Transition map and shadowing lemma for normally hyperbolic invariant manifolds

For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in Hamiltonian systems. This consists in the following steps: (i) computation of the scattering map and of the transition map for a flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows. We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem.


1.
Introduction. Consider a normally hyperbolic invariant manifold for a flow or a map, and assume that the stable and unstable manifolds of the normally hyperbolic invariant manifold have a transverse intersection along a homoclinic manifold. One can distinguish an inner dynamics, associated to the restriction of the flow or of the map to the normally hyperbolic invariant manifold, and an outer dynamics, associated to the homoclinic orbits. There exist pseudo-orbits obtained by alternately following the inner dynamics and the outer dynamics for some finite periods of time. An important question on the dynamics is whether there exist true orbits with similar behavior. In this paper, we develop a toolkit of instruments and 1090 AMADEU DELSHAMS, MARIAN GIDEA AND PABLO ROLDÁN techniques to detect true orbits near a normally hyperbolic invariant manifold, that alternatively follow the inner dynamics and the outer dynamics, for all time. Some of the tools discussed below have already been used in other works. The aim of this paper is to provide a general recipe on how to make a systematic use of these tools in general situations.
The first tool is the scattering map, which is defined on the normally hyperbolic invariant manifold, and assigns to the foot of an unstable fiber passing through a point in the homoclinic manifold, the foot of the corresponding stable fiber that passes through the same point in the homoclinic manifold. This tool, sometimes referred as the homoclinic map, has been used in [10,12,7], and subsequently refined and analyzed in [5]. The scattering map can be defined both in the flow case and in the map case. In Section 2 we recall some background on normally hyperbolic invariant manifolds and Lambda Lemma. In Section 3 we describe the relationship between the scattering map for a flow and the scattering map for the return map to a surface of section. We note that the scattering map is defined in terms of the geometric structure, however it is not dynamically defined -there is no actual orbit that is given by the scattering map.
The second tool that we discuss is the transition map, that actually follows the homoclinic orbits for a prescribed time. The transition map can be computed in terms of the scattering map. Again, we will have a transition map for the flow and one for the return map, and we will describe the relationships between them. The transition map is presented in Section 4.
The third tool is the topological method of correctly aligned windows (see [22]), which is used to detect orbits with prescribed itineraries in a dynamical system. A window is a homeomorphic copy of a multi-dimensional rectangle, with a choice of an exit direction and of an entry direction. A window is correctly aligned with another window if the image of the first window crosses the second window all the way through and across its exit set. This method is reviewed briefly in Section 5.
The fourth tool is a shadowing lemma type of result for a normally hyperbolic invariant manifold, presented in Section 6. The assumption is that a bi-infinite sequence of windows lying in the normally hyperbolic invariant manifold is given, with the consecutive pairs of windows being correctly aligned, alternately, under the transition map (outer map), and under some power of the inner map. The role of the windows is to approximate the location of orbits. Then there exists a true orbit that follows closely these windows, in the prescribed order. To apply this lemma for a normally hyperbolic invariant manifold for a map, one needs to reduce the dynamics from the continuous case to the discrete case by considering the return map to a surface of section, and construct the sequence of correctly aligned windows for the return map. For this situation, the relationships between the scattering map for the flow and the scattering map for the return map, and between the transition map for the flow and the transition map for the return map, explored in Section 3 and Section 4, are useful.
A remarkable feature of these tools is that they can be used for both analytic arguments and rigorous numerical verifications. The scattering map and the transition map can be computed explicitly in concrete systems. The main advantage is that they can be used to reduce the dimensionality of the problem: from the phase space of a flow to a normally hyperbolic invariant manifold for the flow, and further to the normally hyperbolic invariant manifold for the return map to a surface of section. The shadowing lemma also plays a key role in reducing the dimensionality of the problem: it requires the verification of topological conditions in the normally hyperbolic invariant manifold for the return map to conclude the existence of trajectories in the phase space of the flow. In numerical applications, reducing the number of dimensions of the objects computed is very crucial. The potency of these tools in numerical application is illustrated in [4].
