Cone Conditions and Covering Relations for Topologically Normally Hyperbolic Invariant Manifolds

We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. We provide conditions which imply the existence of the manifold within an investigated region of the phase space. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method to obtain an invariant manifold within an explicit range of parameters for the rotating H\'enon map.


1.
Introduction. The goal of our paper is to present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic type properties, in a vicinity of an approximate invariant manifold. In our opinion there are two main advantages of our approach: 1) we do not assume that the given map is a perturbation of some other map for which we have a normally hyperbolic invariant manifold, 2) the assumptions could be rigorously checked with computer assistance if our approximation of the invariant manifold is good enough.
Our results about invariant manifolds are weaker than the standard results for normal hyperbolicity [1,3,7,10]. In fact we do not prove any smoothness result and the existence of fibration of the stable and unstable manifolds of the invariant manifold. We believe though that topological assumptions considered by us should be sufficient to prove normal hyperbolicity in the usual sense. This paper should be considered as a first step towards this end. In a subsequent publication we intend to improve our results in this direction.
In the standard approach to the proof of various invariant manifold theorems all considerations are done in suitable function spaces or sequences spaces, moreover the existence of the invariant manifold for nearby map (or ODE) is always assumed, see for example [3,7,10] and the references given there. Usually these proofs do not give any computable bounds for the size of perturbation for which the invariant manifold exists, with only exception known to us being the result of Bates, Lu and Zeng [1].
In contrast to the above mentioned standard approach, in our method the whole proof is made in the phase space. This method of proof of the existence of invariant manifolds is not entirely new, see for example the proof of Jones [8] in the context of slow-fast system of ODEs. But still he considered the perturbation of some normally hyperbolic invariant manifold. In [2] a somewhat similar approach has been applied to obtain a topologically normally hyperbolic invariant set. The result relied only on covering relations without the use of cone conditions. There it has not been shown that the invariant set is a manifold, hence the result was weaker than the one presented in this paper.
The work is organised as follows. In Section 2 we introduce notations and give preliminaries on vector bundles. In Section 3 we introduce central-hyperbolic sets (ch-sets), covering relations and cones. A ch-set will play the role of a region in which we suspect to find an invariant manifold. Covering relations will ensure the existence of an invariant set within a ch-set. In Section 4 we introduce cone conditions for maps and main results. Cone conditions combined with covering relations will give the existence of a normally hyperbolic-like invariant manifold inside of a ch-set. In Section 5 we show how cone conditions and covering relations can be verified in practice. In Section 6 we discuss how our result relates so far to the classical normally hyperbolic invariant manifold theorem. To demonstrate clearly the strength of our approach, in Section 7 we prove that for the rotating Hénon map considered in [5] for an explicit range of parameters there exists an invariant manifold.

Preliminaries.
2.1. Notation. By Id : X → X we will denote the identity map.
Let R n be equipped with some norm · . For x 0 ∈ R n and r > 0 we define B n (x 0 , r) = {x ∈ R n | x − x 0 < r}. Since most of the time we will be using unit balls centered at 0, we set B n = B n (0, 1). When considering a linear map on A : R n R k the symbol A will always denote the standard operator norm of A, i.e. A = sup x =1 Ax . We will also use A m = inf x =1 Ax .
Let ΠX = X 1 ×X 2 ×· · ·×X n . For i = 1, . . . , n we define projection π i : ΠX → X i by π i (x 1 , . . . , x n ) = x i . Sometimes when the variables in the cartesian product have names, for example (x, y, z), we will use names of variables as indices for projections, hence π y (x, y, z) = y.
Let U = {U i } i∈I and V = {V j } j∈J be coverings of a set X (i.e. X = U = V), we say that covering V is inscribed in U, denoted by V ≺ U, iff for any j ∈ J there exists i ∈ I, such that V j ⊂ U i . If X is a topological space, then we say that covering U is open iff it consists from open sets.
Let U be a topological space. We say that U is contractible, when there exists a deformation retraction of U onto single point space, i.e. there exists continuous map h : [0, 1] → U and x 0 ∈ U , such that h(0, q) = q and h(1, q) = x 0 for all for q ∈ U . For x, y ∈ R n we set [x, y] = {z = tx + (1 − t)y, t ∈ [0, 1]}.

Vector bundles.
We start with recalling the definition of the vector bundle [6].
We will denote such bundle chart by a pair (ϕ, U ).
