4.14) is a vectorspace varying continuously with ξ.

Lemma 4.7. Given δ > 0 sufficiently small, the following properties hold:

1. the set En(ξ) is a subspace with dimension dimEn(ξ) = dimEn for each(n, ξ) ∈ Nα;

2. the map Qn(δαn) � ξ �→ En(ξ) is continuous for each n ∈ N.

Proof. By Lemma 4.3, En(ξ) is a nonempty closed subset of Cn. For eachm ∈ N, the set

⋂mj=n(dpj

F(j, n)−1)Cj contains a subspace Em of dimensiondimEn. By the compactness of the unit ball in R

k, there exists a subspaceE′ ⊂ En(ξ) of dimension dimEn.

Given v ∈ En(ξ), we write v = v1 + v2 with v1 ∈ E′ and v2 ∈ Fn. Sincev, v1 ∈ En(ξ), it follows from (4.14) that (dpn

F(m,n))v and (dpnF(m,n))v1

are in Cm. By Lemmas 4.5 and 4.6, we obtain

‖v2‖′n ≤ 2e−(b+κ)(m−n)‖(dpnF(m,n))v2‖′m

= 2e−(b+κ)(m−n)‖(dpnF(m,n))(v − v1)‖′m≤ 4(1 + γ)e(a−b+κ)(m−n)(‖v‖′n + ‖v1‖′n).

Taking δ > 0 sufficiently small (depending on a, b, q, c and D, for each given γ)we have a − b + κ < 0. Hence, letting m → ∞ we obtain v2 = 0 and thusv ∈ E′. This shows that En(ξ) = E′ is a subspace of dimension dimEn.


For the continuity property, note that v ∈ En(ξ) if and only if

(dpnF(m,n))v ∈ Cm for every m ≥ n.

Consider a sequence (ξk)k ∈ Qn(δαn) converging to a point ξ ∈ Qn(δαn)when k → +∞, and a sequence vk ∈ En(ξk) with ‖vk‖ = 1 for each k ∈ N.Then

(d(ξk,ϕn(ξk))F(m,n))vk ∈ Cm for every k ∈ N, m ≥ n. (4.17)

We first assume that (vk)k converges, to some vector v ∈ En ×Fn. It followsfrom the assumptions in Theorem 4.1 and the Lipschitz property of ϕn in(2.8) that ξ �→ ∂(ξ,ϕn(ξ))F(m,n) is continuous, and hence, for every m ≥ n,

(d(ξk,ϕn(ξk))F(m,n))vk → (d(ξ,ϕn(ξ))F(m,n))v when k → ∞.

It follows from (4.17) that

(∂(ξ,ϕn(ξ))F(m,n))v ∈ Cm for every m ≥ n.

Therefore, v ∈ En(ξ) (see (4.14)). When (vk)k does not converge, let (mk)k ⊂N be a sequence such that (vmk

)k converges, say to v ∈ En × Fn (recall that‖vk‖ = 1 for each k). Proceeding in a similar manner, it follows from (4.17)that

(d(ξ,ϕn(ξ))F(m,n))v ∈ Cm for every m ≥ n,

and v ∈ En(ξ), that is, any sublimit of the sequence (vk)k is in En(ξ). Itfollows from the identities

dimEn(ξk) = dimEn(ξ) = dimEn,

that any sublimit of a sequence of orthonormal bases (with respect to theoriginal norm ‖·‖) of the spaces En(ξk) is also an orthonormal basis of En(ξ).Therefore,

En(ξk) → En(ξ) when k → ∞,

and the map ξ �→ En(ξ) is continuous. �

4.4. Behavior of the Tangent Sets

Given n ≥ 0 and ξ, ξ ∈ Qn(δαn) with ξ = ξ, let

Δξ,ξϕn =(ξ, ϕn(ξ)) − (ξ, ϕn(ξ))

‖(ξ, ϕn(ξ)) − (ξ, ϕn(ξ))‖and

tξϕn ={v ∈ E × F : Δξ,ξmϕn → v for some sequence ξm → ξ

}.

We define the tangent set of the graph of ϕn at ξ ∈ En by

Vn(ξ) = {λv : v ∈ tξϕn and λ ∈ R}.One can easily verify that the function ϕn is differentiable at ξ if and only ifVn(ξ) is a subspace of dimension dimVk(ξ) = dimEn.


Lemma 4.8. Given δ ∈ (0, 1) sufficiently small, for each (n, ξ), (n, ξ) ∈ Nα

and j > l ≥ 0 we have

‖yn+l − yn+l‖′n+l ≤ e−b(j−l)‖yn+j − yn+j‖′n+j + 2ηe−bδq‖xn+l − xn+l‖′n+l,

for some constant η > 0, where yn+l = ϕn+l(xn+l) and yn+l = ϕn+l(xn+l).

