DIFFERENTIABLE CONJUGACY NEAR COMPACT INVARIANT MANIFOLDS

DIFFERENTIABLE CONJUGACY NEAR COMPACT INVARIANT MANIFOLDS CLARK ROBINSON 0. Introduction In this paper 1, we show how the differentiable linearization of a diffeomorphism near a hyperbolic fixed point (a la Sternberg [11]) can be adapted to a neighborhood of a compact invariant submanifold. There are two parts of the stard proof. The first part says that if two diffeomorphisms have all their derivatives equal at a hyperbolic fixed point, then they are C conjugate to one another in a neighborhood. This result is true in a neighborhood of a compact invariant submanifold with little change in the statement or proof. See Theorem 1. The second part says that if a diffeomorphism f satisfies eigenvalue conditions at a hyperbolic fixed point, then there is a C diffeomorphism h such that all the derivatives of g = h -1 f h at the fixed pint are equal to the derivatives of the linear part of f. Near an invariant submanifold, there is no general condition that replaces the eigenvalue condition, so we got only a very much weakened result in this direction. See Theorem 2. However, Theorem 2 does imply that under some conditions the strong stable manifolds of points vary differentiably. See Corollaries 3 4. We were aware that Theorem 1 was true before reading the recent paper of Takens [12]. However, his proof is the easiest to adapt to our setting also save one more derivative than some other proofs. We could just say that Theorem 1 follows from the proof in [12], however for clarity, we repeat the proof with the necessary modifications. The only essential changes are in the definitions of η(δ) O. All other changes are a matter of style. To prove Theorem 2, we adapt the type of proof used for Theorem 1. At a hyperbolic fixed point, this can be solved much more directly by solving for coefficients of polynomials using eigenvalue conditions. See [9], [11], or [12]. For h : M M, let 1. Statement of the theorems j r h(x) = (x, h(x), Dh(x),..., D r h(x)). This is called the r-jet of h at x in local coordinates on the domain. (It is possible to define these without local coordinates, but it really changes none of the ideas in our proofs. See [4].) Key words phrases. dynamical systems, differentiable conjugacy, normally hyperbolic manifold. 1 This paper originally appeared in the Bolletim da Sociedade Brasileira de Matemática, 2 (1971). We have made slight changes in wording in a few places. Also, we have added footnotes to explain certain points. 1

2 CLARK ROBINSON Let V be a compact submanifold of M. Give M a Riemannian metric. Let ρ be the distance between point of M induced by the metric. Let p : T M M be the usual projection. Let T x M = p 1 (x) T V M = p 1 (V ). A diffeomorphism f : M M is called hyperbolic along V if f(v ) = V, there is a splitting T V M = T V E u E s as Whitney sum of subbundles, there is an integer n such that µ x = Df n (x) E s x < 1 λ x = Df n (x) E u x < 1 for all x V, where E s x = E s T x M E u x = E u T x M. For h : M M x V, let D 1 h(x) = Dh(x) T x V, D 2 h(x) = Dh(x) E u x, D 3 h(x) = Dh(x) E s x. A diffeomorphism f is called r-normally hyperbolic along V if λ x D 1 f n (x) k < 1 µ x D 1 f n (x) k < 1 for all x V all 0 k r. This says that f more contracting (resp. exping) normally to V than any contraction (resp. expansion) along V. Let W s (V, f) = { x M : ρ(f j (x), V ) 0 as j } W u (V, f) = { x M : ρ(f j (x), V ) 0 as j }. These are called the stable unstable manifolds of V W ss (x, f) = { y M : there exits a constant c y for f. For x V, let such that ρ(f jn (x), f jn (y)) c y µ x µ f (j 1)n (x) for j 0 } W uu (x, f) = { y M : there exits a constant c y such that ρ(f jn (x), f jn (y)) c y λ x λ f ( j+1)n (x) for j 0 }. These are called the strong stable strong unstable manifolds of x for f. If the diffeomorphism f is C r r-normally hyperbolic, then the papers [6] [7] show that W s (V, f), W u (V, f), V, W ss (x, f), are C r, Also, W s (V, f) = {W ss (x, f) : x V } W u (V, f) = {W uu (x, f) : x V }. T V (W s (V, f)) = T V E s T V (W u (V, f)) = T V E u. A more general theorem of this kind is contained in [8]. Now we define the loss of derivatives that occurs in the conjugation of Theorem 1. Given α, let β = β(f, α) α be the largest integer such that Df n (f n (x)) Df n (x) β D 3 f n (x) α β < 1 for all x V.