For each x ∈ U we define the homeomorphism ϕ x to be the composition A vector bundle atlas Φ on (p, E, B) is a family of vector bundle charts on (p, E, B) with the values in the same R n , whose domains cover B and such that whenever is continuous for all pairs of charts in Φ. A maximal vector bundle atlas Φ is a vector bundle structure on (p, E, B). We then call ξ = (p, E, B, Φ) a vector bundle having (fibre) dimension n, projection p, total space E and base space B. Often Φ will not be explicitly mentioned. In fact we may denote ξ by E. Sometimes it is convenient to put E = Eξ, B = Bξ, etc.
The fibre over x ∈ B is the space p −1 (x) = ξ x = E x . ξ x has the vector space structure.
If the E, B are C r manifolds and all maps appearing in the above definition are C r , then we will say that the bundle (p, E, B, Φ) is a C r -bundle.
One can introduce the notion of subbundles, morphisms etc (see [6] and references given there). The fibers can have a structure: for example a scalar product, a norm, which depend continuously on the base point.
Definition 2.2. We say that the vector bundle ξ is a Banach vector bundle with fiber being the Banach space (F, · ), if for each x ∈ Bξ the fiber ξ x is a Banach space with norm · x such that for each bundle chart (ϕ, U ) the map ϕ For vector bundles ξ 1 , ξ 2 over the same base space one can define ξ = ξ 1 ⊕ ξ 2 by setting ξ x = ξ 1,x ⊕ ξ 2,x . In the following, points in ξ 1 ⊕ ξ 2 will be denoted by a triple (z, x 1 , x 2 ), where z ∈ Bξ 1 = Bξ 2 , x 1 ∈ ξ 1,z and x 2 ∈ ξ 2,z . If ξ 1 and ξ 2 are both Banach bundles, then ξ i ⊕ ξ 2 is also a Banach vector bundle with the norm on ξ 1,z ⊕ξ 2,z defined by (x 1 , x 2 ) z = max( x 1 z , x 2 z ). We will always use this norm on ξ 1 ⊕ ξ 2 . We will also always assume that the atlas on bundle ξ 1 ⊕ ξ 2 respects this structure, namely if (η, U ) is a bundle chart for ξ 1 ⊕ ξ 2 , then its restriction (obtained through projection) to ξ i is also a bundle chart for ξ i for i = 1, 2.
3. Central hyperbolic sets, covering relations and cones. In this section we introduce the setup for our approach. First we introduce the concept of a central hyperbolic set (ch-set). This set will play the role of a compact region in which we expect to find an invariant manifold. We will consider maps defined on ch-sets which have certain contraction and expansion properties. On a ch-set we will have three directions: the direction of topological expansion, the direction of topological contraction, and a third, the central direction, in which the dynamics will be weaker than in the first two directions. The contraction and expansion properties will be expressed in terms of covering relations. The ch-sets will be equipped with cones. The cones will help us to narrow down and specify in more detail the directions of expansion and contraction.
3.1. Central hyperbolic sets. Before we introduce the definition of a central hyperbolic set we will need the following definition.
Definition 3.1. Let ξ u and ξ s be Banach vector bundles, with the base space Λ.
Definition 3.2. Assume that, we have Banach vector bundles ξ u , ξ s , ξ = ξ u ⊕ ξ s . Let u and s be the fiber dimension of ξ u and ξ s , respectively. Let the base space for ξ, denoted by Λ, be a compact manifold without boundary of dimension c.
A central-hyperbolic set D (a ch-set), is an object consisting of the following data 1. |D| a compact subset of Eξ.
2. a homeomorphism φ : ξ u ⊕ ξ s → ξ u ⊕ ξ s such that We will usually drop vertical bars in the symbol |D| and write D instead. The notation u, s and c for the dimensions in Definition 3.2, stand for the unstable, stable and central directions respectively. Their roles will become apparent once we introduce the definitions of covering relations and cone conditions for maps.
For a central-hyperbolic set D we define . The sets D + and D − will later on be the entrance and exit sets for a map defined on D, respectively.
From now on we assume that we work in the Banach vector bundle ξ u ⊕ ξ s with the base space given by a compact manifold without boundary Λ. We will now define an atlas on Λ, which will later be used for the definitions of covering relation and cone conditions. Assume that we have an atlas {(η i , U i )} i∈I for Λ, where I is a finite set and Moreover, we assume that for all i ∈ I there exists (ϕ, W ) ∈ Φξ, such that U i ⊂ W . We fix one such ϕ for each i. Therefore on p −1 (U i ) we have a chart map Therefore the atlas is good if the domains of the chart maps on Λ are also domains for some bundle chart maps. From now we will always implicitly assume that we work with a good atlas.
is a chart from some good atlas. For r > 0 we define N ηi (Z, r), the normal neighborhood, and N ± ηi (Z, r) normal exit and entry sets, as follows If r = 1 then we will often drop it and write N ± ηi (Z) instead.