Proof. We have

yn+l − yn+l = C(n + j, n + l)−1(yn+j − yn+j)

−n+j−1∑m=n+l

C(n + l,m + 1)(hm(xm, ym) − hm(xm, ym)

).

By Lemma 3.2, we also have

‖xn+l‖ ≤ 2Dneal‖ξ‖ ≤ 2Dne

al δβn

2Dn≤ δβn.

Now we show that

‖xn+j − xn+j‖′n+j ≤ 2ea(j−l)‖xn+l − xn+l‖′n+l (4.18)

for each (n, ξ), (n, ξ) ∈ Nα and j ≥ l ≥ 0. Let

ρn+j = ‖xn+j − xn+j‖′n+j and Tm = e−a(m−n−l)ρm.

By the first inequality in (2.3), using (4.2), (4.3) and (4.4), it follows from(3.2) and (3.8) that

ρn+j ≤ supk≥n+j

(‖B(k, n + j)B(n + j, n + l)(xn+l − xn+l)‖e−a(k−n−j))

+n+j−1∑s=n+l

supk≥n+j

(‖B(k, n + j)B(n + j, s + 1)‖

×KδqD1+qn βq

neqa(s−n)ρse

−a(k−n−j))

≤ ea(j−l) supk≥n+l

(‖B(k, n + l)(xn+l − xn+l)‖e−a(k−n−l))

+ KδqD1+qn βq

ne−a

n+j−1∑s=n+l

Ds+1ea(n+j−s)+qa(s−n)ρs

≤ ea(j−l)‖xn+l − xn+l‖′n+l

+ ea(j−l)KδqD1+qn βq

ne−a

n+j−1∑s=n+l

Ds+1e−a(s−n−l)eqa(s−n)ρs.

Therefore,

Tn+j ≤ ‖xn+l − xn+l‖′n+l + KδqD1+qn βq

ne−a

n+j−1∑s=n+l

Ds+1eqa(s−n)Ts.


Letting T = supm∈N Tm, for any sufficiently small δ > 0 (depending on a, qand c) we have

T ≤ ‖xn+l − xn+l‖′n+l + Kδqe−aT

≤ ea(j−l)‖xn+l − xn+l‖′n+l + T/2.

Hence, T ≤ 2‖xn+l − xn+l‖′n+l which yields inequality (4.18).Using (2.3) and proceeding as in (3.8), we find that ‖yn+l − yn+l‖′n+l is

bounded by

supk≤n+l

(‖C(n + l, k)−1C(n + j, n + l)−1(yn+j − yn+j)‖eb(n+l−k))

+ ηδqD1+qn βq

n

n+j−1∑m=n+l

supk≤n+l

(‖C(n + l, k)−1C(n + l,m + 1)‖

× eaq(m−n)‖xm − xm‖eb(n+l−k))

≤ e−b(j−l)‖yn+j − yn+j‖′n+j

+ ηδqD1+qn βq

n

n+j−1∑m=n+l

supk≤n+l

(‖C(k,m + 1)‖eaq(m−n)‖xm − xm‖′meb(n+l−k))


+ ηδqe−bD1+qn βq

n

n+j−1∑m=n+l

Dm+1e−b(m−n−l)eaq(m−n)‖xm − xm‖′m


+ 2ηδqe−bD1+qn βq

n

×n+j−1∑m=l+n

Dm+1eaq(m−n)e−b(m−n−l)ea(m−n−l)‖xn+l − xn+l‖′n+l


+ 2ηδqe−bD1+qn βq

n

∑m≥n

Dm+1eqa(m−n)‖xn+l − xn+l‖′n+l

≤ e−b(j−l)‖yn+j − yn+j‖′n+j + 2ηe−bδq‖xn+l − xn+l‖′n+l,

for some constant η > 0. This completes the proof of the lemma. �

One can now establish a relation between the tangent sets and the in-variant family of cones along each trajectory. We recall that xm = xm(ξ) isdefined by (3.2).

Lemma 4.9. Given δ > 0 sufficiently small, for each (n, ξ) ∈ Nα and m ≥ nwe have Vm(xm) ⊂ Cm.

