Structural Stability of the Inverse Limit of Endomorphisms

Available online at www.sciencedirect.com ScienceDirect

Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 www.elsevier.com/locate/anihpc

Structural stability of the inverse limit of endomorphisms

Stabilité structurelle de l’extension naturelle des endomorphismes

∗ Pierre Berger a, , Alejandro Kocsard b

a LAGA, Université Paris 13, France b Universidade Federal Fluminense, Brazil Received 18 September 2014; received in revised form 3 March 2016; accepted 13 October 2016 Available online 8 November 2016

Abstract We prove that every endomorphism which satisﬁes Axiom A and the strong transversality conditions is C1-inverse limit structurally stable. These conditions were conjectured to be necessary and sufﬁcient. This result is applied to the study of unfolding of some homoclinic tangencies. This also achieves a characterization of C1-inverse limit structurally stable covering maps. © 2016 Elsevier Masson SAS. All rights reserved.

Résumé Nous montrons qu’un endomorphisme a son extension naturelle qui est C1-structurellement stable s’il vériﬁe l’axiome A et la condition de transversalité forte. Ces conditions étaient conjecturées nécessaires et sufﬁsantes. Ce résultat est appliqué à l’étude des déploiements des tangences homoclines. Aussi, cela accomplit la description des recouvrements dont l’extension naturelle est C1-structurellement stable. © 2016 Elsevier Masson SAS. All rights reserved.

Keywords: Inverse limit; Natural extension; Structural stability; Axiom A; Endomorphism; Strong transversality condition

1. Introduction

Following Smale [19], a diffeomorphism f is Cr -structurally stable if any Cr -perturbation f of f is conjugate to f via a homeomorphism h of M:

f ◦ h = h ◦ f .

* Corresponding author. E-mail addresses: [email protected] (P. Berger), [email protected] (A. Kocsard). http://dx.doi.org/10.1016/j.anihpc.2016.10.001 0294-1449/© 2016 Elsevier Masson SAS. All rights reserved. 1228 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

A great work was done by many authors to provide a satisfactory description of C1-structurally stable diffeomorphisms, which starts with Anosov, Smale, Palis (see [13,10]) and finishes with Robinson [18] and Mañé [6]. Such diffeomorphisms are those which satisfy Axiom A and the strong transversality condition. The descriptions of the structurally stable maps for smoother topologies (Cr , Cω, holomorphic...) remain some of the hardest, fundamental and open questions in dynamics. Hence the description of Cr -structurally stable endomorphisms (Cr -maps of a manifold not necessarily bijective) with critical points (points at which the differential is not surjective) is even harder. Indeed, this implies that the critical set must be stable (i.e. the map must be equivalent to its perturbations via homeomorphisms) and so that r must be at least 2. We recall that the description of critical sets which are stable is still an open problem [7]. It is not the case when we consider the structural stability of the inverse limit. We recall that the inverse limit 1 Z set of a C -endomorphism f is the space of the full orbits (xi)i ∈ M of f . The dynamics induced by f on its inverse limit set is the shift. The endomorphism f is C1-inverse limit stable (or equivalently inverse limit of f is C1-structurally stable) if for every C1 perturbation f of f , the inverse limit set of f is homeomorphic to the one of f via a homeomorphism which conjugates both induced dynamics and which is C0-close to the canonical inclusion Z into M . ←− When the dynamics f is a diffeomorphism, the inverse limit set M f is homeomorphic to the manifold M. The C1-inverse limit stability of f is then equivalent to the C1-structural stability of f : every C1-perturbation of f is conjugated to f via a homeomorphism of MC0-close to the identity. The concept of inverse limit stability is an area of great interest for semi-flows given by PDEs, although still at its infancy [16,4]. There were many works giving sufficient conditions for an endomorphism to be structurally stable [9,12,2]. The latter work generalized Axiom A and the strong transversality condition to differentiable endomorphisms of manifolds, and conjectured these conditions to be equivalent to C1-inverse limit stability. A main point of this work was to give evidences that the notion of inverse stability should be independent to the nature of the critical set (stable or not for instance). A similar conjecture was sketched in [15]. We prove here one direction of this conjecture, generalizing [9,12,2,17,18]:

Theorem 1.1 (Main result). Every C1-endomorphism of a compact manifold which satisﬁes Axiom A and the strong transversality condition is C1-inverse limit structurally stable.

The definitions of Axiom A and the strong transversality condition will be recalled in §2.3. Joint with the works of [1] and [2], this proves that C1-inverse limit stable covering maps of manifolds are exactly the C1 covering maps which satisfy Axiom A and strong transversality conditions (see §3.1). On the other hand, our main result applies to the dynamical studies of homoclinic tangencies unfolding as seen in section (see §3.2). The proof of the main result is done by generalizing Robbin–Robinson proof of the structural stability with two new difficulties. We will have to handle the geometrical and analytical part of the argument on the inverse limit space which is in general not a manifold as it is the case for diffeomorphisms (see §6). Also we will have to take care of the critical set in the plane fields constructions and in the inverse of the operator considered (see §7, 8 and 9).

2. Notations and deﬁnitions

Along this article M will denote a smooth Riemannian compact manifold without boundary. The distance on M induced by the Riemannian structure will be simply denoted by d. For any r ∈ N, we denote by Endr (M) the space of Cr endomorphisms of M. By Cr endomorphism of M, we mean a Cr map f of M into M, which is possibly non surjective and can have a non-empty critical set:

Cf := {x ∈ M : Txf not surjective}. We endow Endr (M) with the topology of uniform convergence of the ﬁrst r derivatives. Giveny an f ∈ Endr (M), a subset ⊂ M is forward invariant whenever f() ⊂ , and totally invariant when f −1() = . Note that totally invariance implies forward invariance. P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1229

The set of periodic points of f is denoted by Per(f ) and we write (f) for the set of non-wandering points. Observe that f(Per(f )) = Per(f ) and f((f)) = (f), but in general they are not totally invariant. Now, let K be a compact metric space and E → K a ﬁnite dimensional vector bundle over K. If F ⊂ E is a sub-vector-bundle of E → K, we denote by E/F the quotient bundle. Note that any (Riemannian) norm · E on E naturally induces a (Riemannian) norm on E/F deﬁning

vx + Fx E/F := inf vx + wx E , ∀x ∈ K, ∀vx ∈ Ex. wx ∈Fx On the other hand, observe that any bundle map T : E → E that leaves invariant F (i.e. F is forward invariant for T ) naturally induces a bundle map [T ]: E/F → E/F.

2.1. Inverse limits : → := n Given any set X and an arbitrary map f X X, we define its global attractor by Xf n≥1 f (X) and its inverse limit by ←− Z X f := x = (xn)n ∈ X : f(xn) = xn+1, ∀n ∈ Z . (1) ←− Observe that f(X ) = X , but in general X is not totally invariant (X = f −1(X )), and X ⊂ (X )Z. More- f f ←− f ←− ←− ←− f ←− f f f over, f acts coordinate-wise on X . In fact, we can define f : X → X by f(x) := (f (x )) = (x + ) , and in ←− f f f n n n 1 n this way, f turns out to be a bijection and f a factor of it. Indeed, for every j ∈ Z we can define the j th-projection ←− πj : X f → Xf by πj (x) = xj and then we have ←− πj+1 = πj ◦ f = f ◦ πj . Whenever X is a topological space and f is continuous, we shall consider XZ endowed with the product topology. In ←− ←− this case, X turns out to be closed in XZ and f a homeomorphism. Of course, Per(f ) and (f) are contained in ←− f Xf and X f is compact whenever Xf is compact itself. Z Finally, when X is endowed with a finite distance d, we shall consider X equipped with the distance d1 given by d(x ,y ) d (x, y) := n n . (2) 1 2|n| n∈Z Z The metric space (X , d1) is compact if and only if X is compact itself.

2.2. Structural and inverse limit stability

Two endomorphisms f, g ∈ Endr (M) are conjugate when there exists a homeomorphism h ∈ Homeo(M) satisfying h ◦ f = g ◦ h. More generally, the endomorphisms f and g are inverse limit conjugate whenever there exists a ←− ←− ←− ←− homeomorphism H : M f → M g such that H ◦ f = g ◦ H . Remark that the conjugacy relation implies the inverse limit conjugacy one. A Cr -endomorphism f is Cs -structurally stable (with 0 ≤ s ≤ r) when there exists a Cs -neighborhood U of f such that every g ∈ U is conjugate to f . Analogously, f is Cs -inverse limit stable when every g ∈ U is inverse limit conjugate to f .

2.3. Axiom A endomorphisms

Let f ∈ End1(M) and let ⊂ M be a compact forward invariant set. The set is hyperbolic whenever there exists s a continuous sub-bundle E ⊂ TM satisfying the following properties:

1. Es is forward invariant by Tf, i.e. s ⊂ s ∀ ∈ ; Txf(Ex) Ef(x), x 1230 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

s s 2. the induced linear map [Txf ]: TxM/E → Tf(x)M/E is an isomorphism, for every x ∈ ; (see §2 for notation of quotient bundles and induced maps) ∀ ∈ [ ]−1 3. x , Txf s < 1 and Txf < 1, where the ﬁrst operator norm is induced by the Riemannian structure Ex of M, and the second by its quotient.

s u Remark 2.1. Notice that despite E is contained in TM, in general we cannot deﬁne E as a sub-bundle of the tangent bundle.

However, using a classical cone ﬁeld argument, we show:

u Proposition 2.2. There exists a continuous family (E ) ←− of subspaces of TM such that: x x∈f ←− ∈ u ⊂ u = u 1. for every x f , Ex Tπ0(x)M and Tf(Ex ) E←− , ←− f(x) u u 2. for every x ∈ f , the restriction Tf : E → E←− is invertible and x f(x) −1 (Tf u ) < 1. Ex ←− Given any (small) ε>0 and x ∈ f , we define the ε-local stable set of x by ←− ←− ←− s := ∈ : ≥ n n −n−−−−→+∞→ ∀ ≥ Wε (x,f) y M f ε d1( f (x), f (y)) 0, n 0 , where d1 denotes the distance given by (2); and the ε-local unstable set of x is defined analogously by ←− ←− ←− u := ∈ : ≥ −n −n −n−−−−→+∞→ ∀ ≥ Wε (x,f) y M f ε d1 f (x), f (y) 0, n 0 . u s The geometry of these sets was described in [2]. Let us recall that π0 Wε (x, f) and π0 Wε (x, f) are submanifolds of M (for sufficiently small). The endomorphism f satisfies Axiom A when (f) is hyperbolic and coincides with the closure of Per(f ). ←− ∈ ≥ An Axiom A endomorphism satisfies the strong transversality condition if for every x, y (f)and every n 0, n s ∈ u ∩ −n s the map f u is transverse to π0(W (y)). This means that for every z π0(W (x)) f (π0(W (y))) the π0(W (x)) following holds: n u + s = Tf Tzπ0(W (x)) Tf n(z)π0(W (y)) Tf n(z)M An endomorphism which satisfies Axiom A and the strong transversality condition is called an AS-endomorphism. This notion generalizes the one of diffeomorphism. Let us recall some other examples.