DIFFERENTIABLE CONJUGACY 3 Next, let γ = γ(f, β) β be the largest integer such that Df n (f n (x)) Df n (x) γ D 2 f n (x) β γ < 1 for all x V. Theorem 1. Assume f, g : M M are C α diffeomorphisms, V M is a compact C 1 invariant submanifold such that both f g are 1-normally hyperbolic along V with j α f(x) = j α g(x) for all x V. Let β γ be defined as above with α β γ 1. Assume W s (V, f) is a C β submanifold near V. Then there exist a neighborhood U of V a C β diffeomorphism h : U M such that k = h 1 gh has j β k(x) = j β f(x) for x W s (V, f) U. Also there exists a C γ diffeomorphism h : U M such that (h ) 1 gh (x) = f(x) for x U. Further, h V = id h V = id. The proof is contained in 3. Theorem 2. Let f : M M be a C α diffeomorphism, V M a compact invariant C α submanifold. Assume f contracts along V, i.e., E u is the zero section in the definition of f being hyperbolic along V. Assume that for 1 β α D 1 f 1 (f(x)) Df(x) β 1 D 3 f(x) < 1 for all x V. Then there exists a neighborhood U of V a C β diffeomorphism h : U M such that h V = id g = h 1 fh has D j 3 (pr g)(x) = 0 for 1 j β where pr : U V is a differentiable normal bundle projection. Thus infinitesimally g preserves the fibers of pr : U V. The proof is contained in 4. Corollary 3. Let f be a C α diffeomorphism contracting along V. Assume D 1 f 1 (f(x)) Df(x) α 1 D 3 f(x) < 1 for all x V. Let p r be integers such that D 3 f 1 (f(x)) D 3 f(x) p 1 ( ) r+1 D 1 f(x) α p D3 f(x) < 1. D 1 f(x) Let β = α 1 p r. Then there exists a neighborhood U of V a C β diffeomorphism h : U M such that g = h 1 fh preserves the fibers of pr : U V. Actually, h has all derivatives D j D2h(x) k for x U, 0 j β, 0 j + k α. In particular, the set of W ss (x, f) for x V form a foliation of W s (V, f) U such that each leaf is C α they vary C β. Proof. By applying Theorem 2, we can assume D j 2 (pr f)(x) = 0 for 1 j α x V. Define g 1 : U V by g 1 (x) = f 1 (pr x). In vector bundle charts of pr : U V define g 2 (x) = f 2 (x) Use bump functions to define g = (g 1, g 2 ) : U U. Then g 1 (x) = g 1 (pr x) = f 1 (pr x) for x M j α f(x) = j α g(x) for x V. Theorem 1 gives the C β conjugacy of f g where β = α 1 p r since D 3 f 1 D 1 f α 1 p r D 3 f 1+p+r D 1 f α 1 p r D 3 f 1+r ( ) 1+r D 1 f α p D3 f < 1. D 1 f The extra derivatives of h exist as remarked in the proof of Theorem 1.

