Local Stable Manifold of Of

A NEW PROOF OF THE STABLE MANIFOLD THEOREM FOR HYPERBOLIC FIXED POINTS ON SURFACES

MARK HOLLAND AND STEFANO LUZZATTO

ABSTRACT. We introduce a new technique for proving the classical Stable Manifold theorem for hyperbolic fixed points. This method is much more geometrical than the standard approaches which rely on abstract fixed point theorems. It is based on the convergence of a canonical sequence of “finite time local stable manifolds” which are related to the dynamics of a finite number of iterations.

1. INTRODUCTION

1.1. Stable sets. Let be a two-dimensional Riemannian manifold with Riemannian metric ¡ ,

¦¨§ © and let ¢¤£ be a diffeomorphism. Suppose that is a ﬁxed point. A classical question concerns the effect that the existence of such a ﬁxed point has on the global dynamics

of . In particular we shall concentrate here on some properties of the set of points which are

forward asymptotic to © . ©

Deﬁnition 1. The global stable set © of is

¥ # ¥ %'&(

In general )© can be extremely complicated, both in its intrinsic geometry [14] and/or in

the way it is embedded in [18, 19, 22] It is useful therefore to begin with a study of that part

of which remains in a ﬁxed neighbourhood of for all forward iterations. For and 1)2

$-.0/ let

# &

43 £¡6¢879:!"6©<;=+?> ;A@B;C$ED=/ " 12GF

and 1)2

(

5 5

3 IH 43

JLK

# ©

Deﬁnition 2. For +M, , deﬁne the local stable set of by

2GF

(

5 5

In several situations it is possible to obtain a fairly comprehensive description of the geometrical and dynamical properties of the local stable manifold. In this paper we shall focus on the

simplest setting of a hyperbolic ﬁxed point. We recall that the ﬁxed point © is hyperbolic if the

derivative Q-¢!R has no eigenvalues on the unit circle.

Date: February, 2005. 1991 Mathematics Subject Classiﬁcation. Primary [2000] 37D10. 1

2 MARK HOLLAND AND STEFANO LUZZATTO

£ ¦¨§

Theorem. Let ¢ be a diffeomorphism of a Riemannian surface and suppose that

# #

¡£¢¥¤ ¢¦¡ ¡£¢¥¤¨§©¢

/ + ,

+ ,

constant such that the following properties hold:

) *© ¦

K ¨

(1) is one-dimensional submanifold of M tangent to R ;

¢ ¢

(2) 0 on either side of ;

(4)

(

5 5

H 43

J Remark 1. For ease of exposition, and to illustrate the main ideas of the construction we work in dimension 2. With a small amount of work we can extend out result to hyperbolic ﬁxed points

in dimension . In this case the method of construction will select the one-dimensional stable manifold tangent to the most contracting eigenspace at the ﬁxed point, assuming no multiplicity of the smallest eigenvalues. We comment further on the higher dimensional situation in Section 1.5.

Perhaps the key statement here is that the local stable set is actually a smooth submanifold of

. Notice that the global stable set can be written as a union of preimages of since any point ©

whose orbit converges to © must eventually remain in a neighbourhood of and therefore must 5

eventually belong to ) . Thus we write

(

© The smoothness of then implies that is also a smooth submanifold of . We remark that while the local stable manifold is an embedded submanifold, i.e. a manifold in its own right in the induced topology, the global stable manifold is in general only an immersed submanifold, i.e. a manifold in its intrinsic topology but not in the topology induced by the topology on . This is because it may accumulate on itself and thus fail to be locally connected in the induced topology.

1.2. Main ideas. The proof is based on the key notions of hyperbolic coordinates and finite time local stable manifolds. These are not standard concepts and therefore we describe them briefly here, leaving the details to the main body of the paper. The standard definition of hyperbolicity involves a decomposition of the tangent space into invariant subspaces, which coincide with the real eigenspaces in the case of a fixed point. However this decomposition is specifically related to the asymptotic properties of the dynamics and is not always the most useful or the most appropriate for studying the local geometry related to the dynamics for a finite number of

iterations. Instead, elementary linear algebra arguments show that under extremely mild hyper-

QM¢

bolicity conditions on the linear map , there are well-deﬁned most contracted directions

¦ ¢ ¦¨§

43 which depend in a manner on the point if is . In particular, they deﬁne a fo-

¢ liation of integral curves 43 which are most contracted curves under . They are the natural notion of local stable manifold relative to a ﬁnite number of iterations. A NEW PROOF OF THE STABLE MANIFOLD THEOREM 3