Obviously, for Z ⊂ U i we have Let D be ch-set and let f : |D| → ξ u ⊕ ξ s be a continuous map. We define When using local coordinates around z and f (z), given by charts (η j , U j ) and (η i , U i ) respectively, we will consider functions f ij given by The following commutative diagram illustrates the meaning and mutual relations between maps f , f φ and f ij .
Covering relations for ch-sets. In this section we give the definition of a covering relation for a ch-set. Covering relations will be used later on in the proof of our main result to ensure that we have an invariant set in the interior of a ch-set.  Let D ⊂ ξ u ⊕ ξ s be a ch-set and a map f : D → ξ u ⊕ ξ s be continuous. Assume that {(η i , U i )} i∈I is a good atlas on Λ, where I is some finite set, and {V j } j∈J (J a finite set) is an open covering of Λ, such that for every j ∈ J there exists i 0 , i 1 ∈ I (not necessarily unique) such that We will say that ch-set D f -covers itself, which we will denote by if the following conditions are satisfied for every V j . Let U i0 and U i1 by any sets such that conditions (12) and (13) hold. The conditions are: 1. There exists a continuous homotopy such that the following conditions hold true 2. If u > 0, then there exists a linear map A i1i0j : R u → R u and λ * ∈ η i1 (U i1 ) such that 3. If u = 0, then there exists λ * ∈ η i1 (U i1 ) such that h i1i0j (1, (λ, y)) = (λ * , 0).

CONE CONDITIONS FOR INVARIANT MANIFOLDS 7
The idea behind the definition can be summarized as follows. In local coordinates the map topologically expands the ch-set in the unstable direction and contracts it in the stable direction. In the central direction we do not require any contraction or expansion properties. We just require that the local maps are properly defined on sets N ηi 0 (V j ) (see Figure 1).

Remark 1.
In fact it is enough to require that for a given j there exists a required homotopy for just one pair of U i0 and U i1 satisfying conditions (12) and (13). For the other pairs the homotopy can be obtained using the transition maps from the atlas.
3.3. Cones, horizontal and vertical disks. In this section we introduce horizontal and vertical discs. These will be later on used as building blocks to construct both the invariant manifold and its stable and unstable manifolds in Section 4. Equipping the ch-set with cones will allow us to consider horizontal discs which are appropriately flat and vertical discs which are appropriately steep.
In this section we assume that we have a fixed Banach vector bundle ξ u ⊕ ξ s with a base space Λ, which is a compact manifold without boundary of dimension c.
Definition 3.6. Let Q : R n → R be a quadratic form. We define positive and negative cone by If z ∈ R n then we define positive and negative cones centered at z by In the sequel we will work with cones given by locally defined quadratic forms on ξ u ⊕ ξ s and the sets of interests will be the intersection of positive or negative cones centered at z with N (Λ). Let Q : Let {(η i , U i )} i∈I be a good atlas on Λ and assume that for all i ∈ I we have quadratic forms where α i : R u → R, β i : R s → R and γ i : R c → R are positive definite quadratic forms. We assume that these forms are uniformly bounded in the following sense: there exist M + , M − ∈ R + , such that for every i ∈ I hold Assume that there exists an open covering {V j } j∈J of Λ inscribed in {U i } i∈I , such that V j is contractible for each j and the following two conditions are satisfied 1. for any point q ∈ |D| there exists j ∈ J and i ∈ I such that p(q) ∈ V j , V j ⊂ U i and 2. for any j ∈ J there exists an i ∈ I, such that for all The structure consisting of (D, will be called a ch set with cones. Usually we will refer to this structure by D.

Remark 2.
In fact we may allow to have different γ i in Q i,h and Q i,h , but α i and β i must coincide.
Definition 3.8. Assume that D is a ch-set with cones.
Let q ∈ |D|. We will say that (V j , U i ) is a cone enclosing pair for q, when V j and U i are as in point 1. of Def. 3.7.