4 CLARK ROBINSON Using the methods of the proof of Theorem 1 differently, we can get a stronger statement about the differentiability of the foliation { W ss (x, f) : x V } of W s (V, f). Corollary 4. 2 Let f be a C α diffeomorphism that is 1-normally hyperbolic along V W s (V, f) is C α. Assume that 1 β α 1 satisfies D 1 f 1 (f(x)) D 1 f(x) β D 3 f(x) < 1 for all x V. Then there is a neighborhood U of V such that the set of W ss (x, f) for x V form a C β foliation of W s (V, f) U. The proof is contained in 5. Using the estimates in [9], the proofs of the above theorems should go over to flows. However, beware of the proof of linearization given in [9]. By induction does not works since the variation equation does not satisfy a global Lipschitz constant. We would like to discuss how the above theorems relate to some of the results in [5], [6], [10]. The condition of [6] [7] that f is r-normally hyperbolic is similar but different than the condition we require in Theorem 2 Corollaries 3 4. If f is r-normally hyperbolic, then W s (V, f), W u (V, f), V are C r manifolds. See [6]. Also, for each x V, W ss (x, f) W u (x, f) are C r they vary continuously in the C r topology. Corollaries 3 4 give that they vary differentiably. [10] show that if f is 1-normally hyperbolic, then f is C 0 conjugate to a map g that preserves the fibers of pr : U M V such that g is linear on fibers of pr : U M V. Corollary 3 gives a differentiable conjugacy in the contracting case to a fiber preserving map g, but g is not necessarily linear on fibers. If V is replaced by an exping attractor, then 6.4 in [5] gives conditions under which the stable manifolds of points form a C 1 foliation of a neighborhood. Corollary 4 possibly could be adapted to this setting to give the same answer. The result in [5] only applies to stable manifolds of points, W s (x, f) = { y M : ρ(f j (x), f j (y)) 0 as j }, not the strong stable manifolds of points. 3 Thus, when a submanifold V is an attractor, the results are different. Also, we give a condition that insure higher differentiability. Added in proof: M. Shub points out to me that 6.4 in [5] the C r section theorem prove Corollary 4. 2. Notation definitions Since we are only interested in a conjugacy of diffeomorphisms in a neighborhood of V, we can take a tubular neighborhood of V. Thus, we can consider M as a vector bundle over V, pr : M V. Let p : T M M be the projection of the tangent bundle of M to M. Denote a norm induced by a Riemannian metric on T M by. Let ρ be the distance between points of M induced by. 2 The original paper only stated this theorem for the case when f is contracting along V. 3 W s (x, f) could include some directions within V.

DIFFERENTIABLE CONJUGACY 5 Let L r s(t x M, T y M) be the (linear) space of all symmetric r-linear maps from T x M to T y M. 4 Let J 0 (M, M) = M M J r (M, M) = { (x, y) L 1 s(t x M, T x M) L r s(t x M, T x M) : x, y M }. If h : M M is C r, let j r h(x) = (x, h(x), Dh(x),..., D r h(x)) J r (M, M). This is called the r-jet of h at x. Let π 0 : J 0 (M, M) M be the projection on the first factor, be the natural projection for r 1. Let π r : J r (M, M) J r 1 (M, M) ψ r = π 0 π r : J r (M, M) M. All of these projections are fiber bundles, ψ r : J r (M, M) M is called the r-jet bundle. Define a distance on J 0 (M, M) by ρ 0 ((x 1, x 2 ), (y 1, y 2 )) = max{ ρ(x i, y i ) : i = 1, 2 }. Let the distance on each fiber of π r : J r (M, M) J r 1 (M, M) be the usual one induced by pm T M, ρ r ( (x, y, A 0,..., A r ), (x, y, A 0,..., A r 1, B r ) ) = A r B r = sup{ (A r B r )(v 1,..., v r ) : v i T x M v i = 1 for all i }. By using the distance on the base space, there is an induced (noncanonical) distance on J r (M, M). Given a subset U M, let J r ( (M, U), M ) = ψr 1 (U). Let Γ r ( (M, U), M ) be the space of continuous section of ψ r : J r ( (M, U), M) U. 3. Proof of Theorem 1 I. First we prove that the conjugacy exists along W s (V, f). We use the notation given in 1 2. By the assumptions of Theorem 1, there exists an integer n a 0 < µ < 1 such that Df n (f n (x)) Df n (x) β Df n (x) E s x α β < µ for all x V. Below we construct a conjugacy h between f n g n. Because of its special form, f lim j g nj f nj, h is also a conjugacy between f g. Thus for convenience, we take n = 1. The reader can check the details for n > 1. The constant µ is fixed during the proof. 4 Note: Higher derivatives are only defined in terms of local coordinates. Therefore, cover a neighborhood of V with a finite number of coordinate charts define the jets norms in terms of these coordinate charts.