Proof. We proceed by contradiction. Namely, assume that for a given γ > 0there exists m ≥ n such that Vm(xm) \ Cm = ∅. Then there exists ξ ∈


Qn(δαn) arbitrarily close to ξ for which

‖ym − ym‖′m > γ‖xm − xm‖′m,

where x and x are the sequences given by Lemma 3.2 with xn = ξ and xn = ξ.Now write m = n + l and take j ≥ l. By Lemma 4.8, we obtain

‖xn+l − xn+l‖′n+l ≤ γ−1e−b(j−l)‖yn+j − yn+j‖′n+j

+ 2γ−1ηe−bδq‖xn+l − xn+l‖′n+l.(4.19)

In view of the definition of η, one can choose δ > 0 sufficiently small (de-pending on b, q and c) such that 2γ−1ηe−bδq ≤ 1/2. Hence, it follows from(4.19) that

‖xn+l − xn+l‖′n+l ≤ 2γ−1e−b(j−l)‖yn+j − yn+j‖′n+j .

By (4.18) and (4.3), we obtain

‖xn+j − xn+j‖ ≤ ‖xn+j − xn+j‖′n+j

≤ 4γ−1e(a−b)(j−l)‖yn+j − yn+j‖′n+j

≤ 4γ−1Dn+je(a−b)(j−l)‖yn+j − yn+j‖.

By the definition of ε in (2.1), for each ρ > ε there exists j such thatlogDn+j < ρ(n + j). Therefore,

‖xn+j − xn+j‖ ≤ 4γ−1e(a−b+ρ)(j−l)+ρ(n+l)‖yn+j − yn+j‖.Recall that n and l are fixed. Hence, since a−b+ε < 0 (see (2.9)), take ρ > εsuch that a− b + ρ < 0 there exists j > l such that

‖xn+j − xn+j‖ < ‖yn+j − yn+j‖.But this contradicts the fact that the points (xn+j , yn+j) and (xn+j , yn+j) be-long to the stable manifold, since ϕn+j is Lipschitz with Lipschitz constant 1(see (2.8)). This completes the proof of the lemma. �

4.5. Proof of Theorem 4.1One can now show that the Lipschitz manifold V′

n ⊂ Vn (see (4.1)) is asmooth manifold of class C1.

Proof of Theorem 4.1. We note that Δξ,ξlϕn → v when l → ∞ (for ξl → ξ)if and only if

liml→∞

F(m,n)(ξl, ϕn(ξl)) − F(m,n)(ξ, ϕn(ξ))‖F(m,n)(ξl, ϕn(ξl)) − F(m,n)(ξ, ϕn(ξ))‖ =

(d(ξ,ϕn(ξ))F(m,n))v‖(d(ξ,ϕn(ξ))F(m,n))v‖

for every m ∈ N. This implies that

(d(ξ,ϕn(ξ))F(m, j))Vn(ξ) = Vm(xm). (4.20)

Now let (n, ξ) ∈ Nα. By Lemma 4.9, we have Vm(xm) ⊂ Cm for every m ≥ n.Therefore, in view of (4.20),

Vn(ξ) ⊂ (dF(m,n)(ξ,ϕn(ξ))F(m,n)−1)Cm


for every m ≥ n, and hence, it follows from (4.14) that Vn(ξ) ⊂ En(ξ).On the other hand, for each v ∈ En \ {0} there exists a sequence tl → 0such that Δξ,ξ+tlvϕn converges when l → +∞ (due to the compactness ofthe closed unit ball in R

k). This implies that the first dimEn components ofVn(ξ) project onto En. On the other hand, by Lemma 4.7, the space En(ξ)has dimension dimEn and hence Vn(ξ) = En(ξ).

In particular, Vn(ξ) is a subspace of dimension dimEn. Therefore (seethe beginning of Section 4.4), the function ϕn is differentiable at each pointξ ∈ intQn(δαn). It follows from the continuity of the map ξ �→ En(ξ)(see Lemma 4.7) and the identity Vn(ξ) = En(ξ) that ϕn is of class C1 (sincethe tangent set varies continuously). Hence, V′

n is a C1 manifold for n ∈ N.Moreover, by Lemma 3.5, for each (n, ξ) ∈ Nα we have

ϕn(ξ) = −∞∑l=n

C(l + 1, n)−1hl(xl, ϕl(xl)). (4.21)

By Lemma 4.2, d0hl = 0 for every l, and hence, taking derivatives withrespect to ξ in (4.21), we obtain d0ϕn = 0. Therefore, T0V

′n = En and the

proof of the theorem is complete. �

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Luis Barreira and Claudia VallsDepartamento de Matematica, Instituto Superior TecnicoUniversidade Tecnica de Lisboa, Av. Rovisco Pais, 11049-001 Lisboa, Portugale-mail: [email protected]

[email protected]

Received: May 16, 2012.Revised: September 18, 2012.Accepted: October 19, 2012.

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Stable Manifolds and Nonuniform Hyperbolicity: Optimal Estimates