Examples 2.3.

–The action of any hyperbolic linear matrix in Mn(Z) in the n-dimensional torus is hyperbolic and so is AS. It is not structurally stable whenever the matrix is not in SLn(Z) nor expanding [11]. –The constant map Rn x → 0 ∈ Rn satisﬁes Axiom A and the strong transversality condition. –The map Rn x → x2 + c satisﬁes Axiom A and the strong transversality condition whenever c is such that a (possibly super) attracting periodic orbit exists.

1 1 Remark 2.4. Let us notice that if two endomorphisms f1 ∈ C (M1, M1) and f2 ∈ C (M2, M2) satisfy Axiom A 1 and the strong transversality condition, then the product dynamics f1 × f2 ∈ C (M1 × M2, M1 × M2) also satisfy Axiom A and the strong transversality condition. As an endomorphism is AS iff its one of its iterates is AS, it follows that the following delay dynamics is AS if f ∈ C1(M, M) is AS: n n M (xi)i → (f (xm), x1,...,xm−1, 0,...,0) ∈ M . P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1231

From the latter example and remark, we have the following.

Example 2.5. For every c ∈ R such that x2 + c has an attracting periodic orbit, the following map is AS: ∈ Rn → 2 + ∈ Rn (xi)i (xm c,x1,...,xm−1, 0,...,0) .

We will see that this example appears in the unfolding of generics homoclinic tangency in §3.2.

3. Applications of main Theorem 1.1

3.1. Description of C1-inverse limit stable covering maps

A C1-covering map of a compact, connected manifold M is a surjective C1 endomorphism f of M without critical points. Then, every point of M has the same number p of preimages under f . We remark that every distinguish − ←− neighborhood U ⊂ M has its preimage π 1(U) in the inverse limit M which is homeomorphic to U × Z where 0 f ←− p Zp is a Cantor set labeling the different f -preorbits of U. These homeomorphisms endow M f with a structure of lamination called the Sullivan solenoid [20]. It follows immediately from a theorem due to Aoki, Moriyasu and Sumi [1] that: if an endomorphism f is C1-inverse limit stable and has no critical point in the non-wandering set, then f satisfies Axiom A. By Theorem 2.4 of [2], if f is C1-inverse limit stable and satisfies Axiom A, then f satisfies the strong transversality condition. Together with Main Theorem 1.1, it comes the following description of C1-inverse stable covering maps.

Theorem 3.1. A C1-covering map of a compact manifold is C1-inverse limit stable if and only if it is an AS- endomorphism.

3.2. Application to dynamical study of unfolding homoclinic tangencies

Let M be a manifold of dimension m and let (fμ)μ be a smooth family of diffeomorphisms of M which has a hyperbolic ﬁxed point p with unstable and stable directions of dimensions u ≥ 1 and s ≥ 1 respectively. Hence m = u + s. The following Theorem has been proven in the general case as in [8] (Proposition 1). For more restricted cases see [14] when ,(s u) = (1, 1) and Theorem 1 [21] when (s, u) = (1, 2).

≥ Theorem 3.2 (L. Mora). There exist h u, an open set of families (fμ)μ∈Rh of smooth diffeomorphisms of M, h s u which exhibit at μ0 ∈ R an unfolding of a homoclinic tangency at q ∈ W (p) ∩ W (p), such that there exists a h small neighborhood Nq ⊂ M of q, there exists a small neighborhood Nμ ⊂ R of μ0 covered by submanifolds L of dimension u, satisfying for every n large:

– μ0 belongs to every submanifold L and the intersection of two different such manifolds L is the single point μ0, u – for every L, there is parametrization γn of L by R , such that for every μ = γn(b) ∈ L \{μ0}, there is a chart φμ of Nq , such that the rescaled ﬁrst return map has the form: ◦ n ◦ −1 : Ru × Rs −→ Ru × Rs = Rm φμ fγ(b) φμ u−1 → 2 + + + (x,y) (xu bu bixi,x1,...,xu−1, 0,...,0) Eμ(x,y), i=1 ∞ m m with b = b1, ...bu, x = (x1, ..., xu), y = (y1, ..., ys) and Eμ ∈ C (R , R ) small in the compact-open Cr -topology for every r when n is large.

{ := ∈ R} r In particular, near the curve μn(a) γn(0, ..., 0, a), a , the rescaled ﬁrst return map fμn(a) is C close to the endomorphism: := → 2 + Fa (x1,...,xu,y1,...,ys) (xu a,x1,...,xu−1, 0,...,0). 1232 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

For an open and dense set of parameters a, the map x → x2 + a has an attracting periodic orbit from [5,3]. Then its non-wandering set is the union of an attracting periodic orbit with an expanding compact set. By Example 2.5, we know that Fa is AS. Moreover we can extend Fa to the n-torus which is the product of n-times the one point compactification of R. Its extension is analytic and AS. Hence by Theorem 1.1, the inverse limit of Fa restricted to bounded orbits is conjugate to an invariant compact set of f n |U with n large. μn(a) n In, particular if such an a is fixed and then n is taken large, then there exist an open set Vn of M and a neighborhood ∈ nu| Wn of μn(a) such that for every a Wn, fa Vn has its maximal invariant compact set conjugated to the product of u-times the inverse limit dynamics of x2 + a (restricted to the bounded orbits). For instance, when a = 0, then the non-wandering set of x → x2 consists of the attracting fixed point 0 and the u repelling fixed point 1. On the other hand the non-wandering set of F0 is {0, 1} ×{0}. We remark also that the set of points for which the orbit is bounded is homeomorphic to the square [0, 1]u ×{0} via the first coordinate projection. Hence for a small, the maximal invariant of Fa is a topological u-cube bounded by the stable and unstable manifolds of the hyperbolic continuation of the non-wandering points. It will be useful to keep this example in mind because, even though the most interesting case is when x → x2 + a exhibits positive entropy, this one (with the parameter a sufficiently close to 0) already presents most of the geometric difficulties that appear in the proof of Theorem 1.1.

4. Proof of Main Theorem 1.1

4.1. Sufficient conditions for the existence of a conjugacy ←− ←− 1 We want to find, for every g which is C -close to f , a continuous map h: M f → M g which is close to the canon- ←− Z ←− ←− ←− ical injection M f → M and satisfies h ◦ f = g ◦ h. This is equivalent to find a continuous map h0 : M f → M satisfying ←− h0 ◦ f = g ◦ h0, (C˚1) 0 and which is C -close to the zeroth coordinate projection π0. This means that for every η>0small and every g 1 sufficiently C -close to f , h0 satisfies

sup d(h0(x), π0(x)) ≤ η. (C˚2) ←− x∈Mf

Indeed, one can construct such an h0 from such an h and vice versa writing: ←− n h0 := π0 ◦ h, h := (h0 ◦ f )n∈Z. Let us suppose the existence of such an h. We would like h to be a homeomorphism, so let us ﬁnd sufﬁcient conditions to ensure its injectiveness. In the Anosov case this follows easily from (C˚ 1) and (C˚2). In fact, if two points x and y have the same image by h, then these two f -orbits must be uniformly close by (C˚ ) and (C˚ ), and so they are equal by expansiveness. In the 1 2 ←− wider case of AS dynamical systems, we shall consider the following Robbin metric on M f :

d∞(x,y) = sup d(xi,yi). i∈Z ←− For every x, y ∈ M f , let us observe that: d(xi,yi) d∞(x,y) d (x,y) = ≤ = 3d∞(x,y). (3) 1 2|i| 2|i| i i This metric enabled Robbin [17] to ﬁnd a sufﬁcient condition on h to guarantee its injectiveness. We adapt it to our context. P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1233

←− Proposition 4.14 of [2] gives a geometric interpretation of the metric d∞. After showing that M is a finite union f ←− of laminations, the leaves of which are intersection of stable sets with unstable manifolds of points in f , we proved that d∞-distance between any two of these leaves is positive. Moreover the restriction of d∞ to each leaf is equivalent to a Riemannian metric on its manifold structure. ←− Choosing η>0 small, any continuous map h0 : M f → M satisfying (C˚2) can be written as a perturbation of π0 via the exponential map exp: TM → M associated to the Riemannian metric of M. For the sake of simplicity, let us fix a bundle trivialization TM⊂ M × RN , for some positive integer N. As h satisfies (C˚ ) (with η small), there ←− 0 2 N N exists w : M f → R such that (π0(x), w(x)) ∈ TM⊂ M × R and h (x) = (w(x)). 0 expπ0(x) (4) Now let us extend the Riemannian metric (·, ·x)x∈M of M to an Euclidean norm ( · x)x∈M on the bundle M × RN → M. Let us denote by w ∈[0, ∞] the d∞-Lipschitz constant of w, i.e. w(x) − w(x) := π0(x) w sup . (5) ←− d∞(x,x ) x,x ∈Mf Here is the Robbin condition:

w ≤ η. (C3)

Proposition 4.1 (Robbin [17]). There exists η> which depends only on the Riemannian metric of M, such that for 0 ←− ←− 1 every pair f and g of C -endomorphisms of M, if there exists h: M f → M g satisfying (C˚1), (C˚2) and (C3), then h is injective.