In general, even for a hyperbolic ﬁxed point, the ﬁnite time local stable manifolds will not

coincide with the “real” asymptotic local stable manifold and will depend on the iterate $ . It is natural to expect however that the two concepts are related in the sense that finite time local stable manifolds converge to the asymptotic local stable manifold. In this paper we show that this is indeed what happens. The argument does not require previous knowledge of the existence of the asymptotic local stable manifold, and thus we obtain from the construction an alternative proof of the classical stable manifold theorem. The inspiration for this approach comes from the pioneering work of Benedicks and Carleson on non-uniformly hyperbolic dynamics in the context of the Henon´ map [3], the generaliza- tion of this work to Henon-lik´ e maps by Mora and Viana [17], and the further refinement of these ideas in [16, 26]. In these papers, a sophisticated induction argument is developed to show that a certain non-uniformly hyperbolic structure is persistent, in a measure-theoretic sense, in typical parametrized families of maps. The induction necessarily requires some knowledge of geometrical structures based only on a finite number of iterations. Thus the notion of contractive directions is very natural and very efficient for producing dynamical foliations which incorporate

and reflect information about the first $ iterates of the map. We remark however that in the papers mentioned above, the construction of these manifolds is very much embedded in the overall argument and there is no focus on the intrinsic interest of this particular construction. Moreover, the specific characteristics of the maps under consideration, such as the strong dissipativity and specific distortion bounds, are used heavily in the estimates and it is not immediately clear what precise assumptions are used in the construction. Thus the main purpose of the present paper can be seen as a first step in an attempt to draw on the ideas first introduced in [3] and to formalize them and generalize them into a fully fledged theory of invariant manifolds.

1.3. Structure of the argument. We start the proof with some relatively standard hyperbolicity

and distortion estimates in a neighbourhood of © in Section 2. In sections 3 and 4 we address

the ﬁrst main issue which is the convergence of ﬁnite time local stable manifolds. This will 2

be addressed by carrying2 out several estimates which show that angle between successive most

Q-¢

!K63

contracted direction 43 and depends on the hyperbolicity of . If some hyperbolicity 2GF

conditions are satisﬁed for all $ , this sequence of angles forms a Cauchy sequence and therefore

a limit direction 3 exists. We then show that we can also control the way in which contractive

directions2 depend on the base point and eventually conclude that the sequence of most contracted

begin to show that this limit “object” is indeed the local stable manifold of © by showing ﬁrst

of all that it has positive length. This requires a2 careful control of the size and geometry of the

domains in which the contracting directions 43 are defined, to ensure that the finite time local stable manifold 43 always have some fixed length. In section 6 we then show that this

curve is sufﬁciently smooth. In section 7 we show that points converge exponentially fast to © and, ﬁnally, in Section 8 we prove uniqueness in the sense that the curve we have constructed is the only set of points which satisfy the properties of a local stable manifold. To simplify the notation we shall suppose that both eigenvalues are positive, the other cases are all dealt with in exactly the same way. 4 MARK HOLLAND AND STEFANO LUZZATTO

1.4. Comparison with other approaches. The stable manifold theorem is one of the most basic results in the geometric theory of dynamical systems and differential equations. There exist several approaches and many generalizations [1,2,4–11,13,20,21,23–25,27]. We focus here on what we believe are the two main differences between our approach and existing ones. As far as we know, all existing approaches rely in one way or another on an application of the contraction mapping theorem to some suitable abstract space of candidate local stable manifolds, formulated e.g. as the graphs or sequences with certain properties. A suitable complete metric and an operator depending are then defined in such a way that a fixed point under the operator corresponds to an invariant submanifold with the required properties. The existence and uniqueness of the fixed point are then a consequence of showing that the operator is a contraction and the application of the contraction mapping theorem. Our approach differs significantly from either of the approaches mentioned in at least two ways. First of all we do not use the contraction mapping Theorem, not even in disguise. We show that the sequence of finite time local stable manifolds is Cauchy in an appropriate topology, by relating directly the distance between successive leaves to the hyperbolicity. This is potentially a much more flexible approach and one which may be adaptable to situations with less uniform and perhaps subexponential forms of hyperbolicity, see comments in the next section. Secondly, the finite time local stable manifolds which approximate the real local stable manifold are canonically defined and have an important dynamical meaning in their own right. Again, this aspect of the construction might be useful in other situation and in applications.

1.5. Generalizations and applications. In this paper we present the construction of the local stable manifold in the simplest setting of a hyperbolic fixed point in dimension 2, in order to describe the ideas and the basic strategy in the clearest possible way. However we have no doubt that with some work, the basic argument will generalize in various directions. In work in progress, the authors are extending the techniques to cover the higher dimensional situation and the more general uniformly hyperbolic and projectively hyperbolic cases. When using these techniques to approximate stable manifolds of dimension greater than one, a typical problem is that we cannot always integrate up the vector field of most contracting directions to produce the foliation of finite time stable manifolds. However, we believe the methods can be extended to a general higher dimensional setting but some new ingredient is needed in the proof, which we do not pursue in this present paper. The smoothness results are not yet optimal since it is known that the stable manifold is as smooth as the map, however we believe that the optimal estimates can be recovered with more sophisticated higher order distortion estimates. It is also possible to study questions related to the dependence of the local stable manifolds on the base point or on some parameter by simply including the corresponding additional derivative estimates in the calculations, as already carried out for example in the papers mentioned above in which the original ideas of the integral curves of the most contracting directions was first introduced. Perhaps the most important potential of this method, however, is to situations with very weak hyperbolicity. Indeed, the convergence of the finite time local stable leaves is given by the Cauchy property which follows essentially from a summability condition on the derivative. Thus, A NEW PROOF OF THE STABLE MANIFOLD THEOREM 5 in principle, this may be applicable along orbits for which the derivative does not necessarily ad- mit exponential estimates. This fact is not completely explicit in the present proof since we are dealing with very hyperbolic example, but a future paper [12] by the same authors will provide the minimal abstract conditions under which the convergence argument works. Finally we mention that there is also a vast amount of research literature concerning the explicit, often numerical, approximation of invariant manifolds. We speculate that the approach given here might be more suitable than the classical theorems as a theoretical backdrop to these numerical studies. The finite time local stable manifolds for example can probably be calculated with significant accuracy since they depend only a finite time information on the dynamics and the derivative. Our estimates then give conditions on how close these finite time manifolds are from the real thing.