The main motivation for developing these tools resides with the instability problem for Hamiltonian systems. In the Appendix we describe two models where the above techniques can be applied to show the existence of unstable orbits. The first model is the large gap problem for nearly integrable Hamiltonian systems. The second model is the spatial circular restricted three-body problem.
In conclusion, we provide a practical recipe for finding trajectories with prescribed itineraries for a normally hyperbolic invariant manifold with the property that its stable and unstable manifolds have a transverse intersection along a homoclinic manifold: • Compute the scattering map associated to the homoclinic manifold.
• For some prescribed forward and backwards integration times, compute the corresponding transition map. • If necessary, reduce the dynamics from a flow to the return map via some surface of section. Determine the normally hyperbolic invariant manifold relative to the surface of section, and compute the inner map -the restriction of the return map relative to the normally hyperbolic invariant manifold. • Compute the scattering map and the transition map for the return map.
• Construct windows within the normally hyperbolic invariant manifold relative to the surface of section, with the the property that the consecutive pairs of windows are correctly aligned, alternately, under the transition map and under some power of the inner map. • Apply the shadowing lemma stated in Theorem 6.1 to conclude that there exist orbits that follow closely these windows.

2.
Preliminaries. In this section we review the concepts of normal hyperbolicity for flows and maps, normally hyperbolic invariant manifold for the return map to a surface of section, and we state a version of the Lambda Lemma that will be used in the subsequent sections.
2.1. Normally hyperbolic invariant manifolds. In this section we recall the concept of a normally hyperbolic invariant manifolds for a map and for a flow, following [11,17]. Let M be a C r -smooth, m-dimensional manifold (without boundary), with r ≥ 1, and Φ : M × R → M a C r -smooth flow on M .
that are invariant under dΦ t for all t ∈ R, and there exist a constant C > 0 and rates 0 < β < α, such that for all It follows that there exist stable and unstable manifolds of Λ, as well as stable and unstable manifolds of each point x ∈ Λ, which are defined by for some constants C y , C x,y > 0.
The stable and unstable manifolds of Λ are foliated by stable and unstable manifolds of points, respectively, i.e., W s (Λ) = x∈Λ W s (x) and W u (Λ) = x∈Λ W u (x).
In the sequel we will assume that Λ is a compact and connected manifold. With no other assumptions, E s x and E u x depend continuously (but non-smoothly) on x ∈ M ; thus the dimensions of E s x and E u x are independent of x. Below we only consider the case when the dimensions of the stable and unstable bundles are equal. We The smoothness of the invariant objects defined by the normally hyperbolic structure depends on the rates α and β. Let be a positive integer satisfying 1 ≤ < min{r, α/β}. The manifold Λ is C -smooth. The stable and unstable manifolds W s (Λ) and W u (Λ) are C −1 -smooth. The splittings E s x and E u x depend C −1smoothly on x. The stable and unstable fibers W s (x) and W u (x) are C r -smooth. The stable and unstable fibers W s (x) and W u (x) depend C −1−j -smoothly on x when W s (x), W u (x) are endowed with the C j -topology. In the sequel we will assume that the rates are such that there exists an integer ≥ 2 as above, and that all the manifolds and maps considered below are at least C k -smooth, with 2 ≤ k ≤ .
The notion of normal hyperbolicity for maps is very similar. Let F : M → M be a C r -smooth map on M .
Definition 2.2. A submanifold Λ of M is said to be a normally hyperbolic invariant manifold for F if Λ is invariant under F , there exists a splitting of the tangent bundle of T M into sub-bundles that are invariant under dF , and there exist a constant C > 0 and rates 0 < λ < µ −1 < 1, such that for all There exist stable and unstable manifolds of Λ, as well as the stable and unstable manifolds of each point x ∈ Λ, that are defined similarly as in the flow case, and they carry analogous properties. The smoothness properties of these invariant objects are analogous of those for a flow, if we set 1 ≤ < min{r, (log λ −1 )(log µ) −1 }.

2.2.
Normal hyperbolicity relative to the return map. Let Φ : M × R → M be a C r -smooth flow defined on an m-dimensional manifold M . Denote by X the vector field associated to Φ, where X(x) = ∂ ∂t Φ(x, t) |t=0 . As before, assume that Λ ⊆ M is an l-dimensional normally hyperbolic invariant manifold for Φ. The dimensions of T Λ, E u and E s are l, n, n, respectively, with l + 2n = m.