←− Proof. Let x, x ∈ M f be such that h(x) = h(x ). Note that by (C˚1) and (C˚2), the point πi ◦ h(x) is η-close to πi(x), for every i. Thus πi(x) and πi(x ) are 2η-close for every i ∈ Z. ←− ←− ∈ Z ∞ ≤ ◦ = ◦ i = ◦ i Let i be such that d (x, x ) 2d(xi, xi). We recall that πi h(x) h0 f (x) expxi (w f (x)), and so: ←− ←− i i exp (w ◦ f (x)) = exp (w ◦ f (x )) xi xi The exponential maps exp and exp produce two charts centered at xi and x , and modeled on the vector subspaces xi xi i N −1 Tx M and T M of R . The coordinates change of these charts is the translation by the vector exp x plus a linear i xi xi i RN map L bounded by a constant K times d(xi, xi), where K depends only on the curvature of M. Thus, in , it holds: − ←− ←− exp 1 x + (id + L) ◦ w ◦ f i(x) = w ◦ f i(x ) + o(d(x ,x )). (6) xi i i i ≤ ≤ We recall that d∞(x, x ) 2d(xi, xi) 4η is small. On the other hand by (C3): ←− ←− ◦ i − ◦ i ≤ w f (x) w f (x ) xi ηd∞(x,x ) Thus, replacing each term of equality (6) by these estimates, it holds: d∞(x,x ) − ≤ d(x ,x ) = exp 1 x 2 i i xi i ←− ←− ≤ ◦ i − ◦ i + + w f (x) w f (x ) xi L η o(d(xi,xi)) ≤ ηd∞(x,x ) + Kd∞(x,x )η + o(d∞(x,x )) This implies d∞(x, x ) = 0 and so x = x . 2

On the other hand, in Proposition 5.4 of [2] it is showed the following:

Proposition 4.2. For every AS C1-endomorphism f of M, there exists η> such that for every endomorphism g ←− 0←− sufﬁciently close to f , if there exists a d -continuous and injective h : M → M satisfying (C˚ ) and (C˚ ) with f , g ←− 1 f g 1 2 and η, then h is surjective onto M g. 1234 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

1 Hence if we prove that for every gC -close to an AS endomorphism f , there exists a continuous map h0 satisfying (C˚1), (C˚2) and (C3), then Propositions 4.1 and 4.2 imply that g is inverse limit conjugate to f , and so that f is C1-inverse limit stable. In other words, to prove Theorem 1.1 it remains only to prove the following:

Proposition 4.3. Let f be a C1-AS endomorphism. For every η> and for every endomorphism g sufﬁciently ←− 0 1 C -close f , there exists a continuous map h0 : M f → M satisfying (C˚1), (C˚2) and (C3) with f , g and η.

Therefore the remaining part of this manuscript is devoted to the proof of this proposition, by using the contraction mapping Theorem.

4.2. A contracting map on a functional space ←− N Let be the space of functions w : M f → R which are continuous for d1 and d∞-Lipschitz (i.e. they satisfy (w) < ∞, where (w) is deﬁned as in (5)). We endow with the uniform norm: := v C0 max←− v(x) π0(x). x∈Mf

Equivalent conditions in the space We recall that M × RN ⊃ TM is a trivialization. Moreover we have already N fixed an Euclidean structure ( · x)x∈M on M × R which extends the Riemannian metric (·, ·x)x∈M on TM. Let N px : R → TxM be the orthogonal projection given by · x . 0 Any g sufficiently C -close to f induces the following map from a neighborhood N of 0 ∈ into : − ←−− g (w) := x → exp 1 g ◦ exp p ◦ w ◦ f 1(x) , ∀w ∈ N , f x0 x−1 x−1 where xi = πi(x) for every i, as defined in §2.1. For every η small, and for every g sufficiently close to f , to find h0 satisfying conditions (C˚1), (C˚2) and (C3) is equivalent to find w ∈ satisfying g = f (w) w (C1) ≤ w C0 η (C2)

(w) ≤ η (C3) ←− ∈ ∈ : → Indeed, by (C1) any such w satisﬁes w(x) Tx0 M, for every x M f . It is then easy to remark that h0 x (w(x)) C˚ C˚ expx0 satisﬁes ( 1) and ( 2).

f ∈ ∈ Strategy To solve this (implicit) problem, let us regard the partial derivative of f at 0 with respect to w : ←− f = → → ◦ ◦ −1 D0 f w x Tx−1 f(px−1 w f (x)) .

1 f The first difficulty that appears is the following: if f is only C , in general the map D f does not leave invariant 2 the space . In the C -case, Robbin’s strategy in [17] consists in solving (C1)–(C2)–(C3) by finding a right inverse f − for D f id, and then by following a classical proof of the implicit function theorem which uses the contraction mapping Theorem. A second difficulty which will appear is that Tf is possibly non-invertible, and this will give us many difficulties to construct this right inverse with bounded norm. To eliminate some of these, in Lemma 4.4, we will suppose N twice larger than necessary to embed TM into M × RN .

Robinson trick When f is just C1 and not C2, by classical argument of convoluting f with a smooth molliﬁer we can ∞ 1 construct a family (fδ)δ such that f0 = f , fδ ∈ C (M, M)for every δ>0 and such that the map δ → fδ ∈ C (M, M) is continuous. P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1235

0 → δ RN Then we can consider a continuous family of C -maps (x Fx )δ from M into the space of linear maps of δ = ◦ given by Fx Txfδ px . In such a way, for each δ we get a linear bundle morphism F δ given by ←− ←− F δ : M × RN → M × RN f ←− f (7) (x,v) → ( f(x), F δ (v)) x0 ←− Note that F δ is still over f , i.e. the following diagram commutes: ←− ←− N F δ N M f × R M f × R

←− ←− f ←− M f M f

δ ∈ Lemma 4.4. If N is large enough, then we can suppose moreover that Fx is invertible for every δ>0 and x M.

1 Proof. Put N = 2N. Let fδ be a smooth endomorphism C -close f when δ is small. We extend the projection N N N N px : R → TxM to R = R × R by N N N px : R × R (v1,v2) → px(v1) ∈ R . Also we identify RN to RN ×{0} ⊂ RN . Let us regard: δ : RN = → ◦ + = ◦ + ∈ RN Fx v (v1,v2) (Txfδ px(v1) δv2,δv1) Txfδ px(v) δ(v2,v1) . N For every x ∈ M and δ>0, such a map is invertible, depends smoothly on x and is δ-close to R v → Txfδ ◦ N px(v) ∈ R . 2 ←− δ N Corollary 4.5. The map F is a homeomorphism of M f × R , for every δ>0.

For every v ∈ , the following map is well deﬁned: ←− ←− δ δ −1 F (v) := x ∈ M → F (v( f (x))). This map is continuous, linear and for δ>0it is bijective. δ Moreover, we remark that F (v) belongs to . δ Now, let us suppose the existence of a right inverse J of F − id. This means: δ F − id J = id.

Notice that the existence of a w ∈ satisfying (C1), (C2) and (C3) is equivalent to find a fixed point φ ∈ of the operator δ − − g − ◦ = − g − ◦ F id f id J id ( f id) J such that w = J(φ)satisfies (C2) and (C3). We construct J in §5. From its construction we get

Proposition 4.6. For every >0 and every η>0 sufﬁciently small w.r.t. , there exists δ>0 small enough such that for every gC1-close enough to f , the operator δ − − g − ◦ = δ − g ◦ F id f id J (F f ) J is well deﬁned on a 2η-neighborhood of 0 in and is C0-contracting. δ − g ≤ δ − g ≤ ≤ ≤ Moreover, F Jv η and F Jv , whenever v C0 η and (v) . f C0 f Together with Proposition 4.3, this implies the inverse limit structural stability of AS-endomorphisms. 1236 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

δ 5. Construction of the right inverse J of F − id ←− We recall that f denotes an AS-endomorphism of a compact manifold M. Let (f)be the non-wandering set ←− of f . It is shown in [2] that: ←− ←− Z (f)= M f ∩ (f) . ←− ←− Moreover, the non-wandering set (f)is the disjoint union of compact, transitive subsets ( ) , called basic ←− i i pieces. The family of all basic pieces is finite and called the spectral decomposition of (f). ←− ←− For every basic piece i , we define the stable and unstable sets of i , respectively, by ←− ←− ←− ←− s n W ( i) ={x ∈ M : d(f (x), i) → 0, as n →+∞} ←− ←− ←− ←− u n W ( i) ={x ∈ M : d(f (x), i) → 0, as n →−∞} The geometry of these sets is studied in [2]. ←− ←− ←− ←− ←− ←− ←− u s oGiven tw basic pieces i and j , we write i j if W ( i) intersects W ( j ) \ j . In [2] it is shown that for any AS-endomorphism f , the relation is an order relation. This enables us to enumerate the spectral ←− ←− ←− ←− decomposition ( )q of (f)in such a way that implies i>j. i i=1 ←− i j We recall that a filtration adapted to ( i)i is an increasing sequence of compact sets ←− ∅ = M0 ⊂ M1 ⊂···⊂Mi ⊂···⊂Mq = M f such that for q ≥ i ≥ 1: ←− ←− ←− n f (Mi \ Mi−1) = i and f(Mi) ⊂ int(Mi). n∈Z The existence of such a filtration is shown in Corollary 4.7 of [2]. The following proposition is formally similar to the one used by Robbin [17] or Robinson [18], but it is technically much more complicated and its proof requires to be handled very carefully. New ideas will be needed. The proof will be done in §7, 8, 9 and will use §6. ←− Proposition 5.1. There exist λ ∈ (0, 1), K>0 and an open cover (W )q of M , where each W is a neighborhood ←− i i=1 f i u s × RN → of i , and such that for every δ>0, there exist vector subbundles Ei and Ei of the trivial bundle Wi Wi satisfying the following properties: ←− ∈ ∩ −1 δ s u s u (i) For every x Wi f (Wi), the map F sends Eix and Eix onto E ←− and E ←− respectively. ←− i f(x) i f(x) −1 (ii) For any k ≥ j and every x ∈ Wk ∩ f (Wj ), the following inclusions hold:

F δ(Es ) ⊂ Es←− ,Fδ(Eu ) ⊃ Eu←− . kx j f(x) kx j f(x) s ⊕ u = RN ∈{ } ∈ s u (iii) Eix Eix , for any i 1, ..., q and every x Wi ; the angle between Ei and Ei is bounded from below by K−1. u s (iv) The subbundles Ei and Ei are d1-continuous and d∞-Lipschitz. (v) For every i and any x ∈ Wi , it holds δ u ≥ u ∀ u ∈ u F (v ) v /K, v Eix. ←− ←− ←− (vi) For every q , if x ∈ W then f(x) ∈∪/ W and ∩ ∈Z f n(W ) = . q j>q←− j n q q s ∈ s u ∈ u (vii) For every i, any x in a neighborhood of i which does not depend on δ, and for all v Eix and v Eix, it holds: F δ(vs) ≤λ vs and F δ(vu) ≥ vu /λ. P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1237

u s RN The subbundles Ei and Ei can be considered as functions from Wi to the Grassmannian GN of . Property (iv) means that they are d1-continuous and d∞-Lipschitz. The Grassmannian GN is a manifold with as many connected components as possible dimension for RN -subspaces, i.e. N + 1.