Acknowledgements. In writing this article M. Holland acknowledges the support of the EPSRC, grant no. GR/S11862/01.

2. LOCAL HYPERBOLICITY AND DISTORTION ESTIMATES

The ﬁrst step in the proof is to use the smoothness of ¢ to show that some hyperbolic structure

exists in a neighbourhood of the point © . First of all we introduce some notation which we will

$-.0/

use extensively throughout the paper. For and we let

£ £ K K

¢¡Q-¢ ¡ " ¤¡ 6Q-¢ ¡ "

and

£ QM¢

denote the maximum expansion and the maximum contraction respectively of . Then

§¦

QM¢ 43

denote the hyperbolicity of . If a sequence of neighbourhoods is ﬁxed, as it will be

"©¨ ¨

below, we shall often use the notation ¨ and to denote the maximum values of these

1 2

quantities in 43 . Also, when expressing relationships between these quantities which hold for

all in 43 we shall often omit explicit reference to . We shall also use the same convention

for other functions of to be introduced below. We shall also use the notation

7 K

¢¡QM¢ ¡ "

# ( ( (

" " $ D / for @E We shall use to denote a generic constant which is allowed to depend only

on the diffeomorphism ¢ . For simplicity we shall allow this to change even within one inequality when this does not create any ambiguities or confusion. The following Lemma follows from standard estimates in the theory of uniform hyperbolicity.

We refer to [15] for details and proofs.

# # # #

, > , +8 , " $N. @ . "

Lemma1 1. such that , such that for all and all

" we have:

(

¤©§ ¤¨§ ¤ ¤

A 67 . . D 67 . 67 . . K D 7

7 7 6 MARK HOLLAND AND STEFANO LUZZATTO

Moreover, we also have that

¥ ( ¥

; ;

; and

¤£ 7 ¤£ 7 7 7

¥¡ 7 7¢¡¨ We will also give some estimates on the higher order derivatives which follow easily from the

hyperbolicity properties given above.

, >

Lemma 2. such that 43 we have

(

§ ¦ § ¦ §

Q ¢ ¡ ; ¡Q ¨§ © QM¢ ¡ ;

¡ and

( ( (

Q-¢ Q

Proof. Let and let 43 . Let denote differentiation of

7 K 7 K

with respect to the space variables. By the product rule for differentiation we have

( ( (

Q 'Q Q ¢

K K

( ( ( ( ( (

(1) (

K 7 !K 7 7 K

7¢¡¨

Taking norms on both sides of (1) and using the fact that

( ( ( ( ( (

7 K

CQ-¢ " Q-¢!7 "

K 7 !K 7 K

£ £

Q Q Q-¢ 'QB§ ¢ Q-¢

and by the chain rule, 7 , we get

7 K

§ § §

£ £

¡ ; ¡QM¢ ¡ ¡Q ¢ ¡ ¡Q ¢ ¡ ¡Q-¢

7¢¡¨

§ §

; ¡Q ¢ ¡

7¢¡¨

(

;

7¢¡¨

Q ¢ ¡ The last inequality follows from the fact that ¡ is uniformly bounded above. Then, by

Lemma 1 we have

K K K

(

§ § §

; ; ¡Q ¢ ¡ ; ;

7 7 7

7¢¡¨ 7¢¡¨ 7¡¨

This proves the ﬁrst part of the statement. To prove the second, we argue along similar lines,2

§© Q-¢ Q§ © Q-¢ M Q

this time letting . Then we have, as in (1) above,

( ( ( ( ( ( ( ( ( (

7 K

Q § © Q-¢

Moreover we have that £ ,

7 !K 7 7 K 7 !K K K

( ( (

7¡¨

£ £

§ © QM¢ Q CQ §© Q-¢ 6Q§ © Q-¢ Q-¢ 7

and, by the chain rule, also 7 .