We will say that a pair (V j , U i ) is a cones chart pair iff condition 2 in Def. 3.7. is satisfied.
is both a cones chart pair, as well as a cone enclosing pair for q (see Figure 2).
We now move on to the definitions of horizontal and vertical discs.
, then we will say that b is a horizontal disk for the pair (Z, U i ).
Definition 3.10. Let D be a ch-set with cones, Z ⊂ U i and let b : B u → D be a horizontal disk for (Z, U i ). We say that b is a horizontal disc satisfying the cone condition for (Z, U i ) if the following condition holds If condition (26) holds for all U j , such that |b| ⊂ U j , then we say that b is a horizontal disk in D. Figure 2. A vertical cone (A) and a horizontal cone (B), together with a projection of the cones chart pair (V i−2 , U i−4 ) and two vertical cones (one centered at q and the other at a point from ) onto the θ, y coordinates (C). The idea behind Definition 3.10 is that when we consider a horizontal disc in local coordinates and attach a horizontal cone to any of its points, then the entire disc will be contained in the cone (see Figure 3).
Analogously we define a vertical disk as follows.
, then we will say that b is a vertical disk for (Z, U i ).
Definition 3.12. Let D be a ch-set with cones, Z ⊂ U i and let b : B s → D be a vertical disk for (Z, U i ). We say that b is a vertical disc satisfying the cone condition for (Z, U i ) if the following condition holds If condition (28) holds for all U j , such that |b| ⊂ U j , then we say that b is a vertical disk in D.
The following lemma was proved in a slightly different setting in [11, Lemma 5] Then b can be represented as the graph of the function in the following sense: there exists continuous functions d c : An analogous lemma is valid for vertical disks satisfying cone conditions. The lemma below shows that for a graph of a function over "unstable coordinate" with its range in Then there exists homotopy h : From the contractibility of V j it follows that there exists λ * ∈ V j and a continuous homotopy g : We define homotopy h by setting An analogous lemma is valid for vertical disks.
4. Cone-conditions for maps and main theorems. Our main result is given in Theorem 4.7. There we will show that for a map which satisfies cone conditions (see Def. 4.1) there exists an invariant manifold together with its stable and unstable manifolds inside of the ch-set. For the proof of Theorem 4.7 we will first show that if a map satisfies cone conditions then an image of a horizontal disc which satisfies cone conditions is also a horizontal disc which satisfies cone conditions. This fact will be used to construct vertical discs of points whose forward iterations remain inside of the ch-set. Using these discs we will construct our invariant manifold together with its stable and unstable manifolds. We start by defining cone conditions for a map. We say that f satisfies (forward) cone conditions on D if there exists an m > 1 such that: For any x ∈ D and any (V j0 , U i0 ) which is a cone enclosing pair for We will now define backward cone conditions for the inverse map. In order to do so we will need the following notations. (18) and (19). We define =⇒ D, with reversed roles of the coordinates u and s. We say that f satisfies backward cone conditions if f −1 satisfies forward cone conditions on (D, , with reversed roles of the coordinates u and s.
We will now show that an intersection of the ch-set with an image of a horizontal disc which satisfies cone conditions is a horizontal disc which satisfies cone conditions.
and that f satisfies (forward) cone conditions. Let b : B u → D. Assume that for any x ∈ B u we have a (V j(x) , U i(x) ) which is a cone enclosing pair for b(x), and that b is a horizontal disc which satisfies cone condition for (V j(x) , U i(x) ). Then is nonempty and for any q ∈ β there exists a (V j(q) , U i(q) ) which is a cone enclosing pair for q, such that β is a horizontal disc which satisfies cone condition for (V j(q) , U i(q) ).
Proof. Without any loss of generality we can assume that φ = Id and therefore The existence of such sets is a direct consequence of Def. 3.5. From Def. 3.10 it follows that there exists a homotopy h : Let us denote by g the homotopy g = h i1i0j appearing in Def. 3.5 and also let A = A i1i0j . To study the set f (b(B u )) and its behavior under the homotopy h and g it is enough to use charts (η i1 , U i1 ) in the range and (η i0 , U i0 ) in the domain. We will show now that for any z ∈ B u there exits x ∈ B u , such that Later we will prove the uniqueness of such x. This will allow us to define map z → f (b(x(z))), which we will show to be the horizontal disc we are looking for.