Remark 5.2. A main difﬁculty in this proposition is that K does not depend on δ, whereas the norm of the inverse of F δ blows up as δ approaches 0 whenever f has critical points. Hence the proof of this proposition will not be symmetric in u and s.

s u At this point is important to remark that the vector subbundles Ei and Ei depend on δ. q We will prove in Corollary 6.2 the existence of a partition of the unity (γi)i subordinated to (Wi)i=1, where each γi is d1-continuous and d∞-Lipschitz. ∈ s : RN → s u u : RN → u Given any x Wi , let πix Eix denote the projection parallely to Eix and πix Eix the projection s parallely to Eix. For v ∈ and σ ∈{s, u}, put: ∞ −1 σ := σ · := − δn s := δn u vi πi (γi v), Jis(v) F (vi ) and Jiu(v) F (vi ). (8) n=0 n=−∞ We can now deﬁne: J := Jiσ . (9) i,σ Let us deﬁne := s u C sup sup πix , πix . 1≤i≤q x∈suppγi By Property (iii) the constant C is bounded from above independently of δ.

∈ ∈ s Lemma 5.3. There exists a constant D independent of δ such that for every j, for all x Wj and u Ejx (resp. ∈ u ≥ ≤ u Ejx), for every n 0 (resp. n 0): | | (F δ)n(u) ≤Dλ n u ←− ←− ∈ i ∈ Proof. For every x Wj , since (Wk)k is a cover of M f , there exists a sequence (ni)i such that f (x) Wni for every i. From property (vi), the sequence (ni)i must be decreasing. −k k As x ∈ W , we can suppose that n = j. By property (ii), for every k ≥ 0, F δ sends Eu into Eu and F δ sends j 0 jx nk Es into Es . jx nk ←− Let (V ) be the neighborhoods of respectively ( ) on which (vii) holds. Since the non-wandering set contains k k ←− k k ←− ←− ←− m−1 the limit set and M f is compact, there exists m ≥ 0 such that there is no x ∈ M f such that (x, f(x), ··· f (x)) i are all outside of ∪kVk. We can suppose Vk included in Wk for every k. Consequently, for every x, all the terms f (x) i ∈ ∈ s ∈ u of the sequence (f (x))i but qm are in Vni . From (vii), it follows that for all x Wi and u Eix (resp. u Eix), for every n ≥ 0 (resp. n ≤ 0): | | (F δ)n(u) ≤Dλ n u , = δ δ| u −1 qm 2 with D max( F , (F Eix) ) , which is bounded by a constant independent of δ by (v).

From Lemma 5.3, it holds that for every v ∈ : 2CDq Jv 0 ≤ v 0 , (10) C 1 − λ C where C, D and λ are independent of δ>0 small. Moreover we easily compute the following. 1238 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

δ Proposition 5.4. The map J is the right inverse of F − id:

δ (F − id) ◦ J = id.

To prove main Theorem 1.1, it remains only to prove Propositions 4.6 and 5.1. Tow sho Proposition 5.1, we will develop some analytical tools in the next section. On the other hand, we are ready to prove Proposition 4.6.

δ − g ∈ Proof of Proposition 4.6. Let us start by computing (F f ) at 0 : δ − g = δ − −1 ◦ ◦ (F f )(0)(x) Fx− (0) expx g expx− (px−1 (0)) 1 0 1 (11) −1 =−exp (g(x− )). x0 1 δ − g = In particular, this implies F (0) dC0 (f, g). f C0 δ − f 1 ∈ On the other hand, (F f ) is a C map defined on a neighborhood of 0 . Thus we can compute its derivative at the origin: ←−− D F δ − f (v)(x) = F δ − T f (p ◦ v ◦ f 1(x)) (12) 0 f x−1 x−1 x−1 ←− 0 for every v ∈ and every x ∈ M f . In particular, the operator norm subordinate to the C norm satisfies: δ f D0 F − → 0, as δ → 0. f C0 δ f At a neighborhood of 0, the derivative of F is constant, whereas the one of φf is continuous. Furthermore, for g 1 g f C -close to f , Dφf is close to Dφf . Hence, for every μ > 0, there exists a small η(μ) > 0 such that for any g sufficiently close to f in the C1-topology ∈ ≤ ≤ and any w with w C0 η(μ) and δ η(μ), it holds δ g Dw F − ≤ μ. (13) f C0 Then, putting together (11), (13), we get δ − g ≤ + (F )J v dC0 (f, g) μ Jv C0 f C0 ≤ + dC0 (f, g) J C0 μ v C0 .

By (10), J C0 is bounded independently of δ, we put = 1 1 μ0 inf min , > 0. (14) δ small 2 2 J C0 0 Hence for every η, δ<η(μ0), for every g moreover η/2-C -close to f it holds for v ∈ : ≤ ⇒ δ − g ≤ η + 1 v C0 η (F )J v v C0 <η. f C0 2 2 δ − g Which is the second statement of the Proposition. Also inequalities (13) and (14) implies that (F f )J contracts the C0-norm by a small factor when η, δ are small and g is close to f , which is the first statement of the Proposition. We remark that as far as δ ≤ η(μ0), which does not depend on η, we can suppose η ≤ η(μ0) as small as we want which satisfies the same property, if g is sufficiently close to f . δ − g ∈ It remains only to estimate ((F f )(J v)) for v . To do that, we prove the following lemma similar to the Robin’s computation §6 of [17]: P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1239

Lemma 5.5. For σ ∈{s, u}, there exist a constant A which depends on f but not on δ, and a constant Bδ which depends on δ such that for every v ∈ , for any i and σ = s, u: σ ≤ σ + σ (Jiσ vi ) A(vi ) Bδ vi C0 . (15)

σ As the norm of (vi ) is dominated by (v) times a constant independent of δ, it holds by taking the constants A and Bδ larger: ≤ + (J v) A(v) Bδ v C0 . (16) Put L := F δ − exp−1 ◦g ◦ exp ◦p . We have: x x−1 x0 x−1 x−1 δ − g − δ − g = − + − (F f )J (v)(x) (F f )J (v)(y) Lx(J (v)(x) J (v)(y)) (Lx Ly)J (v)(y) Hence: δ − g ≤ + + ((F f )J (v)) L C0 (A(v) Bδ v C0 ) (L) J(v) C0 , where (L) depends on δ.

By (13), we can suppose g sufﬁciently close to f and δ small enough so that L C0 is 1/(2A) contracting on a small neighborhood of 0. From this: δ g (v) Bδ ((F − )J (v)) ≤ + + (L) J 0 v 0 . f 2 2A C C Hence for every >0, for every η such that:

Bδ −1 η ≤ ( + (L) J 0 ) 2A C 2 ≤ ≤ If v C0 η and (v) and g sufﬁciently close to f , it holds: δ − g ≤ 2 ((F f )J (v))

Proof of Lemma 5.5. We prove the case σ = s, since the other case σ = u is similar. For n ≥ 0, we evaluate: δn s − δn s F (x,vi (x)) F (y,vi (y)) ≤ δn ◦ s s − s + δn ◦ s s − δn ◦ s s F πi (x,vi (x) vi (y)) F πi (x,vi (y)) F πi (y,vi (y)) By Remark 5.3, there exists a constant D which does not depend on n nor δ such that: δn| s ≤ n F Ei Dλ Hence: δn ◦ s s − s ≤ n s F πi (x,vi (x) vi (y)) DCλ C(vi )d∞(x,y) On the other hand, n n F δ ◦ π s(x, ·) − F δ ◦ π s(y, ·) i i n−k−1 ←− ←− ≤ F δ |Es ←− · F δ( f k(x), ·) − F δ( f k(y), ·) · Tfk|Es , k+1 y nk f (x) k ←− k+1 where nk is such that f (x) ∈ Wn . This is less than: k − − ←− ←− CDλn k 1 · F δ( f k(x), ·) − F δ( f k(y), ·) ·CDλk. k Hence there exists a constant K(δ) which depends only on f and δ such that: δn ◦ s · − δn ◦ s · ≤ · n F πi (x, ) F πi (y, ) n K(δ)λ d∞(x,y) Consequently: δn s − δn s ≤ 2 n s + · n F (x,vi (x)) F (y,vi (y)) (DC λ (vi ) n K(δ)λ v C0 )d∞(x,y) Summing over n we conclude. 2 1240 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

6. Analysis on Mf ←− 0 Let us introduce a few notations. Let N be an arbitrary Riemannian manifold. We recall that C (M f , N) denotes ←− ∞ ←− the space of d -continuous maps φ : M → N. Also Lip (M , N) denotes the space of d∞-Lipschitz maps φ : ←− 1 f f M f → N. Let us define: ←− ←− ←− ∞ := 0 ∩ ∞ Mor0 (M f ,N) C (M f ,N) Lip (M f ,N). ←− ←− 0 0 We endow C (M f , N) with the uniform distance given by the Riemannian metric of N. Note that C (M f , N) is a Banach manifold. Actually its topology does not depend on the Riemannian metric of N. The aim of this section is to ←− ←− ∞ 0 prove the denseness of Mor (M f , N)in C (M f , N). To do this, we will use a new technique based on convolutions. ∞ 0 Let ρ ∈ C (R) be a non-negative bump function with support in (−1, 1). Let μ be any Lebesgue measure on M such that μ(M) = 1, and let μ˜ = μ be the induced probability on MZ. ←− Z n For every map g from M f into R , for every r>0, we define gr by: ←− d1(x,y) g : M x → g(y) · ρ dμ(y˜ ). r f r MZ The following result plays a key role: ←− n ˜ Lemma 6.1. Let φ : M f → R be a continuous function with respect to the distance d1. Let φ be a continuous Z Z extension to (M , d1). Let 1 be the function on M constantly equal to 1 ∈ R. For every r>0, the functions 1r and ˜ φr (defined as above) satisfy:

φ˜ and φ˜ / are well deﬁned. (i) r r 1r ←− ˜ ∞ Rn (ii) φr is d1-continuous and d∞-Lipschitz, i.e. it belongs to Mor0 (M f , ). ˜ 0 ˜ (iii) The function φr /1r is C -close to φ whenever r is small. ˜ (iv) The support of φr is included in the r-neighborhood of the support of φ.

The following are immediate corollaries of this lemma: ←− ←− ⊂ ∞ R Corollary 6.2. For every open cover (Ui)i of M f , there exists a partition of unity (ρi)i Mor0 (M f , ) subordinate to it. ←− ←− ∞ 0 Corollary 6.3. The subset Mor0 (M f , N) is dense in C (M f , N). ←− Remark 6.4. Both above corollaries are also true if we replace M f by any compact subset E of it.