7 K 7 A NEW PROOF OF THE STABLE MANIFOLD THEOREM 7

This gives

(

7 K

£ 7 £ 7 £

Q ¨§ © Q-¢ ¨§ © QM¢ Q § © Q-¢ ¨§ © QM¢ Q-¢

(2) £

7¡¨

By the multiplicative property of the determinant we have the equality

(

7 K

£ £

§© Q-¢ § © Q-¢87 § © Q-¢ ¦ § © LQ-¢

Therefore we get

¡Q § © Q-¢ ¡

¢ ¢

£ £

¡Q §© Q-¢ ¡ ; § © LQ-¢

¢ ¢

§ © LQ-¢

7¡¨

(

; ;

7¢¡¨

The ﬁrst inequality follows by taking norms on both sides of (2), the second inequality follows

2 2

¢ ¢

£ £

Q ¨§ © QM¢ ¡ §© Q-¢

from the fact that ¡ and are uniformly bounded above and below

3 3

§ © Q-¢ £

and the fact that , the third inequality follows from an application of Lemma

¡ 1.

3. HYPERBOLIC COORDINATES In the context of hyperbolic ﬁxed points (or general uniformly hyperbolic sets) we are used to thinking of the eigenspaces (or the subspaces given by the hyperbolic decomposition) as providing the basic axes or coordinate system associated to the hyperbolicity. However this

is not necessarily the most natural splitting of the space. Indeed, the hyperbolicity condition

¥ ¥

/ ¦ ;0/ QM¢

(notice that we always have ) implies that the linear map maps the

£ ©

¢¤£¦¥ ¢ Q-¢ §¢ ¨£¦¥

unit circle to an (non-circular)2 ellipse with well deﬁned ma-

" 43

jor and minor axes. The unit vectors 43 which are mapped to the minor and major axis re-

spectively of the ellipse, and are thus the most contracted and most expanded vectors respectively,

¡ ¡Q-¢ § "!#"%$& ¡ ¦ ¡' ) are given analytically as solutions to the differential equation

which can be solved to give the explicit formula

- ¦/- - ¦ -

(

,. , ,. * ,

§ §

K K

!()$+*)

(3)

¦- ¦ - - -

, , D ,0. D=,.

§ § § §

§ §

K K

8 MARK HOLLAND AND STEFANO LUZZATTO

2 2

43 3

In particular, and are always orthogonal and clearly do not in general correspond to the Q-¢

stable and unstable eigenspaces of . We shall adopt the notation

2 2

43 3

CQ-¢879 Q-¢879 436

436 and

7 7

where everything is of course relative to some given point . Notice that

£ £

¢¡ Q-¢ K ¡ K ¡ ¡ ¢¡Q-¢ 43 ¡

and 2

(

£ £

¢¡Q-¢ ¡ ¢¡ ¡ ¡QM¢ 6 43 ¡

2 2

Thus is a natural way of expressing the overall hyperbolicity of at .

3 43

We also deﬁne angles by

2 2 2

43 43 !K63

43P¢¡¨ 43 " !K63 " £¡¨ "

7 7 7

and

2 2 2 2 2 2

(

43 ) 43 " " $ 436" 3P D #"%$+ 3 " 436

2 2 2 2 2 2

(

3 43 43 43 43 43

D #" $& " ) " " $& "

Notice that

2 2 2

(

43 K63 7 3

7¡!K

2 2 2 2

and 2

3 43 43 43 43

Q D Q " $& " Q

which implies in particular that

2 2 2 2

(

43 3 K63 7 3

¡Q ¡ ; ¡Q ¡ ; ¡Q ¡ ¡Q¤

(4) ¡

7¡!K

where the constant depends only on the choice of the norms. The notion of most contracted and most expanded directions are central to the approach to the

stable manifold theorem given here. We2 start with a lemma concerning the convergence of the

¥ %

! $ 3

sequence of most contracted direction as .

2 2 2

, >M$-.0/ ¥ :!

!K63 43 43

Lemma 3. such that and and , we have

¥ (

¢ ¢

3 ;

2 2 2

2 2 2 2 2 2

C+ ¢ + § ¢ §< /

!K63 !K63

Proof. Write 43 where by normalization. Linearity implies that

!K63 !K63 !K63 !K63 43 43

¡ § ¡ § ¢§ ¡ ¡ § + § ¡ C+ ¢ ¡ 2

and orthogonality2 implies that

!K !K !K !K !K !K

43 43

+ § § ¢ § § ¡ " ¡ ¢¡ ¢¡ !()$ 6¢ ¦ +

3 K

where . Since we get

!K !K

2 2

43 43

¡ ¡ § D § ¡ ¡ ¦

(

¢ ¢

!K !K

!()$¦ 43 ;

D ¡ ¡

§©¨

¡ ¦ / D ¡

!K !K

A NEW PROOF OF THE STABLE MANIFOLD THEOREM 9

¢ © ¦

¡Q-¢ ¡¡

Notice that we can choose $ large enough depending only on the map, so that

¦* $ .C$ $ .C$

/ for all . Then, for we have

¢ ¢

" ; ()$ ; ¡ ¡ ¦

where we have used the boundedness of the derivative to obtain the last equality. Since $

$0; $

depends only on ¢ , we can take care of all iterates by simply adjusting the constant

The next result says gives some control over the dependence of the most contracting directions

on the base point.

2 2 2

, >M$-.0/ > ¥ :!

!K63 43 43

Lemma 4. such that and , we have

2 2

(

¡Q¤ 43 ¡ ; ¡Q ¡ ;

and 43 The proof of Lemma 4 is relatively technical and we postpone it to the ﬁnal Section. First

though we prove another estimate which will be used below.