6 CLARK ROBINSON We define the following numbers a x = Dg 1 (x) { ρ(f(x), V ) ρ(x, V ) 1 A x = lim{ A y : y W s (V, v) V y x } b x = Df(x) Note for x V, There exist neighborhoods A x Df(x) E s x < 1, a 1 f(x) A x < 1 B x, a f(x) B β x A α β x < µ < 1. η(δ) = { x W s (V, f) : ρ(x, V ) < δ } O of { (m, m) : m V } in M M for x M for x W s (V, f) V for x V for x W s (V, f). such that (i) f(η(δ)) η(δ) (ii) if x η(δ) (f(x), y) O, then a y Bx β Ax α β < µ. For simplicity of notation, let J r = J r ((M, η(δ)), M) = ψr 1 (η(δ)) ΓJ r be the continuous sections of the bundle ψ r : J r η(δ). We define a second norm on the fibers of π r : J r J r 1 (possibly infinite) as follows: 5 for π r c 1 = π r c 2, σ r (c 1, c 2 ) = sup{ ρ r (c 1 (x), c 2 (x)) ρ(x, V ) (α r) : x η(δ) V }. If c ΓJ 0, then we can identify it with the map c 0 : η(δ) M such that c(x) = (x, c 0 (x)). Let Let Φ r : ΓJ r ΓJ r Φ 0 : ΓJ 0 ΓJ 0 be defined by Φ 0 (c)(x) = (x, g 1 (c 0 (f(x)))) = j 0 (g 1 c 0 f)(x). be defined by Φ r (c)(x) = j r (g 1 (c 0 (f(x)))) = j r (g 1 hf)(x) First we prove Φ r contracts along fibers of π r. Lemma 1. Let 0 r β, c 1, c 2 ΓJ r Then, where j r h(f(x)) = c(f(x)). with π r c 1 = π r c 2, σ r (c 1, c 2 ) <, π 1 π r c i (f(x)) O for all x η(δ), i = 1, 2. σ r (Φ r (c 1 ), Φ r (c 2 )) µ σ r (c 1, c 2 ). 5 The map on sections is not a contraction in the usual metric. It needs a factor related to moving closer to the invariant manifold.

DIFFERENTIABLE CONJUGACY 7 Proof. Assume r 1. σ r (Φ r c 1, Φ r c 2 ) = sup{ ρ r (Φ r c 1 (x), Φ r c 2 (x)) ρ(x, V ) (α r) : x η(δ) V } sup{ ρ r (c 1 (f(x)), c 2 (f(x))) a y B r x ρ(x, V ) (α r) : x η(δ) V π 1 π r c i (f(x)) = (f(x), y) }. This last inequality is true using the formula for higher derivatives of a composition of functions the fact that π r c 1 = π r c 2. Then, this is sup{ ρ r (c 1 (f(x)), c 2 (f(x))) µ ρ(f(x), V ) (α r) : µ σ r (c 2, c 2 ). x η(δ) V } When r = 0, ρ(x, V ) α µ ρ(f(x), V ) α. The details are left to the reader. Let I r ΓJ r be defined by I r (x) = j r (id)(x) = (x, x, id x, 0,..., 0) where id : M M is the identity map id x : T x M T x M is the identity map. Let C 0 = σ 0 (Φ 0 I 0, I 0 ). C 0 is finite because j α f(x) = j α g(x) for all x V V is compact. Let D 0 = C 0 (1 µ) 1. Let 0 r be the zero section of π r : J r J r 1. Let F 0 = { c ΓJ 0 : σ 0 (c, I 0 ) D 0 } F r = { c ΓJ r : π r c F r 1 σ r (c, 0 r π r c) D 0 } for r 1. Since σ 0 (c, I 0 ) D 0 for c F 0, there exists a δ > 0 smaller than above if necessary, such that for c F 0 x η(δ), then c(f(x)) O. Lemma 2. Let 0 r β. Then Φ r : ΓJ r ΓJ r maps F r into itself. Proof. We prove the lemma by induction. F 1 = is invariant by Φ 1. Assume F r 1 is invariant by Φ r 1. Let c F r. Then σ r (Φ r c, 0 r π r Φ r c) σ r (Φ r c, Φ r 0 r π r c) + σ r (Φ r 0 r π r c, 0 r π r Φ r c) µ σ r (c, 0 r π r c) + σ r (Φ r 0 r π r c, 0 r π r Φ r c). For r = 0, this last term is µ D 0 + C 0 D 0. For r > 0, it is <. Lemma 3. Let 0 r β. Then Φ r : F r F r is continuous in terms of σ r. Proof. We