To study equation (40) we consider a parameter dependent map H : [0, 1]×B u → R u given by (41) Using the local Brouwer degree we will study the parameter dependent equation Any solution of (42) with t = 0 is a solution of problem (40) and vice versa. Observe first that the local Brouwer degree (see Section 8 for the properties of the degree) of H t on B u at 0, denoted by deg (H t , B u , 0), is defined because from Def. 3.5 it follows that Hence 0 / ∈ H t (∂B u ).
(44) Therefore from the homotopy property of local degree it follows that For t = 1 we have and since A is a nonsingular linear map, therefore hence deg(H t , B u , 0) = 0 and by the solution property of the local degree, there exists an x in B u solving (42). Therefore there exists a solution for (40).
Observe that this solution is unique. We can take any x 0 , x 1 ∈ B u , x 0 = x 1 , such that f (b(x 0 )) ∈ D and f (b(x 1 )) ∈ D. We will use the notation q = f (b(x 0 )). By Def. 4.1 we can choose i(q), j(q) such that (V j(q) , U i(q) ) is a cone enclosing pair for q and f (p −1 (V j(x0) ) ∩ |D|) ⊂ p −1 (U i(q) ).
(48) From the cone condition for b and map f it follows, that This immediately implies that Therefore, we have a well defined map v : B u → B u , given as the solution of implicit equation . Take now any x 0 such that q = f (b(x 0 )) ∈ |D|, and take a cone enclosing pair (V j(q) , U i(q) ) for q which satisfies (48-49). We will show that d is a horizontal disc which satisfies cone condition for (V j(q) , U i(q) ). From (49)) it follows that for any Observe that this implies that the mapd =η i(q) • d is Lipschitz. Namely, for any x 1 , x 2 ∈ B u from (50) we have therefore we obtain which ensures continuity of d. To finish the argument that d is a horizontal disk for (V j(q) , U i(q) ) we need the homotopy. This homotopy is obtained by applying Lemmas 3.13 and 3.14.
From Lemma 4.3 we obtain by induction the following lemma. and that f satisfies (forward) cone conditions. Let b : B u → D. Assume that for any x ∈ B u we have a (V j(x) , U i(x) ) which is a cone enclosing pair for b(x), and that b is a horizontal disc which satisfies cone condition for (V j(x) , U i(x) ). Then β = {y : y = f n (b(x)) and f i (b(x)) ∈ D for i = 1, . . . , n for some x ∈ B u } is nonempty and for any y ∈ β there exists a (V j(y) , U i(y) ) which is a cone enclosing pair for y, such that β is a horizontal disc which satisfies cone condition for (V j(y) , U i(y) ).
Using Lemma 4.4 we will now show that on a horizontal disc which satisfies cone conditions we have a unique point whose forward iterations remain inside of the ch-set.
and that f satisfies (forward) cone conditions. Let b : B u → D. Assume that for any x ∈ B u we have a (V j(x) , U i(x) ) which is a cone enclosing pair for b(x), and that b is a horizontal disc which satisfies cone condition for (V j(x) , U i(x) ). Then there exists a unique point x * ∈ B u , such that f n (b(x * )) ∈ int D, n = 0, 1, 2, . . . .
Proof. Without any loss of generality we can assume that φ = Id and therefore From Lemma 4.4 it follows that for any n ∈ N there exists a point x n ∈ B u such that f k (b(x n )) ∈ D for k = 0, 1, . . . , n.
Because B u is compact and D is closed, we can pass to a convergent subsequence and obtain a point x * ∈ B u , such that In fact we have because from the definition of covering relation it follows that points from D + are not in f (D) and points from D − are mapped out of D.
Let us show that such a point x * is unique. Let us assume that we have two points x 0 , x 1 ∈ B u , x 0 = x 1 satisfying (52). From Lemma 4.4 it follows that for any n = 0, 1, 2, . . . points q l,n given by q l,n := f n (b(x l )) ∈ int D, l = 0, 1 belong to b n , a horizontal disc satisfying cone conditions for a cone enclosing pair (V jn , U in ) for q 0,n . From cone conditions for f and b n we have Hence Q in,h (η n (q 0,n ) −η n (q 1,n )) > m n Q 0,h (η i0 (q 0,0 ) −η i0 (q 1,0 )).
We will now show that points whose forward iterations remain inside of the ch-set form vertical discs which satisfy cone conditions.
In addition, b is a vertical disc which satisfies cone conditions for any cones chart pair (V j , U i ) such that λ 0 ∈ V j . What is more, if p(q) = λ 0 and f n (q) ∈ D for all n ∈ N, then q ∈ b(B s ).