# #

, > > ;A@B;C$

Lemma 5. such that 43 and we have

( ¥

; ¡ ¡ ;

7 7

¡ 7

Proof. Using the linearity of the derivative we write

2 2 2 2

(

¡ ¡ ¡Q-¢87 43 ¡ ; ¡Q-¢879 43!D 7 3 ¡ ¡Q-¢87 7 3 ¡

7 3

¡Q-¢ ¡ ¡ ¡

7 3

Then we have 7 and

2 2 2 2 2 2

(

¡QM¢879 438D 7 3 ¡ ; ¡Q-¢87¡ ¡ 3!D 7 3 ¡ ; ¡ 438D 7 3 ¡ 7

By Lemma 3 we have

2 2 2

K K

¥ (

¡ 43!D 7 3¡ ; *3 ;

¥¡ 7 ¡ 7 This give the ﬁrst inequality. The second follows from Lemma 1. ¡

4. FINITE TIME LOCAL STABLE MANIFOLDS

The smoothness of the ﬁeld of most contracted directions implies, by standard results on the 2

existence of local integrals for vector ﬁelds, the existence of most2 contracted and most expanded

$ .

3 leaves. Thus for every we consider the integral curves 43 through to the ﬁeld of

contractive directions which are well deﬁned in 43 .

These ﬁnite time local stable manifolds have not been used much in dynamics even though from a certain point of view they are even more natural than the traditional local stable manifolds. Indeed, the local stable manifolds captures information about some local geometrical structure related to the asymptotic nature of the dynamics, while perhaps not containing the right kind of information regarding geometrical structures associated to some ﬁnite number of iterations of the

map. As far as the2 dynamics for some ﬁnite number of iterates is concerned, the ﬁnite time local

stable manifold 43 is the natural object to look at, since it is a canonically deﬁned submanifold 2

which is locally most contracted under a ﬁnite number of iterations2 of the map.

2 3

We shall use the notation to denote the two pieces of on either side of . A priori

we have no means of saying how successive leaves 43 are related nor, importantly, what their 2

length are. Indeed, the neighbourhoods 43 are in principle shrinking in size and thus the length of the leaves 43 could do the same. The heart of our argument however is precisely to show that

this does not happen. First we apply2 the pointwise convergence results of the previous section to

2GF

show that the sequence of curves 43 is a uniformly convergent Cauchy sequence of curves

and thus in particular converges pointwise to a limit curve 3 . In Section 5 we show that the

2GF

arclength of the curves 43 on both sides of is uniformly bounded below, implying that

the limit curve 3 has positive length. Then, in the ﬁnal Sections we complete the proof by

2 2

3 2

showing that is the local stable manifold of and has the required properties.2

¡ ¡

43 !K63

2 2 2 2

!K63

Let and , be parametrizations by arclength of the two curves 43 and

¡¤£ ¡¤£

& &

¡ ¡ ¡¤£ ¡ ¡ ¡¤£

3 !K63 3 !K63

respectively, with 2 and choose so that both and are

¢ ¡ ¡

!K63 both contained in . We shall prove below that we can choose uniformly in . For the

moment we obtain an estimate for the distance between the two curves.

¢ ¢ ¢

, >M$-.0/ D ; ;

Lemma 6. such that and we have

2 2

¡¦¥ ¡¤£§¥

¥ ¥

¢ ¢

¡ ¡

3 !K63

D ; ¨ ; ¨

2 2 2 2

2 2

¡ ¡

¢ ©¨ ©¨

¡ ¡ ¡

43 43 !K63 K63

D ¡ ¡

43 !K63

2 2

Pr2 oof. Writing and we have

2 2

43 !K63 !K63

¡ D ¡¡

!K63

43 . Thus, by the triangle inequality, this gives

2 2

2 2 2 2

¢ ¢

¡ ¡

43 !K63

3 43 43 !K63

D ; ¡ D ¡

2 2 2

(5) 2

(

¡ 3 !K63 D !K63 !K63 ¡¡

By the Mean Value Theorem and Lemma 4 we have

2 2 2 2 2 2 2

¢ ¢

¡ 43 43 D 43 !K63 ¡ ; ¡Q 43 ¡ 43 D !K63

2 2

(6)

¢ ¢

; 3 D !K63 "

and, by Lemma 3 we have

2 2 2 2 2

¥ (

¢ ¢

!K63 !K63 ¡ ; 43 ; ¨ 43 !K63 D

(7) ¡

From (6) and (7) we get

2 2 2 2 2 2

¥ (

¢ ¢

!K63 !K63 ¡ ;A 3 D !K63 43 43 D

(8) ¡

A NEW PROOF OF THE STABLE MANIFOLD THEOREM 11

Substituting (8) into (5) and using Gronwall’s inequality gives

2 2

¡¦¥

( ¥ ¥

¢ ¢ ¢ ¢

¡ ¡

3 !K63

¡ ; ¨ D ¨ ; 43 D !K63

(9)