Proof of Lemma 6.1. Let us start by proving (i). As φ˜ and ρ are continuous on a compact space, they are bounded. ˜ Z = ˜ ∈ Z | As μ(M ) 1, the functions φr and 1r are well deﬁned. Let x M . There exists δ>0 such that ρ Bd1 (0, δ/r) is ∈ Z ˜ greater than δ. For every x M , the μ-volume of the ball Bd1 (x, δ/r) is greater than: N δ μ B x , > 0, i 6r −N −|n| δ where N is any natural number satisfying | |≥ 2 diam(M) ≤ . ←− n N 2r := {˜ : ∈ } Thus, m inf μ(Bd1 (x, δ/r)) x M is positive and 1r >mδ. Consequently, φr /1r is everywhere well deﬁned. Z ˜ Let us proof (iii). As (M , d1) is compact, the function φ is uniformly continuous: for every δ>0, there exists ˜ ←− r>0 such that the image by φ of any d1-ball of radius r has diameter less than δ. Thus for every x ∈ M f : P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1241 d1(x,y) |φ˜ (x) − φ(x˜ ) · 1 (x)|≤ δ · ρ μ(y˜ ) ≤ δ · 1 (x) r r r r MZ Let us proof (ii). We remark that if a function is d -Lipschitz, then it is d∞-Lipschitz, and so it belongs to ←− 1 ←− ∞ Rm ˜ ∈ Mor0 (M f , ). Then, let us prove φr is d1-Lipschitz. For every x M f : d1(x,y) d1(x ,y) φ˜ (x) − φ˜ (x ) = φ(y˜ ) · ρ − ρ dμ(y˜ ) r r r r MZ As ρ is smooth, its derivative is bounded by some L, and so: d1(x,y) d1(x ,y) L L ρ − ρ ≤ |d (x,y) − d (x ,y)|≤ d (x,x ) r r r 1 1 r 1 Consequently: L |φ˜ (x) − φ˜ (x )|≤ d (x,x ) |φ(y˜ )|dμ(y˜ ) r r r 1 MZ ˜ ˜ Thus, since φ is bounded and μ˜ is a probability, we get that φr is d1-Lipschitz as desired. 2

7. Proof of Proposition 5.1

Let f be an AS endomorphism of a compact manifold M.

7.1. Preliminaries

Distance on Grassmannian bundles We endow the space of linear endomorphisms of RN with the operator norm N N N · induced by the Euclidean one of R . We recall that the Grassmannian GN of R is the space of d-planes of R , ≤ ≤ ∈ for 0 d N. Given two planes P, P GN let πP and πP be their associated orthogonal projections. The metric dG on GN is deﬁned by: dG(P, P ) = πP − πP

Angle between planes Two planes P and P of Rn make an angle greater than η if for all u ∈ P \{0} and v ∈ P \{0}, the angle between u and v is greater than η (for the Euclidean norm), in particular they are in direct sum. ←− ←− Deﬁnition of Es We recall that for any (x, a) ∈ M × RN , F δ(x, a) = ( f(x), F δ (a)), where x = π (x). f x0 0 0 s 0 The stable direction Ex of F at x is given by s := ⊕ s Ex Ker px0 Tx0 W (x0,f), (17) ←− s where W (x0, f)is the stable set of x0; its intersection with a neighborhood of π0(M f ) is an immersed manifold (see Proposition 4.11 [2]). ←− s ∈ We remark that Ex depends only on x0, for every x M f . In order to construct the plane ﬁelds of Proposition 5.1, we will have to take care of the critical points of f . The unique control that we have on them is the strong transversality condition. This condition implies, in particular, that ←− for every x ∈ M f it holds s N E←− + Tf(Tx M)= R . (18) f(x) 0 s q s Therefore we shall construct the distributions (Ei )i=1 “close” to E in GN . Let us explain how we will proceed, and what does it mean. 1242 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

←− 0 Topology on plane ﬁelds of nested domains of deﬁnition For a subset C ⊂ M f , we denote by C (C, GN ) the space of d1-continuous maps from C into GN . When C is compact, we endow this space with the uniform metric: d(g,g ) = max d(g(x),g (x)). x∈C

0 ¯ Given a plane ﬁeld E ∈ C (C, GN ) and η>0, we denote by B(E, η) (resp. B(E, η)) the open (resp. closed) ball centered at E and radius←− η. 0 0 Let W be subset of M f and V a neighborhood of W . Let EW ∈ C (W, GN ) and EV ∈ C (V, GN ) be two plane ﬁelds. We say that EW is compact-open close to EV if for any compact subset C ⊂ W , there exists a small compact neighborhood N of C in V such that the graph of EV |N is close to the graph of EW |C for the Hausdorff distance on compact subsets of Mf × GN induced by d1 + dG. This will be explained in greater details for its application case in Remark 7.2.

7.2. Splitting Proposition 5.1 into the stable and unstable ﬁelds

We are going to illustrate the geometrical part of the proof of Proposition 5.1 by depicting the construction for the following example. Let f : (x, y, z) ∈ R3 → (x2, y2, 0). This map is AS and can be extended to an AS endomorphism of the compactiﬁcation (R ∪{∞})3 of R3 equal to the 3-torus. On this compact manifold, its inverse limit is homeo- 3 δ δ δ morphic to [0, ∞] , via the projection π0. Since this map is invariant via the symmetries (x, y, z) → (x x , y y , z z ), − (δ , δ , δ ) ∈{−1, 1}3, we will focus only on the restricted dynamics on π 1([0, 1]3) which is the inverse limit of f x y z 0 ←− restricted to the set of points with bounded orbit. The restricted non-wandering set f is formed by 4ﬁxed points (0, 0, 0)Z, (0, 1, 0)Z, (1, 0, 0)Z, (1, 1, 0)Z. Let us split Proposition 5.1 into two propositions. ←− ←− q s q Proposition 7.1. There exist neighborhoods (Vi)i=1 of respectively (W ( i))i=1 in M f , and for every small δ, there are functions

s : → Ei Vi GN satisfying the following properties for every i: ←− −1 (i) for every x ∈ Vi ∩ f (Vi) the following inclusion holds:

F δ(Es ) ⊂ Es←− . ix i f(x) ←− −1 (ii) for every k ≥ j, for every x ∈ Vk ∩ f (Vj ) the following inclusion holds:

F δ(Es ) ⊂ Es←− . kx j f(x) ←− s s| s (iii) Ei is compact-open close to E W ( i), when δ is small. s (iv) Ei is of constant dimension, d1-continuous, and locally Lipschitz for the metric d∞.

Fig. 1 depicts an example of such plane ﬁelds.

Remark 7.2. From the deﬁnition given in §7.1, Property (iii) means that for every i, for every compact subset C of ←− s W ( i), for every >0, there exists a compact neighborhood U of C in Vi such that for every δ sufﬁciently small s| s| ≤ dH (Graph(Ei U),Graph(E C)) , ←− where dH (·, ·) denotes the Hausdorff distance of compact subsets of M f × GN induced by the distance d1 + dG. We notice that U depends on C and but not on δ small enough. P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1243

s : → 2 2 Fig. 1. Plane ﬁelds Ei in the example given by f (x,y,z) (x ,y , 0).

s Remark 7.3. Property (iv) means that Ei is of constant dimension, d1-continuous, and that for every compact subset δ C of Vi there exists a constant LC such that: s s ≤ δ ∀ ∈ dG(Eix,Eiy) LCd∞(x,y), x,y C.

In the diffeomorphism case, to obtain the existence of (Eu ) it suffices to first push forward by F δ each of the plane ←− ←−iδ i ←− field Es on ∪ f n(V ) (which is a neighborhood of W u( ), and then to apply the same proposition to f −1. In our i n ←−i i case, even though f is invertible, the bundle map F 0 is not. However, in Lemma 4.4, we saw that F δ is invertible for s u every δ>0. Nonetheless, the norm of the inverse of this map depends on δ, and so the angle between Ei and Ei as well. However in Proposition 5.1 such an angle must be bounded by a constant which is independent of δ (and this is necessary in the proof of Proposition 4.6). ←− s Hence we must redo a similar construction, still in a neighborhood of each W ( i) since it is the only place where we control the singularities. Another difference in the construction of Eu is the following: to construct the plane field Eu we will not be allowed i ←− i s 0 to pull back, since the critical set might intersect W ( i), and a pull back by F would contain critical vectors which belong to Es , this would contradict the angle condition (iii) for δ>0. Hence the construction of Eu must be done in ←− i s compact set in a small neighborhood of W ( i) via push forward.

q ←− ←− Proposition 7.4. There exist K>0, an open cover (Wi)i=1 of M f , where each Wi contains i and is included in u × RN → Vi , such that for every δ>0, there exists a subbundle Ei of Wi Wi satisfying the following properties for i ∈[1, q]: ←− ∈ ∩ −1 δ u u (i) For every x Wi f (Wi), the map F sends Eix into E ←− . ←− i f(x) −1 (ii) For every j ≥ i, for every x ∈ Wi ∩ f (Wj ) the following inclusion holds:

F δ(Eu ) ⊃ Eu←− . jx i f(x) s ⊕ u = RN ∈ s u −1 (iii) Eix Eix , for every x Wi , the angle between Ei and Ei is bounded from below by K . u (iv) The subbundle Ei is of constant dimension, d1-continuous and d∞-Lipschitz. (v) For every x ∈ Wi , it holds δ u ≥ u ∀ u ∈ u F (v ) v /K, v Eix. ←− ←− ←− n (vi) For every q , if x ∈ Wq then f(x) ∈∪/ j>q Wj and ∩n∈Z f (Wq ) = q .

Fig. 2 depicts an example of such plane ﬁelds. 1244 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

u : → 2 2 Fig. 2. Plane ﬁelds Ei in the example given by f (x,y,z) (x ,y , 0).

Fig. 3. A ﬁltration for the example given by f : (x,y,z)→ (x2,y2, 0).