Proof. Without any loss of generality we can assume that φ = Id , hence b φ = b for all vertical or horizontal discs and f φ = f . Let us choose λ 0 ∈ Λ and some q ∈ p −1 ({λ 0 }) ∩ |D|. Let (V j , U i ) be a cones chart pair and a cone enclosing pair for q. Let us fix a y ∈ B s and let b 0 : B u → D be a horizontal disc defined by Since (V j , U i ) is a cones chart pair, for any x ∈ B u we have a (V j(x) , U i(x) ) which is a cone enclosing pair for b 0 (x). This means that we can apply Lemma 4.5. Hence there exists a uniquely defined x * (λ 0 , y) ∈ B u , such that We define a map b : We will show that the map b is a disc satisfying the assertion of our theorem. We need to show that b is a vertical disc which satisfies cone conditions for (V j , U i ). We will first show that for any two points y 1 , y 2 ∈ B s such that y 1 = y 2 we have (62) Let us assume that (62) does not hold. Then since π λ (b(y 1 )) = π λ (b(y 2 )) = λ 0 we have From the fact that f satisfies cone conditions we have Let us take any horizontal disc satisfying the cone condition for (V j1 , U i1 ) passing through f (b(y 1 )) and f (b(y 2 )). From Lemma 4.5 it follows that since f n (b(y i )) ∈ int D for i = 1, 2 and all n = 0, 1, 2, . . . that f (b(y 1 )) = f (b(y 2 )), but this is in a contradiction with (64). Hence (62) holds. From (62) we have From (65) and (20) we have which together with (61) b guarantees its continuity. This by Lemma 3.14 means that b is a vertical disc. Condition (62) now ensures that it satisfies cone conditions.
We now come to our main result. Theorem 4.7 will give us the existence of a normally hyperbolic invariant manifold together with its stable an unstable manifolds. At this stage, the assumptions of the theorem might seem somewhat abstract. In Section 5 we will show how these assumptions can be verified in practice, and in Section 6 we will highlight how the theorem relates to the classical version of the normally hyperbolic invariant manifold theorem [7].
be a ch-set with cones. Let f : ξ u ⊕ ξ s → ξ u ⊕ ξ be a homeomorphism. If f satisfies forward and backward cone conditions, then: 1. There exists a continuous function such that p(χ(λ)) = λ for all λ ∈ Λ, and 2. There exist C 0 submanifolds W u and W s such that W u consists of all points whose backward iterations converge to χ(Λ), and W s consists of all points whose forward iterations converge to χ(Λ).

Remark 3.
Let us note that the stable and unstable manifolds W u and W s from Theorem 4.7 are different from the vector bundles ξ u and ξ s . The bundles ξ u and ξ s provide only approximate coordinates in which we look for W u and W s , in practice W u will be close to ξ u and W s close to ξ s , but they need not precisely match.
Proof of Theorem 4.7. Observe first that vertical cones for the inverse map coincide with horizontal cones for the forward map (see (18), (35)) Take λ 0 from Λ. From the fact that f satisfies cone conditions, by Theorem 4.6 we have a vertical disc b : B s → D ∩p −1 (λ 0 ) of forward invariant points (b is vertical for the ch-set (D, ). Since f −1 satisfies cone conditions, by Theorem 4.6 we also have a "vertical" disc b − : with reversed roles of u and s). We can define The function is χ properly defined since by (67) b − is a horizontal disc for the chset (D, {η i , U i , Q i,h , Q i,v } i∈I , {V j } j∈J ). A horizontal disc and a vertical disc which satisfy cone conditions and are contained in p −1 (λ 0 ) intersect at a single point.
We now have to show that χ is continuous. Take any sequence λ n → λ 0 . By the fact that D is compact we can take a convergent subsequence χ(λ vn ) → z. We need to show that z = χ(λ 0 ). For any k ∈ N, by continuity of functions f, f −1 and closeness of D, we know that lim n→∞ f k (λ vn ), lim n→∞ f −k (λ vn ) ∈ D, hence z ∈ inv(f, D). The fact that z = χ(λ 0 ) follows from the uniqueness of the choice of χ(λ 0 ). Now we will construct the manifold W s . For any λ from Λ, by Theorem 4.6 we have a vertical disc b λ : B s → D ∩ p −1 (λ) of forward invariant points. We can define We need to show that W s is C 0 . We take any Cauchy sequence x n from W s . From the compactness of D we know that x n converges to some x 0 ∈ D. We will show that x 0 ∈ W s . Since for any k, n > 0 we have f k (x n ) ∈ D, by continuity of f we have that f k (x 0 ) ∈ D. Letting λ 0 = π λ (φ(x 0 )), we can see that x 0 ∈ p −1 (λ 0 ), which by Theorem 4.6 means that x 0 lies on the unique vertical disc b λ0 of forward invariant points in p −1 (λ 0 ), hence by (68) we have x 0 ∈ W s .