use the chain rule for higher derivatives of a composition. σ r (Φ r c 1, Φ r c 2 ) = sup{ρ r (Φ r c 1, Φ r c 2 ) ρ(x, V ) (α r) : x η(δ) V } (constant) sup{ D i g 1 (y 2 ) ρ j1 (c 1 (f(x)), c 2 (f(x))) ρ jr (c 1 (f(x)), c 2 (f(x))) D k1 f(x) D kj f(x) ρ(x, V ) (α r) : x η(δ) V, π 1 π r c 2 (f(x)) = (f(x), y 2 ), 1 i r, j = j 1 + + j r, k 1 + + k j = r } + (constant) sup{ ρ j (g 1 (y 1 ), g 1 (y 2 )) D j1 c 1 (f(x)) D ji c 1 (f(x)) D k1 f(x) D kj f(x) ρ(x, V ) α r) : π 1 π r c 1 (f(x)) = (f(x), y 1 ) }.

8 CLARK ROBINSON Here the constants depend only on the binomial coefficients. We look at the first supremum leave the second to the reader. It is (constant) sup{ σ j1 (c 1, c 2 ) σ ji (c 1, c 2 ) ρ(f(x), V ) (iα j) ρ(x, V ) (α r) : (constant) σ r (c 1, c 2 ) r. 1 i r, 1 j r } These last two constants include the supremum of derivatives of f g 1. From this it follows that Φ r is continuous. By Lemma 1, Φ 0 : F 0 F 0 is a contraction in terms of σ 0. Thus, there is a unique attractive fixed point, c 0. Attractive means that for each c F 0, σ 0 (c 0, Φ j 0 (c)) 0 as j. Assume that F r 1 has an attractive fixed point for 1 r β. By Lemma 3, Φ r is continuous. By Lemma 2, Φ r : F r F r contracts along fibers of π r : F r F r 1 by a factor of µ. By the fiber contraction theorem (Theorem 1.2 in [5]), Φ r has a unique fixed point in F r it is attractive. Let id : M M be the identity diffeomorphism I β (x) = j β (id)(x). Then Φ β (I β ) converges ( in the uniform topology of sections of ψ β : J β η(δ) ) to a section c ΓJ β. Let c(x) = (x, c 0 (x),..., c β (x)) with c i (x) L i s(t x M, T c0(x)m) for 1 j β. By the uniform convergence, it follows that c i : η(δ) L i s(t x M, T y M) : x η(δ), y M } is C β i. Thus, the conditions of the Whitney Extension Theorem are satisfied. See [1] for a statement of the theorem. There exists a C β function h : M M such that for x η(δ), j β h(x) = c(x), Dh(x) = id x for x V so h is a local diffeomorphism in a neighborhood of V. Thus, h 1 gh is defined in a neighborhood of V in M j β (h 1 gh)(x) = j β f(x) for x η(δ) W s (V, f). This completes the conjugacy of f g along W s (V, f). Remark. In Corollary 3, we noted that more derivatives of the conjugacy existed along the fibers. In that setting, we have C α diffeomorphisms such that D j 2 (pr g)(x) = 0 for 1 j α x V f(pr x) = pr f(x). Here pr : M V is a normal bundle. For α r β, let J β,r be the bundle of maps with all derivatives D j D2h(x) k for 0 j β 0 j + k r. Let ρ β,r be the associated norm. Let π r : J β,r J β,r 1 be as before. For π r c 1 = π r c 2, let σ β,r (c 1, c 2 ) = sup{ ρ β,r (c 1 (x), c 2 (c)) ρ(x, V ) (α r) : x η(δ) V }. If c 1, c 2 J β,r π r c 1 = π r c 2, then ρ β,r (Φ β,r c 1 (x), Φ β,r c 2 (x)) ρ(x, V ) (α r) ρ β,r (c 1 (f(x)), c 2 (f(x))) a y B α x A r β x ρ(x, V ) (α r). A little checking is necessary to show this depends only on ρ β,r not ρ r. (f preserves fibers.) Then this is µ ρ β,r (c 1 (f(x)), c 2 (f(x))). Lemma 1 follows. The other details are left to the reader.