From the two latter propositions, we easily deduce:

Proof of Proposition 5.1. By Propositions 7.1 and 7.4, we have immediately properties (i)–(ii)–(iii)–(iv)–(v)–(vi) of Proposition 5.1. To prove property (vii), we remark that by Proposition 7.1 (i) and (iii) together with the hyper- ←− ←− bolicity of , the bundle Es is contracted by F δ over a neighborhood of , for every i. Moreover, by properties i ←− i ←− u u| δ (i)–(iii)–(iv) of Proposition 7.4, the bundle Ei is close to E i , and so expanded by F on a neighborhood of i (see Proposition 2.2). 2

8. Proof of Proposition 7.1

q ←− q Let us recall that (Mj )j=1 is a ﬁltration adapted ( j )j=1 (see §5 for details). An example of such a ﬁltration is depicted Fig. 3. We are going to construct (Es) by (increasing) induction on i. Here it is the induction hypothesis at the step i: i i ←− ←− ←− ≤ i i s ∩ Ni i Ni For every Ni 0, there exist neighborhoods (Vj )j=1 of respectively (W ( j ) f (Mi))j=1 in f (Mi), there are functions s : i → Ej Vj GN which satisfy the following properties for every j ≤ i: ←− ∈ i ∩ −1 i (i) for every x Vj f (Vj ) the following inclusion holds:

F δ(Es ) ⊂ Es←− . jx j f(x) ≥ ∈ i ∩ i (ii) for every k j, for every x Vk Vj the following inclusion holds: P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1245

s ⊂ s Ekx Ejx. s s| s ∩ i (iii) Ej is compact-open close to E W (j ) Vj , when δ-is small. s (iv) Ej is of constant dimension, d1-continuous, and locally Lipschitz for the metric d∞.

i Remark 8.1. The induction hypothesis at step i ≥ 0 states that for any Ni ≤ 0 there exists a covering of (V )j≤i ←− ←− j of f Ni (M ) which satisﬁes the above properties. We recall that M is a compact included in ∪ ≤ W s( ). But in i ←− i j i j ←− s i s general, for i

s We recall that each Ej depends on δ. During the induction, several parameters will be fixed. The order is the following at the step i. First an arbitrary negative integer Ni is given. Then η>0is chosen. Depending on Ni and η, we will suppose δ small. The induction hypothesis is used with δ and Ni−1 chosen large in function of Ni and η. ←− ←− ←− Step i = 1Let N ≤ 0, and put K := f N1 (M ). We notice that W s( ) is an open set of M . Hence we put 1 ←− 1 1 1 f V 1 := K . Note that f(K) ⊂ K . 1 1 1 1 ←− s s Let K1 x → E be the restriction to K1 of a smooth approximation of the continuous map E |W ( 1) : ←− x s W ( 1) → GN given by Corollary 6.3. s Observe that E is uniformly close to E |K1. Moreover it is d1-continuous and d∞-Lipschitz. We recall that the 0 Banach manifold C (K1, GN ) was defined in §7.1. ¯ ⊂ 0 For all η>0 and δ>0, the following is well defined on the closed ball BC0 (E , η) C (K1, GN ) with image in 0 C (K1, GN ): δ# ¯ δ−1 F := B 0 (E ,η) P → x → F (P←− ) , with x := π (x). C x0 f(x) 0 0 k By hyperbolicity, for δ small enough and E sufficiently close to Es, there exists some k ∈ N such that F δ# is ¯ λ-contracting and sends the closed ball BC0 (E , η) into itself. s δ# Let E1 be the unique fixed point of F in BC0 (E , η). By definition, condition (i) is satisfied. Condition (iii) follows from the fact that η can be taken small when δ is small. s δ It remains only to show that (iv) holds. First let us recall that E ∈ B 0 (E , η) and x ∈ M → F ∈ L (R) is of 1 C 0 x0 N 1 → δ δ class C , and so, K1 x Fx is d∞-Lipschitz. Since F is moreover invertible, there exists Lδ,k such that for all ∈ ∈ 0 x, y K1 and P BC0 (E , η) it holds δk−1 δk−1 d F (P←− ), F (P←− ) ≤ L d∞(x,y), (19) x f k(y) y f k(y) δ,k where F δk := F δ ◦···◦F δ and x := π (x) the ith coordinate of x. x xk−1 x0 i i k On the other hand, the map F δ# is pointwise λ-contracting:

−1 −1 d F δk (P←− ), F δk (P←− ) ≤ λd(P←− ,P←− ). (20) x f k(x) x f k(y) f k(x) f k(y) Consequently, adding (19) and (20) we get δ−k δ−k d F (P←− ), F (P←− ≤ L d∞(x,y) + λd(P←− ,P←− ). (21) x f k(x) y f k(y) δ,k f k(x) f k(y)

For every d∞-Lipschitz distribution P let us denote by (P ) its Lipschitz constant. It holds for every k: 1246 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

N N +1 2 2 Fig. 4. Construction of Z2 = f 2 (M2) \ int f 1 (M1) in the example given by f : (x,y,z)→ (x ,y , 0).

d(P←− ,P←− ) ≤ (P )d∞(x,y). (22) f k(x) f k(y) Thus by (21) and (22): δk−1 δk−1 d F (P←− ), F (P←− ≤ (L + λ(P ))d∞(x,y) x f k(x) y f k(y) δ,k Consequently the closed subset of BC0 (E , η) formed by sections with d∞-Lipschitz constant smaller or equal than −1 δ#k any ≥ (1 − λ) Lδ,k is forward invariant under F . We recall that E is d∞-Lipschitz. Hence if ≥ (E ), this ¯ subset is non empty (it contains E ), thus there exists a ﬁxed point d∞-Lipschitz in BC0 (E , η). By uniqueness, the → s ∈ 2 ﬁxed point K1 x E1x GN is d∞-Lipschitz.

Step i − 1 → i Let Ni be an arbitrary negative integer. Put: ←− ←− s Ni Ki := W ( i) ∩ f (Mi). Let us begin as in the step i = 1. s| : → → We can extend d1-continuously the section E Ki Ki GN to an open neighborhood of Ki. Let x Ex be a smooth approximation given by Corollary 6.3 of such a continuous extension. ←− N The section E is well deﬁned on a small neighborhood Zi of Ki in f i (Mi) of the form: ←− ←− Ni Ni−1+1 Zi := f (Mi) \ int f (Mi−1), Ni−1 ≤ 0 ←− ←− N n Indeed note that Ki = f i (Mi) \∪n≤0 f (Mi−1), so Zi is close to Ki whenever −Ni−1 is large enough (see Fig. 4).

0 s Observe that E |Ki is C -close to E |Ki , d1-continuous and d∞-Lipschitz. Hence, for every η small, for every Z and δ small enough, for E sufficiently close to Es, the following is well i ←− ¯ 0 0 −1 defined on the ball B 0 (E , η) ⊂ C (Zi, GN ) with image in C (Zi ∩ f (Zi), GN ): C ←− − ←− δ# ¯ −1 δ 1 F := B 0 (E |Z ,η) P → Z ∩ f (Z ) x → F (P ( f(x))) . C i i i x0 s By hyperbolicity, for η small and then for Zi and δ small enough and E sufficiently close to E , there exist some k ∈ N, such that the following map: ⎡ ⎤ ←− ←− δ#k ¯ ⎣ −l δ −k k ⎦ F := B 0 (E |Z ,η) P → f (Z ) x → F (P ( f (x))) , C i i x0 0≤l≤k ←− ¯ | ¯ | ∩ −l is contracting and sends the closed ball BC0 (E Zi, η) into BC0 (E l≤k f (Zi), η). As the target space is not the same as the source space, we cannot conclude to the existence of a fixed point. We are going to extend the sections in the image of F δ# by a section constructed by the following lemma shown below:

˜ ∈ | Lemma 8.2. There exist a sequence of negative integers (Nj )j

←− ←− + − := Nj \ Nj−1 1 ⊂ i 1 (1) Zj f (Mj ) int f (Mj−1) int Vj . ∀ ∈ ∩ ˜ ⊂ s (2) x Zj Zi, E(x) Ej (x).

˜ δ# ˜ s ˜ δ# ˜ Gluing E to F E and definition of Ei We remark that E and F E are well defined on: ←− ←− i := Ni \ Ni−1 Vi f (Mi) f (Mi−1). ←− By Corollary 6.2, there exists a partition of the unity (ρ, 1 − ρ) ∈ Mor∞(M )2 subordinated to the cover ←− ←− ←− 0 f N − −1 N − ( f i 1 (int Mi−1), M f \ f i 1 (Mi−1)). ∈ i RN ˜ # ˜ For x Vi , let px and px be the orthogonal projections of onto respectively E(x) and F E(x). Put 0 := { + − ; ∈ ˜ } E (x) (ρ(x)px (1 ρ(x))px)(u) u E(x) . i → 0 ∈ ∈ i We notice that Vi x E (x) GN is d1-continuous and d∞-Lipschitz. Furthermore, for x Vi close to ←− ←− − Ni−1 0 ˜ ∈ i \ Ni−1 1 0 f (Mi−1), the plane E (x) is equal to E(x) and for x Vi f (int Mi−1), the plane E (x) is equal δ# ˜ to F Ex . We define

←− − := { ∈ 0 i : = 0 i \ Ni−1 1 η P C (Vi ,GN ) P E on Vi f (Mi−1) and ←− ∩k j i − 0 − } the restriction P and E to j=0 f (Vi ) are η C close . We remark that the following map is continuous: ⎡ ⎧ ⎤ ⎨ ←− − E0(x) if x ∈ V i \ f Ni−1 1(M − ) δ ⎣ i i i 1 ⎦ F := P ∈ η → V x → ←− i ⎩ δ−1 F x (P ( f (x))) otherwise. ←− δ#k ¯ | ¯ | ∩ j δk As F is contracting and sends BC0 (E Zi, η) into BC0 (E j f (Zi), η), the map F is contracting and sends η into itself. s δ Let Ei be the fixed point of F . s = s By definition, Ei satisfies property (i). Similarly to the step i 1, the section Ei satisfies Properties (iii) and (iv). s i−1 i However all the sections (Ej )j≤i need to be extended from (Vj )j≤i to (Vj )j≤i which remains to be constructed.

i s ˜ Construction of (V )j≤i and extension of (E ) For every j

− ←− ←− + − ←− + ˜ ⊃ i 1 ∩ Ni−1 \ Nj−1 1 = i 1 \ Nj−1 1 Vj Vj f (Mi−1) int f (Mj−1) Vj int f (Mj−1). ←− ←− ←− ←− ˜ s N − s N − +1 Hence Vj is a neighborhood of W ( j ) ∩ f i 1 (Mi−1) since W ( j ) does not intersect f j 1 (Mj−1). Put

←− − ←− ←− i := ∪ | i n ˜ i = Ni \ Ni−1 Vj n≥0( f Vi ) (Vj ) where Vi f (Mi) f (Mi−1)

≤ i s ∩ Ni Ni Lemma 8.3. For every j i, the set Vj is a neighborhood of W (j ) f (Mi) in f (Mi). 1248 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

←− Proof. As the case i = j is obvious, we suppose j

s i We extend Ej on Vj by: ←− i n ˜ s δ−n s ∀x ∈ V , ∀n ≥ 0 minimal such that f (x) ∈ Vj ,E = F (E ←− ). j jx x j f n(x)

s i Induction hypotheses (i), (iii) and (iv) for j

s ≤ Proof of Lemma 8.2. We are going to project E onto each Ej , j i. The following is a consequence of the Lambda- lemma and the strong transversality condition.