Let Σ be an (m − 1)-dimensional local surface of section, i.e., Σ is a C 1 submanifold of M such that X(x) ∈ T x Σ for all x ∈ Σ. Let Λ Σ = Λ ∩ Σ. Then Λ Σ is a (l − 1)-dimensional submanifold in Σ, assuming that the intersection is non-empty.
Assume that each forward and backward orbit through a point in Λ Σ intersects again Λ Σ . Since X(x) ∈ T x Σ for all x ∈ Σ, then the intersection of the forward and backward orbits with Σ are transverse. Also, X(x) ∈ T x Λ Σ for all x ∈ Λ Σ . Additionally, assume that the function is a continuous function. Following [11], we will refer to Λ Σ with these properties as a thin surface of section.
By the Implicit Function Theorem, τ can be extended to a C 1 -smooth function in a neighborhood U Σ of Λ Σ in Σ such that Φ(x, τ (x)) ∈ Σ for all x ∈ U Σ . The Poincaré first return map to Σ is the map F : Let Λ X Σ ⊆ Λ be the union of the orbits of the flow through points in Λ Σ . Since Λ X Σ is a C 1 -submanifold of Λ, and is invariant under Φ, then is a normally hyperbolic invariant manifold for the flow Φ. The theorem below implies that the manifold Λ Σ is normally hyperbolic for the return map F . The invariant sub-bundles T Λ, E u , E s associated to the normal hyperbolic structure on Λ correspond to sub-bundles T Λ Σ , E u Σ , E s Σ in the following way. Let π : T M = span(X) ⊕ T Σ → T Σ be the projection onto T Σ. Then T Λ Σ = π(T Λ ), E u Σ = π(E u ), and E s Σ = π(E s ). Note that the surface of section described above is only a local surface of section. It is in general very difficult, or even impossible, to obtain a global surface of section for a flow. However, one can obtain global surfaces of section for Hamiltonian flows on 3-dimensional strictly convex energy surfaces [18].
2.3. Lambda Lemma. We describe a Lambda Lemma type of result for normally hyperbolic invariant manifolds that appears in J.-P. Marco [19].
By a normal form in a neighborhood V of Λ in M we mean a C k -smooth coordinate system (c, s, u) on V such that V is diffeomorphic through (c, s, u) with a product Λ × R n × R n , where Λ = {(c, s, u) | c ∈ Λ, u = s = 0}, and 3. Scattering map. In this section we review the scattering map associated to a normally hyperbolic invariant manifold for a flow or for a map, and discuss the relationship between the scattering map for a flow and the scattering map for the corresponding return map to some surface of section.
3.1. Scattering map for continuous and discrete dynamical systems. Consider a flow Φ : M × R → M defined on a manifold M that possesses a normally hyperbolic invariant manifold Λ ⊆ M . As the stable and unstable manifolds of Λ are foliated by stable and unstable manifolds of points, respectively, for each x ∈ W u (Λ) there exists a unique x − ∈ Λ such that x ∈ W u (x − ), and for each x ∈ W s (Λ) there exists a unique x + ∈ Λ such that x ∈ W s (x + ). We define the wave maps Ω + : W s (Λ) → Λ by Ω + (x) = x + , and Ω − : W u (Λ) → Λ by Ω − (x) = x − . The maps Ω + and Ω − are C -smooth.
We now describe the scattering map, following [7]. Assume that W u (Λ) has a transverse intersection with W s (Λ) along a l-dimensional homoclinic manifold Γ. The manifold Γ consists of a (l − 1)-dimensional family of trajectories asymptotic to Λ in both forward and backwards time. The transverse intersection of the hyperbolic invariant manifolds along Γ means that Γ ⊆ W u (Λ) ∩ W s (Λ) and, for each x ∈ Γ, we have , Let us assume the additional condition that for each x ∈ Γ we have where x − , x + are the uniquely defined points in Λ corresponding to x. The restrictions Ω Γ + , Ω Γ − of Ω + , Ω − , respectively, to Γ are local C −1 -diffeomorphisms. By restricting Γ even further, if necessary, we can ensure that Ω Γ + , Ω Γ − are C −1 -diffeomorphisms. A homoclinic manifold Γ for which the corresponding restrictions of the wave maps are C −1 -diffeomorphisms will be referred as a homoclinic channel.