5. THE ASYMPTOTIC LOCAL STABLE MANIFOLD Lemma 6 shows that the ﬁnite time local stable manifolds are exponentially close as long

as they are both of some positive length. It does not however imply that this length can be

¡ ,

guaranteed. To show this we ﬁx ﬁrst a constant and a sequence where

¡ £¢

" ¨

and the constant is chosen as in Lemma 6. The constant will be the target length of the stable

¤¦¥

+ $ *© 43

curve, and this will depend on but not on . Then we let © denote the neighbourhood

condition

2 1)2

¤©¥

(

¢ ¢

and ©

2 2

2 K63

Notice that the contractive directions K63 are well deﬁned in by Lemma 1. There-

*©

fore the time 1 local stable manifold K63 exists and, by taking sufﬁciently small, we can

¨§ *©

guarantee that condition is satisﬁed. In particular the length of K63 is actually of the

order + . We choose the constant so that

(

¡ ¡

£¢

/ + ¦* and

Thus it just remains to prove the general inductive step:

E, $-.0/

Lemma 7. > sufﬁciently small and ,

(

¨§ §

2 2

§ ©

!K63

Proof. We assume condition and start by proving that exists and has the required

¤¥

2 2 2

43 !K63

length. By the assumption that © , the vector ﬁeld of contractive direction

¤¥

!K63 43 !K63

is well deﬁned in the whole of © . Therefore certainly exists and the 2

question is whether it satisﬁes the condition on the length. The fact that is does follows2 by Lemma

¤©¥

!K63

6 and our choice of the sequence . Indeed, Lemma 6 implies that © for all

¢ ¢ ¢ ¢

; where can be chosen as .

¤¦¥

' *©

!K63

To show the second part of the statement, it is sufﬁcient to show that ©

implies i.e.

¡*© " ¢ :! ;=+ > @B;C$ /

2 2

¤©¥

!K63 !K63 First of all, since © , we can choose some such that

¡ . Then, by the triangle inequality we have

(

¢879:! ; ¡©P" ¢87 ¡¢879 " ¢87 :! (10) ¡©P" 12 MARK HOLLAND AND STEFANO LUZZATTO

We estimate2 the quantities on the right hand side of (10) in the following way. First of all, since

M *© ¢ ©

!K63 7

, the distance of the iterates from can be calculated by

2 2

!K63 !K63

¡*©P"4¢87 <; ¡ ¡¡ ; ¡ ¡ +M; ¨

(11)

7 7

!K63 where the integral is taken along and the last inequality follows from Lemma 5.

Let us now estimate the second term on the right hand side. We claim the following.

(

+ ¦* ¨ $-.0/ $ / [email protected]/ ¨

Claim 7.1. For all and all we have

7 !K © To prove this claim, recall that we can choose the size + of the neighbourhood of in which the

entire construction takes place arbitrarily small, in order to ensure that the constant is arbitrarily

small, where is such that

¤¨§ ¤ ¤ ¤©§

7 7 K 7 7

. D . . . D A .

7 7

This gives

¤ ¤¨§ ¤

¤©§

¨ ; A 67 ; " ¨ ;

¤¨§ ¤

D ¢ D

/ "

if is small enough. To complete the proof of the claim, we just use the fact that and

¡ ¡

+ ¦)* § + ¦)* " ¨ ¨

so that .

!K 7

$ / [email protected]/

Claim 7.2. For all $-.0/ and all we have

(

¡!6¢879 " ¢!7 :! ; ¨ ¡!" !<; ¨

7 7 !K

2 2

¤¥

@ ; $ / © £

43 7 3

We show by induction that for each , we have © , and therefore

by the Mean value theorem, and Claim 7.1:

(

¡¢87 "4¢87 :! ; ¨ ¡!!" !<; ¨ ;=+ ¦*

7 7 !K

This will complete the proof of the lemma since substituting these inequalities into (10), we get

+" ¢87 ! ; ¨ ¨

(12) ¡*©P"

7 7 !K

@B;C$ /

and this is ;=+ for all .

Returning to the proof of Claim 7.2, let us verify the inductive statement for @ . We obtain:

¡ "© ; ;=+"

2 2 2

¤¥ ¤¥

(

3 K63 43 © and therefore © Assuming true for , and taking we

have by equation (11) and Claim 7.1:

¡*©P" ¢ ! ; ¨ ¨ "

¤ ¡

; 67 ¨ ¨ +"

!K 7

( ¡

¥£ / and hence the inductive statement holds for @ A NEW PROOF OF THE STABLE MANIFOLD THEOREM 13

6. SMOOTHNESS

2GF

¦ K

Lemma 8. 3 is with Lipschitz continuous derivative.