DIFFERENTIABLE CONJUGACY 9 II. Now we can assume f g are C β j β f(x) = j β g(x) for all all x W s (V, f) = W s x near V. For x M, define the following numbers: For x V, a x = Df 1 (x) { ρ(f 1 (x), W s ) ρ(x, W s ) 1 for x / W s b x = lim{ b y : y / W s y x } for x W s, B x = Dg(x). a 1 x < 1 < b 1 x B f 1 (x) B f 1 (x) a γ x b β γ x < µ < 1. By using a bump function, we can make g(x) = f(x) at points x such that ρ(x, V ) δ. ( g is then defined on all of M. ) Also, g can be left unchanged at points x with ρ(x, V ) δ /2. Let η(δ) = { x M : ρ(x, W s ) < δ } η (δ) = { x M : ρ(x, V ) < δ }. By taking δ smaller if necessary O to be a small neighborhood of { (x, x) : m W s } in M M, we can insure that for x η (δ) (f 1 (x), y) O, it follows that B y a γ x b β γ x < µ. Let Φ r be induces by h g h f 1. That is, in the earlier definition replace f by f 1 g 1 by g. Continue as before taking sections c of ψ r : J r (η(δ), M) η(δ) such that c(x) = j r id(x) for x η(δ) η (δ). Lemmas 1, 2, 3 apply to these sections. The lim j Φ j γ(i γ ) gives the γ-jet of the conjugacy h on η(δ). 4. Proof of Theorem 2 In this section, we assume T V M = T V E s. The bundle F 1 = T V is differentiable. Since we do not assume the bundles are invariant, we can approximate E s by F 3 that is differentiable. Write D i h(z) = Dh(z) F i. We assume in the theorem that D 1 f 1 (f(z)) Df(z) β 1 D 3 f(x) < µ < 1 for all z V. Denote a normal bundle projection by pr : M V. For c J r ( (M, V ), V ), we write c(z) = (z, c 0 (z),..., c r (z)) with c k (z) L k s(t z M, Fc 1 ). 0(z) Let F r be the set of sections c of J r ( (M, V ), V ) such that, for each z V there is a C r function h : M V such that h V = id c(z) = j r h(z). This is equivalent to assuming that for each z V, (i) π 1 π r c(z) = (z, z) (ii) c k (z) (F 1 F 1 ) = D k (id)(z) where id : V V is the identity function. Let f V = f V : V V fv -1 = (f V )-1 : V V. Define Φ r : F r F r by Φ r c(z) = j r (f -1 V hf)(z) where j r h(f(z)) = c(f(z)). By abuse of notation, Φ r c(z) = j r (fv -1cf)(z). Lemma 4. Let 0 r β c 1, c 2 F r be such that π r c 1 = π r c 2. Then ρ r (Φ r c 1, Φ r c 2 ) µ ρ r (x 1, c 2 ).