Definition 3.1. Given a homoclinic channel Γ, the scattering map associated to Γ is the Figure 1. In the sequel we will regard S Γ as a partially defined map, so the image of a set A by If we flow Γ backwards and forward in time we obtain the manifolds Φ −tu (Γ) and Φ ts (Γ) that are also homoclinic channels, where t u , t s > 0. The associated wave , respectively. The scattering map can be expressed with respect to these wave maps as We recall below some important properties of the scattering map.
Proposition 1. Assume that dim M = 2n + l is even (i.e., l is even) and M is endowed with a symplectic (respectively exact symplectic) form ω and that ω |Λ is also symplectic. Assume that Φ t is symplectic (respectively exact symplectic). Then, the scattering map S Γ is symplectic (respectively exact symplectic).
Proposition 2. Assume that T 1 and T 2 are two invariant submanifolds of comple- In the case of a discrete dynamical system consisting of a diffeomorphism F : M → M defined on a manifold M , the scattering map is defined in a similar way. We assume that F has a normally hyperbolic invariant manifold Λ ⊆ M . The wave maps are defined by Ω + : Assume that W u (Λ) and W s (Λ) have a differentiably transverse intersection along a homoclinic l-dimensional C −1 -smooth manifold Γ. We also assume the transverse foliation condition (2).
A homoclinic manifold Γ for which the corresponding restrictions of the wave maps are C −1 -diffeomorphisms is referred as a homoclinic channel. , Ω The scattering map can be expressed with respect to these wave map as The scattering map for the discrete case satisfies symplectic and transversality properties similar to those in Proposition 1 and Proposition 2 for the continuous case.

3.2.
Scattering map for the return map. Let Φ : M × R → M be a C r -smooth flow defined on an m-dimensional manifold M , and X be the vector field associated to Φ. Let Λ ⊆ M be an l-dimensional normally hyperbolic invariant manifold for Φ. Assume that Σ is a local surface of section and Λ Σ = Λ ∩ Σ satisfies the conditions in Subsection 2.2.
Consider Γ a homoclinic channel for Φ. First, we assume that Γ has a non-empty intersection with Σ. Note that Γ is a (l − 1)-parameter family of orbits; we further assume that each trajectory intersects Σ transversally. Since Γ is a homoclinic channel, each orbit intersects Σ exactly once. Let Γ Σ = Γ ∩ Σ. It is easy to see that Γ Σ is a homoclinic channel for F . Thus, we have a scattering map S Γ for Γ associated to the flow Φ, and we also have a scattering map S ΓΣ for Γ Σ associated to the map F .
We want to understand the relationship between S Γ and S ΓΣ . Associated to the homoclinic channels Γ and Γ Σ there exist wave maps Ω Γ ± : Γ → Λ and Ω ΓΣ ± : Γ Σ → Λ Σ , respectively. These maps are diffeomorphisms. Let x ∈ Γ Σ , and let We want to relate x − withx − , and x + withx + . These points are all in Λ.
We want to express these maps in terms of the dynamics restricted to Λ.