2GF

¦ K

Proof. To see that 3 is , we just substitute (9) into (8) to get

2 2

¡¦¥ ¥

¥ ¥

¡ ¡

43 !K63

¡ 3 D !K63 ;+ ¨ ¡ ; ¨

¥ ¥ #

D +-; ;=+ ¨

for every . Since exponentially fast, this implies that the sequence 43 is

uniformly Cauchy in . Thus by a standard result about the uniform2GF convergence of derivatives,

¦ K they converge to the tangent directions of the limiting curve 3 and this curve is . To prove that the tangent direction are Lipschitz continuous functions, notice that by the triangle

inequality we have

F F

¢ ¢

!D :¡ ;

F 2 2 2 2 F

¢ ¢ ¢ ¢ ¢ ¢

:! D 43 :! 43 :! D 43 : 3 ! D :

2GF F

2 F 2

" $0. $ :! D 3

for any two given points and any . By Lemma 3 we have

¥ ¥

¢ ¢ ¢

! ; : LD : ;

43 3

2 2

2 , and, by Lemma 4 and the Mean Value

7 7

7 J 7 J

¢ ¢ ¢ ¢ ¢ ¢

:! D : ; ¡Q ¡ BD ; BD

43 43 43

Theorem, . This gives

F F

¥ (

¢ ¢ ¢ ¢

:! D ; BD

7 J

¥ ¥ # ¥ %

$ $ ! D

SinceF this inequality holds for any and as it follows that

74J

¢ ¢ ¢

: ; D

7. CONTRACTION

Finally we want to show that our limiting curve behaves like a stable manifold in the

¥ %

sense that it contracts as $ . Let denote a parametrization by arclength of the leaf

Lemma 9. There exists a , which can be taken arbitrarily small with , such that for any

¢ ¢

, .0D + $-.0/

+M. and we have

(

¢ ¢ ¤ ¢ ¢

¡ ¡ ¡ ¡

¢ D ¢ ;A D

2 2GF 2

3 3

Proof. Writing 43 we have by the linearity of the derivative

¡ ¡

¡ ¢ ¡¡ D ¢ ¡

2 2GF 2

(

43 ¡¡ 38D 436 Q-¢ ¡Q-¢

2 2GF 2

¡Q-¢ ¡ ¡ D ¡ ;

43 3 3

Since and by Lemma 3, we get

74J

¢ ¢

¢ ¢ ¢

¡ ¡

; ¨ ¨ ¡ ¢ DA¢ ¨ D

7 K

7 J

14 MARK HOLLAND AND STEFANO LUZZATTO

¥ ¥ ¥

¨ ¨ ¨

Now is an exponentially decreasing sequence and therefore ; and thus,

7 7

reasoning exactly as in the proof of Lemma 6 we have 74J

¤¨§ ¤

; ¨ ; ¨ ¨ ;A ¨

¤©§

D ¢

7 J

, ,

for some which can be chosen arbitrarily small by choosing small. Moreover we

¢ ¢

¢ ¤ ¤ ¢¡ ¢ ¢

¡ ¡

¨ ; ; D D +

also have . Finally, using the fact that if is

§ K small we conclude the proof.

8. UNIQUENESS

Finally we need to show that the local stable manifold we have constructed is unique in the

£¢ *© © + +

sense that there is some neighbourhood of of size (perhaps smaller than ) such that

+B, +M,

for which 1

(

5 5

*©PO ¥¢ © ©

JLK

£¢ ©

Proof. Suppose, by contradiction that there is some point C which belongs to the set

5 5

refer the reader to [15, 23] for full details. The property we need is the existence of an invariant

tangent directions lie in this expanding coneﬁeld will remain admissible for as long as it remains

¤ §

¥¢ © D

in and will grow in length at an exponential rate bounded below by . Moreover the

expanding cones do not contain the direction determined by the stable eigenspace R at the ﬁxed

point . By construction the curve is tangent to R at and therefore, for small enough,

5 5 5

) the tangent vectors to ) are not contained in any unstable cone at any point of . Thus

is in some sense transversal to the unstable coneﬁeld.

§¢ *© ©

This transversality implies that for any A there exists an admissible curve joining

) ©

¥¢ *© © ©

(or even ¢ ). By construction, the point never leaves this neighbourhood

and therefore the point must at some point leave. ¡

9. PROOF OF LEMMA 4 Q-¢

Since is a linear map, we have

¥ (

()$ 43P ()$

(13)

!K A NEW PROOF OF THE STABLE MANIFOLD THEOREM 15

Differentiating on both sides and taking norms we have

2 2

¥ ¥

43 43

()$ ¡ Q ()$ ¡ ¡Q ¡Q 43 ¡ ; ¡

2 2 2

!K !K

(14)

¥ ¥

43 43 43

/ (/$ ; ¡ Q¤ !()$ ¡ ¡Q ¡

2 2

!K !K !K

43 43

Q¤ ; A

We shall show in three separate sublemmas, that ; , , and

¥ ¥ ¥

!K !K

; ¦

Q . Using the fact that is also bounded, and substituting these estimates

into (14) yields the estimate in Lemma 4.

¢ ¢

;

Sublemma 10.1. .