Claim 8.1. For Z small enough (that is −N − large enough), for δ small enough, for every j

←− ←− ←− Proof. By the strong transversality condition, on W u( ) \ the stable direction Es is transverse to TWu( ). ←− ←− i i i This is true in particular on W u( ) ∩ W s( ). By hyperbolicity and the strong transversality condition, for every ←− ←− i j u s s s (x )n in W ( i) ∩ W ( j ) approaching x ∈ i , every accumulation plane P of (E )n contains the plane E . This n ←− ←− ←− xn x s ∈ = Ni ∩ s implies that for every (xn)n in W ( j ) approaching x Ki f (Mi) W ( i), every accumulation plane P of s s s (E )n contains the plane E . The claim follows since E is close to E |Ki for δ small and Zi small. 2 xn x ←− s ∩ s We will perform orthogonal projections of Ei on compact subsets of each Zi W ( j ). Let us implement these compact subsets. ←− −1 s First let us notice that cl(Zi \ f (Zi)) is a compact subset of ∪j

←− ←− N N − +1 Claim 8.2. For every k ≤ i − 1, the set Zk := f k (Mk) \ f k 1 (Mk−1) has its closure included in the interior of i−1 s Vk . Moreover, for δ small enough, the distance between the orthogonal projection pk onto Ek satisﬁes: − ≤ ∀ ∈ pk(x)(Ex) Ex η/3i, x Zk. (23) ←− s Proof. For k ≤ i −1, suppose Nk constructed. Then for −Nk−1 large, the set Zk is close to the compact set W ( k) ∩ N f k (Mk), and so it is included in Vk. Moreover, by Remark 7.3 and Claim 8.1 for −Nk−1 large and δ small, inequality (23) holds. 2

ˆ i−1 ∈ ˆ Let Zk be a neighborhood of Zk in Vk such that for all x Zk,

η p (x)(E ) − E ≤ . k x x 2i ∞ ←− ∈ Ni−1 [ ] ˆ c By Corollary 6.2, there exists a dump function ρ Mor0 ( f (Mi−1), 0, 1 ) equal to 1on Zk and to 0on Zk. We j 0 construct (Px )j

j−1 Px ={ρj (x) · qj (x)(u) + (1 − ρj (x)) · u : u ∈ Px }

ˆ = i−1 ∈ ∩ ˜ ⊂ s Let E Px . By induction hypothesis (ii), for every x Zi Zj , it holds Ex Ejx. ˆ ˜ | 2 By deﬁnition of Zk, the section E is in BC0 (E Zi, η/2) and is d∞-Lipschitz.

9. Proof Proposition 7.4

The proof of Proposition 7.4 is done by decreasing induction on q ∈[1, q]. We recall that Proposition 7.1 con- ←− s s structed sections (Ej )j on neighborhoods (Vj )j of respectively (Wj ( j ))j , which satisfy properties (i)–(ii)–(iii) and (iv). Here is the induction hypothesis. ←− q s For every q ≤ q, there exist K>0 and an open cover (Wi) = of ∪j≥q W ( j ), where each Wi is a neighborhood ←− i q i u ∈ 0 of i included in Vi and such that for every δ>0, there exists a function Ei C (Wi, GN ) satisfying the following properties for i ∈[q, q]: ←− ∈ ∩ −1 δ u u (i) For every x Wi f (Wi), the map F sends Eix into E ←− . ←− i f(x) −1 (ii) For every j ≥ i, for every x ∈ Wi ∩ f (Wj ) the following inclusion holds:

Eu ⊂ Eu←− . ix j f(x) s ⊕ u = RN ∈ s u −1 (iii) Eix Eix , for every x Wi ; the angle between Ei and Ei is bounded from below by K . u (iv) The subbundle Ei is of constant dimension, d1-continuous and d∞-Lipschitz. (v) For any x ∈ Wi , it holds

δ u ≥ u ∀ u ∈ u F (v ) v /K, v Eix. ←− ←− −1 q q (vi) It holds cl( f (∪ Wi)) ⊂∪ Wi . Moreover, for every j ≥ i, if x ∈ Wj then f(x) ∈∪/ k>j Wk and ←− ←− q q n ∩n∈Z f (Wj ) = j .

q ←− q We continue to denote by (Mj )j=1 a ﬁltration adapted to ( j )j=1 (see §5 for details and Fig. 3). At each step q of the induction we will work with a small η and we will suppose an integer −Nq large and δ small both depending on η. 1250 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

←− ←− Step q = q The subset = W s( ) is compact. Moreover there exists an arbitrarily small compact neighbor- ←− q q ←− ←− −N hood Wq of q which satisfies (vi) for i = q . Indeed, consider Wq of the form M f \ f (Mq−1). Hence we can suppose that W is included in V . q q ←− ←− Lete η>0b small, in particular smaller than the angle between Es|(f)and Eu|(f). → Let x Ex be the restriction to Wq of a smooth approximation of a continuous extension of the continuous map ←− ←− Eu| : → G given by Corollary 6.3. This means that on the one hand, E is d -continuous and d∞-Lipschitz, q q N 1 ←− and that for every small, if Wq is sufficiently small then for every x ∈ Wq there exists y ∈ q -close to x such that u the distance between Ex and Ey is η small. ←− ←− s u ∈ | u By hyperbolicity of q , the angle between Ey and Ey is uniformly bounded from below on y q and Tf Ey is bijective. By property (iii) of Proposition 7.1 and Remark 7.2, there exists K large such that for every η>0 small, for every Wq sufficiently small, for all δ ≥ 0 small, and for every y ∈ Wq the following holds:

s −1 (a) the angle between Eqy and Ey is greater than K , (b) for every plane P making an angle with Ey smaller than η, it holds:

∀u ∈ P, F δ(u) ≥ u /K. ←− u 0 Indeed, for x ∈ q , every vector u in E is expanded by F . We can now proceed as in the step i = 1of the proof of Proposition 7.1. Since for every δ>0, the map F δ is bijective, the following is well defined ¯ δ 0 F := B 0 (E ,η) P → x → F (P←− ) ∈ C (W ,G ). # C x0 f −1(x) q N u k Moreover, for δ, Wq small enough and E close enough to E , there exists some k ∈ N such that F is contracting ¯ # and sends the closed ball BC0 (E , η) into itself. s Let Eq be the unique fixed point of F# in BC0 (E , η). In this way, condition (i) is clearly satisfied. Properties (iii) and (v) follow from respectively Properties (a) and (b) above. Property (ii) is empty. To prove property (iv), we proceed as in the proof of Proposition 7.1, step i = 1. 2

+ → q Step q 1 q Let us suppose the neighborhoods (Wi)i=q+1 constructed so that

– property (vi) holds, ←− – Wi is a neighborhood of i ,

Let us proceed again as in the proof of Proposition 7.1 step i − 1 → i. ←− ←−− q := s \ 2 := ∪ ≤ We remark that Cq W ( q ) f (Oq ) is compact, with Oq i=q+1Wi . Moreover for every Nq 0, the following is a compact set containing Cq :

←− ←−− := Nq c \ 2 Yq f (Mq−1) f (Oq ).

Moreover, when −Nq is large, Yq is close to Cq for the Hausdorff metric. By Yq small we mean −Nq large. First, we assume −Nq large enough so that the set Yq is included in Vq . By strong transversality and property (iii) of Proposition 7.1, for every η>0, there exists K large such that for every δ and Yq small, it holds:

∀ ∈ ∀ ∈ RN \{ }: | s | ⇒ δ ≥ x Yq , u 0 (u, Eqx) >η F (u) u /K. (24) s 0 s Indeed, if x ∈ Cq , a unit vector u making an angle at least η with E has its image by F not in E←− , by the strong x f(x) tranversality condition. Hence the norm if its image is bounded from below by a certain 1/2K. Consequently for δ and Yq small inequality (24) holds. P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1251

0 For η>0, let Uη be the closed subset of C (Yq , GN ) made by sections P such that for every x ∈ Yq the angle s between Px and E is at least η. q x ←− 0 For all η, δ and Yq , the following is well deﬁned with image in C (Yq ∩ f(Yq ), GN ): ←− F := U P → x ∈ Y ∩ f(Y ) → F δ (P←− ) . # η q q x0 f −1(x) ←− ≥ 0 ∩k i Similarly, for every k 0, for all η, δ and Yq , the following is well deﬁned with image in C ( i=0 f (Yq ), GN ): ←− k F k := U P → x ∈∩k f i(Y ) → F δ (P←− ) . # η i=0 q x0 f −k(x)

←− ←−− + ←− ∩k i = Nq c \ 2 k We remark that i=0 f (Yq ) f (Mq−1) f (Oq ) is close to q , when k is large and Yq is small (that is s u −Nq large). We assume η>0 smaller than the angle between E |q and E |q . Hence by hyperbolicity, for Yq and δ sufﬁciently small, there exists k such that F k is contracting for the C0-metric. Note that k does not depend on η. # ←− k Moreover, by hyperbolicity, if k is large enough and δ small enough, for every x ∈ f (Cq ), for every Px η-close to s δ−k s E , the plane F (Px) is η-close to E←− . Hence every x ∈ Cp, every P making an angle greater than η with q x x f −k(x) s δk s E , the plane F (Px) makes an angle greater than η with E ←− . x x q f k(x) − k 0 ∩k i Consequently, for Nq large enough, F# takes its values in the subspace of C ( i=0f (Yq ), GN ) formed by ∈∩k i s sections P such that for every x i=0f (Yq ), Px makes an angle with Eqx greater than η>0. However, the target space of F# is not the same as the source space. So we cannot conclude to a ﬁxed point. We are going to complement the sections in the image of F# by sections obtained by the following lemma shown below: ˜ Lemma 9.1. For δ and Yq small enough, there exists a d1-continuous and d∞-Lipschitz section E ∈ Uη such that for every j>q: ←− ˜ δ u x ∈ Yq ∩ f(Wj ), Ex ⊂ F (E ←− ). j f −1(x)

˜ ˜ u ˜ ˜ Gluing E to F#E and deﬁnition of Ei We remark that E and F#E are well deﬁned on:

←− ←−− ←− := Nq c \ 1 ⊂ ∩ Wq f (int Mq−1) cl( f (Oq )) Yq f(Yq ). We remark that induction hypothesis (vi) is satisﬁed. ←− By Corollary 6.2, there exists a partition of the unity (ρ, 1 − ρ) ∈ Mor∞(M )2 subordinated to the cover ←− 0 f −1 c (Oq , int f (Oq ) ). RN ˜ ˜ ∈ Let px and px be the orthogonal projections of onto respectively E and F#E. For x Wq , put 0 := { + − ; ∈ ˜ } Ex (ρ(x)px (1 ρ(x))px)(u) u Ex .