10 CLARK ROBINSON Proof. ρ r (Φ r c 1, Φ r c 2 ) sup{ D 1 f -1 V (f(z)) (c 1 (f(z)) c 2 (f(z))(df(z)) r : z V } sup{ D 1 f 1 1 (f(z)) ρ r(c 1, c 2 ) D 3 f(z) Df(z) r 1 : z V } since c 1 r (F 1 F 1 ) = c 2 r (F 1 F 1 ). Then ρ r (Φ r c 1, Φ r c 2 ) µ ρ r (c 1, c 2 ). As in the proof of Theorem 1, we can apply the fiber contraction principle to find c F β such that Φ r (c) = c. Let s ΓJ β ( (M, V ), M) be given by s(z) = (c(z), j β (pr 3 z)), i.e., the components of s in F 3 in the range is like the jet of the identity function on fibers zero derivatives in the directions along V. (This has meaning at the jet level but not as maps.) By the uniform convergence of Φ k β (jβ pr) to c, it follows that s satisfies the conditions of the Whitney Extension Theorem. There exists a C β h such that j β h(z) = s(z) for z V. The map h is a diffeomorphism on a neighborhood of V because of the form of the derivatives at points of V. Let g = hfh 1 g 1 = pr g. At the level of jets for z V, j β (fv -1h 1f)(z) = j β (h 1 )(z) j β (pr g h)(z) = j β (h 1 f)(z), so j β (g 1 )(z) = j β (pr g)(z) = j β (f V pr)(z) has the derivatives zero in the direction of F 3 as claimed. 6 This completes the proof of Theorem 2. 5. Proof of Corollary 4 Proof. Since W s (V, f) is C α, we can restrict the map to this space assume f is contracting along V. By applying Theorem 2, we can assume D j 3 (pr f)(x) = 0 for 1 j β x V. Define g 1 : U V by g 1 (x) = f 1 pr(x). In the proof of Theorem 1, replace a x = Dg 1 (x) by a x = Dg1 1 (x) where g1-1 = (f V ) -1 : V V. Next consider jets in J r = J r (η(δ), V ) instead of J r (η(δ), M). Define Φ r : ΓJ r ΓJ r by Φ r (c)(x) = j r (g 1 hf)(x) where j r h(f(x)) = c(f(x)). As in the earlier proof, we can find a c such that Φ β c = c c satisfies the conditions of the Whitney Extension Theorem. There exists a C β 1 function h : M V such that g 1 hf = h. The map h is a projection onto V defines a C β foliation. Since hf = g 1 h, it follows that f preserves the foliation. Since the foliation is tangent to E s, it follows it is W ss (x, f). References [1] R. Abraham J. Robbin, Transversal Mappings Flows, Benjamin, 1967. [2] G. Belickii, On the local conjugacy of diffeomorphisms, Sov. Math. Dokl., 11 (1970), No. 2, pp. 390 393. (English Translation). [3] J. Dieudonné, Foundations of Analysis, Academic Press, 1960. [4] M. Hirsch, Differential Topology, Springer-Verlag, New York, Heidelberg, Berlin, 1976. [5] M. Hirsch C. Pugh, Stable manifolds hyperbolic sets, Proceeding of Symposia in Pure Mathematics, 14 (1970), Amer. Math. Soc., pp. 133 164. [6] M. Hirsch, C. Pugh, M. Shub, Invariant manifolds, Bulletin of Amer. Math. Soc., 76 (1970), pp. 1015 1019. 6 The argument of this paragraph is written with a few more details than the original paper.

DIFFERENTIABLE CONJUGACY 11 [7] M. Hirsch, C. Pugh, M. Shub, Invariant manifolds, Lecture notes in Mathematics, 583 (1977), Berlin New York. [8] M. Hirsch, J. Palis, C. Pugh, M. Shub, Neighborhoods of hyperbolic sets, Inventiones Math, 9 (1970), pp. 112 134. [9] E. Nelson, Topic in Dynamics I, Flows, Mathematical Notes, Princeton Press, 1969. [10] C. Pugh M. Shub, Linearizing normally hyperbolic diffeomorphisms flows, Inven. Math., 10 (1970), pp. 187 198. [11] S. Sternberg, On the structure of local homeomorphisms of Euclidean n-space, II, Amer. J. Math., 80 (1958), pp. 623 631. [12] F. Takens, Partially hyperbolic fixed points, Topology 10 (1971), pp. 133 147. Department of Mathematics, Northwestern University, Evanston IL 60208 E-mail address: clark@math.northwestern.edu