Proof. Follows immediately by substituting the results of Lemma 3 into (13) ¡

¡Q¤ ;

Sublemma 10.2. ¡

2 2 2 2 2 2 2

3 43 !K63 !K63 43 43 !K63

¡Q ¡ ¢¡Q D DAQ ¡?;¤¡Q ¡

Proof2 . Writing we have

!K !K !K !K !K !K !K

Q ¡

¡ Our strategy therefore is to obtain estimates for the terms on the right hand side. First

of all we write

QM¢

¢¡

¦ Q

2 2

" "4¦ Q Q-¢

where and are the matrix entries for the derivative evaluated at . Since

" Q-¢ 3

3 correspond to (resp.) maximal contracting and expanding vectors under

£ ¡Q-¢ "! " $& ¡

we may obtain them precisely by solving the differential equation .

£¥¤ Q-¢ By solving this differential equation (and by solving a similar one for the inverse map ) we

get

¦ Q *§¦ *

()$+*) £G

¦ D DAQ

§ § § §

and

¦ * Q * ©

()$+*) £GD

Q ¦ D D

§ § § §

2 2

¦ " © "

Notice the use of " as a shorthand notation for the expression in the quotients. Now

43 43

" Q

are respectively maximally expanding and contracting for , and so we have the

identity

Q D

- - -

Q Q

det

D ¦

Then, using the quotient rule for differentiation immediately gives

¨ ¨

(

¡ ¡Q 43 ¡ ¡Q

(15) and

¦ ©

§ § § §

Claim 10.1.

¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢

and

16 MARK HOLLAND AND STEFANO LUZZATTO

" "4¦ "4Q

Proof. For the ﬁrst set of estimates observe that each of the partial derivatives of

¢ ¢ ¢ ¢

Q-¢ ; ¡Q-¢ ¡ ¦ ¦ Q ; * ¡QM¢ ¡ §

is . Then . The same reasoning gives the

¡¦ ¡ ¦ Q

estimates in the other cases. To estimate the derivatives, write

¢ ¢ ¢

¦ Q ; * ¡Q ¢ ¡

. Now and similarly for the other terms.

Claim 10.2.

(

§ § § § § § §

D © ¦

§ " §

Proof. Notice ﬁrst of all that are eigenvalues of

- -

6Q ¡ !Q

¡ ¡

¦ Q Q

§ §

¦ Q

¦ Q ¦¨§ Q §

¤ ¤

§ " § §LD § § ¦ § Q §

In particular are the two roots of the characteristic equation

§ § § § §

6¦ Q LD ¨ ¦ Q

and therefore, by the general formula for quadratic

§ § § § 6¦ § QB§ D § § § ¦ § Q §

equations, we have § and

(

From this one can easily check that ¦

( ¡

¨§ § § D § §

$ Substituting the estimates of Claims 10.1-10.2 into (15) and using the fact that $¤. and

Lemma 2, this gives

¡Q-¢ ¡¤£ ¡QB§ ¢ ¡ ¡Q-¢ ¡¤£ ¡QB§4¢ ¡

¡Q 43 ¡ " ¡Q ¡ ;)/ * ; ¦¥

§ §

§ (

¡Q ¢ ¡

; ;

2 2 2 2

43 43 43 43

Q ) "! " $&

To estimate we write , so that

2 2

!K !K !K !K

43 3

(

¦ Q #" $&

2 2

K K

()$&

43 43

#"%$&

K K

Notice that we have

§ §

£ £ £ 43

¡ ¡Q ¡ ; * ¡Q-¢ ¡ ¡ ; * ¡Q ¢ ¡ ¡Q-¢ ¡ ¡Q-¢ ¡ ¡Q ¡

1 1

¡ ¡Q ¡

and similarly for ¡ and . Therefore, writing

1 1

§ §

¡Q ¡

§ §

with 2

¡ § ¡

§ §

. "

¡Q ¡

¡ ¡

and substituting the estimate obtained, we get the desired result.

¡Q ¡ " ¡Q ¡Q ; § ¡ ;

Sublemma 10.3. ¡ and

A NEW PROOF OF THE STABLE MANIFOLD THEOREM 17

£ 43 £

Q Q ¡ ¡ Q

2 2 2

Proof. We ﬁrst estimate 2 . The corresponding estimate for is identical.

£ £

43 43

Q ¡Q Q § ¢ Q-¢ Q ¡ ; ¡QB§ ¢ ¡

2 2 2 2 2

43 3

First of all one has,2 and hence

( (

£ £

43 3 43 43 43

Q ¡ ¡Q ¡ ; ¡Q ¡Q-¢ ¡ ¡Q ¡ ¡ ¡ Q ¡ ¡

43 K

Since one has that

Q ¡ ;

By the previous lemma and the distortion conditions we get ¡ . Using the fact that

§ © Q-¢

and the quotient rule for differentiation, we get

¥ (

'Q Q CQ D

¡ ¡

§ § §

¥ ¡

¡ ; Q

By the estimates above we then get ¡ .

This completes the proof of the ﬁrst inequality in the statement of Lemma 4. The second one follows immediately from the ﬁrst and the inequality in (4).

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MATHEMATICS DEPARTMENT, UNIVERSITY OF SURREY, GUILDFORD, GU2 5XH, UK E-mail address: [email protected]

MATHEMATICS DEPARTMENT, IMPERIAL COLLEGE, LONDON SW7, UK E-mail address: [email protected] URL: www.ma.ic.ac.uk/˜luzzatto