∈ → 0 ∈ ∈ We notice that x Wq Ex GN is d1-continuous and d∞-Lipschitz. Furthermore, for x Wq close to ←−− 1 0 ˜ ∈ \ 0 ˜ f (Oq ), the plane Ex is equal to Ex and for x Wq Oq , the plane Ex is equal to F#Ex . We put

0 0 η := {P ∈ C (Wq ,GN ) : P = E on Wq \ Oq and ←−− + s \ 2 k } P and Eq makes an angle greater than η on Wq f (Oq ) . We put: ⎡ ⎧ ⎤ ←−− ⎨ 0 ∈ \ 1 Ex if x Wq f (Oq ) δ := ∈ → ⎣ ∈ → ⎦ F P η x Wq ⎩ F δ (P←− ) otherwise. x f −1(x) 1252 P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253

δ 0 δ δ We notice that the map F takes its values in C (Wq , GN ). Moreover, from the properties of F# , the map F is λ-contracting and takes its values in η. u δ Let Eq be the fixed point of F . u = s By definition, Eq satisfies property (i). Similarly to the step i 1, the section Ei satisfies Property (iv). k ∈ δ − Also, Property (iii) is satisfied for every P F η, if Nq is large enough and δ small enough. Hence it holds u − u for Eq . Likewise by (24), if Nq is large enough and δ small enough, property (v) holds for Eq . This gives a bound on K. Such a bound at this step does not depend on δ small enough. ←− −1 Let us check Property (ii). We only need to check that for j>q, for x ∈ f (Wq ) ∩ Wj it holds:

Eu ←− ⊂ F δ(Eu ). q f(x) jx ←− That is for every x ∈ Wq ∩ f(Wj ) it holds: u δ u E ⊂ F (E ←− ). q x j f −1(x)

←− ←− ←−− ←− ←− ∈ ∩ = Nq c \ 1 ∩ ∩ \ Let x Wq f(Wi) f (int Mq−1) cl( f (Oq )) f(Wj ). In particular x belongs to Wq f(Oq ) ←−− f 1(O ). q ←− −1 u ˜ δ ˜ If x belongs to W ∩ O \ f (O ), then E is a linear sum of vectors included in E and F (E←− ). We q q q q x x f −1(x) ˜ δ u ˜ δ u recall that E is included in F (E ←− ) by Lemma 9.1. Also E←−− is included in F (E ←− ) with j ≥ i such x −1 f 1(x) −2 ←− i f (x) j f (x) −2 δ ˜ δ u u δ u that f (x) ∈ W . By (ii), F (E←−− ) is included in F (E ←− ). Hence E is included in F (E ←− ). j f 1(x) −1 q x −1 ←− i f (x) i f (x) u δ ˜ δ2 ˜ If x belongs to W ∩ f(O ) \O , then E is a linear sum of vectors included in F (E←− ) and F (E←− ). q q q q x f −1(x) f −2(x) δ ˜ δ u ˜ δ u As in the previous case, F (E←−− ) is included in F (E ←− ). Similarly, E←−− is included in F (E ←− ) with f 1(x) −1 f 2(x) −3 ←− i f (x) j f (x) −3 δ2 ˜←− δ3 u δ u u j ≥ i such that f (x) ∈ Wj . By (ii), F (E − ) is included in F (E ←− ) ⊂ F (E ←− ). Hence E is f 2(x) j f −3(x) i f −1(x) q x included in F δ(Eu←− ). i f −1(x)

˜ Proof of Lemma 9.1. We want to construct E ∈ Uη which is d1-continuous and d∞-Lipschitz and such that for every j>q: ←− u δ u x ∈ Yq ∩ f(Wj ), E ⊂ F (E ←− ). q x j f −1(x) By property (v), the section ←− δ u x ∈ f(Wj ) → F (E ←− ) j f −1(x) is d1-continuous and d∞-Lipschitz. ←− For every j>q, let W be an open neighborhood of j with closure in Wj and such that ∪j>q W contains ←− j j s ∪j>q W ( j ) for every q ≥ q and (vi) is satisﬁed. ←− ←− For every j>q, let ρj be a dump function equal to 1 on f(W) ∪ W and with support in f(Wj ) ∪ Wj . We j j←− ←− − ∪ c ∩ c remark that (ρj , 1 ρj ) is a partition of the unity subordinate to the cover ( f(Wj ) Wj , f(Wj ) Wj ). N δ u s Let pj (x) be the projection of R onto F (E ←− ) parallely to E . j f −1(x) jx ˜ s | Let Eq be the orthogonal of Eq Yq . ˜ By Claim 8.1, for every j, for Yq and δ-small enough, the plane Eq makes an angle greater than a certain η>0 s with Eqδ(x). Hence its projection by pj remains of constant dimension and so is d1-continuous and d∞-Lipschitz. ˜ We now construct inductively (Ej )j≥q . ˜ Let j ≥ q and let us suppose the section Ej ∈ Uη, d1-continuous and d∞-Lipschitz, constructed so that for every i ∈ (q, j): P. Berger, A. Kocsard / Ann. I. H. Poincaré – AN 34 (2017) 1227–1253 1253

←− ˜ δ u –for every x ∈ f(W), the plane Ejx is included in F (E ←− ). i i f −1(x) ∈ ˜ u –for every x Wi , the plane Ejx is included in Eix.

Put: ˜ ˜ Ej+1x := ρj+1(x) · pj+1(x)(u) + (1 − ρj+1(x)) · (u) : u ∈ Ejx . ˜ ˜ We deﬁne E = Eq . We remark that by (ii): ←− u δ u ∀x ∈ Yq ∩ f(W+ ), E ⊂ F (E ←− ). j 1 q x j f −1(x) 2 Hence by replacing (Wj )j by (Wj )j , Lemma 9.1 is proved.

Conﬂict of interest statement

There are no known conﬂicts of interest associated with this publication and the manuscript has been read and approved by both authors.

Acknowledgements

This work has been Partially supported by the Balzan Research Project of J. Palis and the project BRNUH of Université Sorbonne Paris Cité. A.K. was partially supported by CNPQ-Brazil and FAPERJ-Brazil. We are grateful to A. Rovella for helpful discussions. We would also like to thank the anonymous referee for his/her valuable comments that have helped to improve the presentation of the manuscript.

References

[1] N. Aoki, K. Moriyasu, N. Sumi, C1-maps having hyperbolic periodic points, Fundam. Math. 169 (1) (2001) 1–49. [2] P. Berger, A. Rovella, On the inverse limit stability of endomorphisms, Ann. non Linéaire IHP 30 (2013) 463–475. [3] Jacek Graczyk, Grzegorz Swi´ atek, ˛ The real Fatou conjecture, Ann. Math. Stud., vol. 144, Princeton University Press, Princeton, NJ, 1998. [4] R. Joly, G. Raugel, Generic Morse–Smale property for the parabolic equation on the circle, Ann. Inst. Henri Poincaré, vol. 27, Elsevier, 2010, pp. 1397–1440. [5] Lyubich Mikhail, Dynamics of quadratic polynomials, I, Acta Math. 178 (2) (1997) 185–247; Lyubich Mikhail, Dynamics of quadratic polynomials, II, Acta Math. 178 (2) (1997) 247–297. [6] R. Mañé, A proof of the C1 stability conjecture, Inst. Hautes Études Sci. Publ. Math. (66) (1988) 161–210. [7] J. Mather, Notes on topological stability, Bull. Am. Math. Soc. (N.S.) 49 (4) (2012) 475–506. [8] Leonardo Mora, Homoclinic bifurcations, fat attractors and invariant curves, Discrete Contin. Dyn. Syst. 9(5) (2003) 1133–1148. [9] R. Mañé, C. Pugh, Stability of endomorphisms, in: Dynamical Systems, Warwick, 1974, in: Lect. Notes Math., vol. 468, Springer, Berlin, 1975, pp. 175–184. [10] Jacob Palis, On Morse–Smale dynamical systems, Topology 8(4) (1969) 385–404. [11] Feliks Przytycki, Anosov endomorphisms, Stud. Math. 58 (3) (1976) 249–285. [12] F. Przytycki, On -stability and structural stability of endomorphisms satisfying Axiom A, Stud. Math. 60 (1) (1977) 61–77. [13] J. Palis, S. Smale, Structural stability theorems, in: Global Analysis, Berkeley, CA, 1968, in: Proc. Symp. Pure Math., vol. XIV, Amer. Math. Soc., Providence, RI, 1970, pp. 223–231. [14] Jacob Palis, Floris Takens, Fractal dimensions and inﬁnitely many attractors, in: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, in: Camb. Stud. Adv. Math., vol. 35, Cambridge University Press, Cambridge, 1993. [15] J. Quandt, Stability of Anosov maps, Proc. Am. Math. Soc. 104 (1) (1988) 303–309. [16] J. Quandt, On inverse limit stability for maps, J. Differ. Equ. 79 (2) (1989) 316–339. .[17] J.W Robbin, A structural stability theorem, Ann. Math. (2) 94 (1971) 447–493. [18] C. Robinson, Structural stability of C1 diffeomorphisms, J. Differ. Equ. 22 (1) (1976) 28–73. [19] S. Smale, Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967) 747–817. [20] Dennis Sullivan, Linking the universalities of Milnor–Thurston, Feigenbaum and Ahlfors–Bers, in: Topological Methods in Modern Mathe- matics, Stony Brook, NY, 1991, Publish or Perish, Houston, TX, 1991, pp. 543–564. [21] Joan Carles Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergod. Theory Dyn. Syst. 21 (1) (2001) 249–302.