The construction of W u is analogous.
We will now show that for any x 0 ∈ W s f n (x 0 ) converges to χ(Λ) as n goes to infinity. Let us consider the limit set of the point x 0 If we can show that ω(f, x 0 ) is contained in W u ∩W s = χ(Λ), then this will conclude our proof. We take any q = lim k→∞ f n k (x 0 ) from ω(f, x 0 ). We need to show that q ∈ W u ∩ W s . By continuity of W s we know that q ∈ W s . Suppose now that q / ∈ W u . This would mean that there exists an n > 0 for which f −n (q) / ∈ D. Since we have that f −n (q) ∈ ω(f, x 0 ), but this contradicts the fact that ω(f, x 0 ) ⊂ D.
Showing that all backward iterations of points in W u converge to χ(Λ) is analogous.

5.
Rigorous numerical verification of covering and cone conditions. In this section we will show how assumptions of Theorem 4.7 can be verified. We set up our conditions so that they are verifiable using rigorous numerics. To verify them it is enough to obtain estimates of the derivatives of the local maps on the ch-set. We will show that the assumptions of Theorem 4.7 follow from explicit algebraic conditions on these estimates. An example of how this works in practice will be shown in Section 7.
We start with the definition of an interval enclosure of a derivative.
Definition 5.1. Let U ⊂ R n and f : U → R n be a C 1 function. We define the interval enclosure of df on the set U as for all i, j = 1, . . . , n .

5.1.
Verification of the covering condition. In this section we show how covering relations on ch-sets follow from algebraic conditions on the derivatives of local maps.

5.2.
Verification of cone conditions. In this section we will show how to verify cone conditions using interval enclosures of derivatives of local maps. We start with a technical lemma.
Proof. Using the fact that we can compute The theorem below gives conditions which imply cone conditions.
Assume that for any x ∈ D and any (V j0 , U i0 ) which is a cone enclosing pair for x, if f (x) ∈ D then there exist i 1 ∈ I, j 1 ∈ J such that (V j1 , U i1 ) is a cone enclosing pair for f (x) and f (p −1 (V j0 ) ∩ |D|) ⊂ p −1 (U i1 ). What is more for any we assume that we have ||A 11 || ≤ C, (The C, M, µ, A, α, β, ε u , ε s and ε c can depend on j 0 , i 0 , j 1 and i 1 ). If there exists an m > 1 such that Proof. We only need to prove condition (33). Take any

Let us define a matrix
. By Lemma 5.3 we have 1. T f expands ξ u more sharply than T f r expands T Λ, 2. T f contracts ξ s more sharply than T f r contracts T Λ.
Theorem 6.1. [7] Let f be r-normally hyperbolic at Λ. Through Λ pass stable and unstable manifolds invariant by f and tangent at Λ to T Λ ⊕ ξ s , ξ u ⊕ T Λ. They are of class C r . The stable manifold is invariantly fibred by C r submanifolds tangent at Λ to the subspaces ξ s . Similarly for the unstable manifold and ξ u . These structures are unique, and permanent under small perturbations of f .
We will now highlight how our result obtained in Theorem 4.7 (see also Theorems 5.2, 5.4 and Remark 5) relates to Theorem 6.1.
In our approach we do not need to start with a normally hyperbolic invariant manifold. We start with a region (ch-set) D in which we suspect the manifold to be contained. Theorem 4.7 gives us both the existence an invariant manifold, and of the stable and unstable manifolds inside of D.
Using Theorem 6.1 it is not straightforward to obtain an estimate on the size of the perturbation under which the structures persist. This is due to the fact that the proof is conducted by the use of the implicit function theorem on infinite dimensional functional spaces. In contrast, our proof is designed so that this estimate is explicitly given (see Theorems 5.2,5.4, and Section 7 for an example how this is done in the case of the rotating Hénon map). This was possible due to the fact that the proof relies only on topological arguments, and is conducted in the phase space of the system. Theorem 6.1 is in many ways stronger than our result. Due to the fact that our proof relies only on topological arguments we have lost the C r regularity results and our manifolds are only C 0 . As of yet the method also does not give us the fibration of the stable and unstable manifolds. This issue will be addressed in forthcoming work. We also point out that in the method of verification of cone conditions given by Theorem 5.4 (see also Remark 5) we assume that we have strong expansion properties for the first iterate of the map. This is not required for classical normal hyperbolicity, where it is enough that one has strong expansion and contraction properties for some higher iteration of the map (for more details see [7]). This means that we can apply our results in the classical setting, but to do so we need to consider a higher iterate of the map, and take the higher iterate to be the function considered in Theorem 4.7.
7. Rotating Hénon map. In this Section we will apply Theorem 4.7 to obtain an invariant manifold for the rotating Hénon map. We will obtain an explicit estimate of the region in which the manifold is contained. The size of this region depends on the size of the perturbation. The smaller the perturbation is the more exact our estimate. 7.1. Statement of the problem. We will consider the rotating Hénon map θ = θ + ω (mod 1), The dynamics of (80) with a = 0.68 and b = 0.1 has been investigated by Haro and de la Llave in [5], for a demonstration of a numerical algorithm for finding invariant manifolds and their whiskers in quasi periodically forced systems.
In this section we will prove that for the the same parameters a and b for all ε ≤ 1 2 , there exists an invariant C 0 manifold of (80) which is homeomorphic to T 1 and is contained in a set where (x − , y − ) is a fixed point for the (standard) Hénon map, 7.2. The unperturbed map. We start by investigating the case of ε = 0. We will ignore the coordinate θ and concentrate on a map F (x, y) = (1 + y − ax 2 , bx).
We will consider the following Jordan forms of the matrix DF (x − , y − ) where κ = 1/(λ 2 − λ 1 ). The constants τ, η serve the purpose of an appropriate rescaling of the stable and unstable directions in the local coordinates, and will be chosen later on. When we will consider the perturbed Hénon map in Section 7.4, for a given ε > 0 we will use the maps Φ −1 ε and Φ ε . We introduce local coordinates of hyperbolic expansion and contraction around the point (x − , y − ) as The map F in the local coordinates is and its derivative dF is equal to where For anyx,ỹ ∈ [−1, 1] we have the following estimates, which will be used later on for the verification of the covering and cone conditions . Now we turn to the inverse map. The inverse map to F is and has a derivative In the local coordinates (82) the inverse map is and its derivative dF −1 is equal to where 7.3. The atlas. Let us start by introducing a notation F ε : T 1 × R 2 → T 1 × R 2 for the perturbed Henon map (80) F ε (θ, x, y) = θ + ω, 1 + y − ax 2 + ε cos(2πθ), bx .
For computational reasons it will be convenient for us to choose Λ = R/v for some large (later specified) number v ∈ N, v ≥ 9. The v will play the role of the rescaling parameter from Remark 4. We choose p : T 1 × R 2 → R/v as p(θ, x, y) := vθ. For i, j ∈ {0, 1, . . . , v} we define V j , U i ⊂ Λ as U i := i + (0, 9) mod v, and define maps η i : U i → R as For all i we define quadratic forms Q i,h , Q i,ν as Q i,h (θ, x, y) := x 2 − y 2 − θ 2 , Q i,ν (θ, x, y) := x 2 − y 2 + θ 2 .
Let us note that for different ε we will have different sets |D|.
With the above notations we can see that (D, {η i , U i , V i , Q i,h , Q i,ν }) is a ch-set with cones. 7.4. Verification of the covering conditions. We will show that (with the roles of the stable and unstable coordinates reversed for the inverse map). We will first apply Theorem 5.2 to establish (89). From the fact that F ε is a rotation on T 1 and from the definition of the sets V j , U i (see (87)), for any V j ⊂ U i0 there exists i 1 such that F ε (p −1 (V j )∩|D|) ⊂ p −1 (U i1 ). We will choose the parameters ε u , ε s for the estimates (70) and (71) to be independent from the choice of j, i 0 , i 1 .
To simplify notations we will obtain these estimates using the map (F ε ) φ rather than (F ε ) i1i0 . We can do so since η i0 , η i1 are identity maps (see (88)).
From the fact that F −1 ε is a rotation on T 1 and from the definition of the sets V j , U i (see (87)) we have that for any V j ⊂ U i0 there exists i 1 such that F −1 ε (p −1 (V j ) ∩ |D|) ⊂ p −1 (U i1 ).