Let V be a flow box atx − (for definition see [21]). This means each trajectory through a point y ∈ V intersects Σ exactly once. Then there exists a differentiable functionτ : V → R defined by τ (z) = 0 if z ∈ Σ and Φτ (y) (y) ∈ Σ for each y ∈ V . The functionτ can be extended in a unique way on each trajectory passing though V . Due to the relationship between the invariant bundles for the flow and the invariant bundles for the map described in If there exist two such points, y − and y − , to them they correspond two points y, y in Γ Σ such that y ∈ W u F (y − ) and y ∈ W u F (y − ). The points y, y should belong to the same unstable fiber W u F (ŷ − ). Then it means that y, y are on the same trajectory. As they are also in Γ and Γ is a homoclinic channel, than y = y and y − = y − . In summary, the projection map P Γ − : Now we can formulate the relationship between the scattering map S Γ associated to the flow Φ, and the scattering map S ΓΣ associated to the map F . Proposition 3. Assume that Γ is a homoclinic channel for the flow Φ, and Γ Σ = Γ ∩ Σ is the corresponding homoclinic channel for the map F . Let S Γ be the scattering map corresponding to Γ, and let S ΓΣ be the scattering map corresponding to Γ Σ . Then: 4. Transition map. The scattering map for a flow Φ is geometrically defined: However, there is no trajectory of the system that goes from near x − to near x + . Instead, the trajectory of x approaches asymptotically the backwards orbit of x − in negative time, and approaches asymptotically the forward orbit of x + in positive time. For applications we need a dynamical version of the scattering map. That is, we need a map that x - x + x - x + takes some backwards image of x − into some forward image of x + . We will call this map a transition map. The transition map depends on the amounts of times we want to flow in the past and in the future. The transition map carries the same geometric information as the scattering map. Since in perturbation problems the scattering map can be computed explicitly, the transition map is also computable. The notion of transition map below is similar to the transition map defined in [3], however, their version is not related to the scattering map.

4.1.
Transition map for continuous and discrete dynamical systems. Consider a flow Φ : M × R → M defined on a manifold M that possesses a normally hyperbolic invariant manifold Λ ⊆ M . Assume that W u (Λ) and W s (Λ) have a transverse intersection, and that there exists a homoclinic channel Γ. Given t u , t s > 0, the time-map Φ ts+tu is a diffeomorphism from Φ −tu (Γ) to Φ ts (Γ). Using (3) we can express the restriction of Φ ts+tu to Φ −tu (Γ) in terms of the scattering map as given by where S Γ : U − → U + is the scattering map associated to the homoclinic channel Γ. We use this to define the transition map as an an approximation of Φ ts+tu provided that t u , t s are sufficiently large.
is the scattering map associated to the homoclinic channel Γ.
Alternatively, we can express the transition map as The symplectic property and the transversality property of the scattering map lend themselves to similar properties of the transition map.
In the case of a dynamical system given by a map F : M → M , the transition map can be defined in a similar manner to the flow case, and enjoys similar properties. As before, we assume that Λ ⊆ M is a normally hyperbolic invariant manifold for F .
is the scattering map associated to the homoclinic channel Γ.

4.2.
Transition map for the return map. We will consider the reduction of the transition map to a local surface of section. Let Σ be a local surface of section and Λ Σ = Λ ∩ Σ. By Theorem 2.3, Λ Σ is normally hyperbolic with respect to the first return map to Σ. Assume that Γ intersects Σ as in Subsection 2.2, and let , approaches asymptotically Λ as t u → ∞, and intersects Σ infinitely many times. Similarly, Φ ts (x) lies on W u (Φ ts (x + )), approaches asymptotically Λ as t s → ∞, and intersects Σ infinitely many times.
Let υ > 0 be a small positive number. We define t u = t u (x) to be the smallest time such that Φ −tu(x) (x) ∈ Σ, and the distance between Φ −tu (x) and Φ −tu (x − ), measured along the unstable fiber W u (Φ −tu (x − )), is less than υ. Let N u > 0 be such that Φ −tu (x) = F −Nu (x). Similarly, we define t s = t s (x) to be the smallest time such that Φ ts (x) ∈ Σ, and the distance between Φ ts (x) and Φ ts (x + ), measured along the stable fiber W s (Φ ts (x + )), is less than υ. Let N s > 0 be such that At this point, we have a transition map S Γ tu,ts associated to the flow Φ and to the homoclinic channel Γ for the flow, and a transition map S ΓΣ Nu,Ns associated to the map F and to the homoclinic channel Γ Σ for the map.
We have that Φ −tu (Γ) and Φ ts (Γ) are both homoclinic channels for the flow Φ, and F −Nu (Γ) and F Ns (Γ) are both homoclinic channels for the map F . Let us consider the projection mappings P The relationship between the transition map for the flow Φ and the transition map for the return map F is given by the following: Proposition 4. Assume that Γ is a homoclinic channel for the flow Φ, and Γ Σ = Γ∩Σ is the corresponding homoclinic channel for the map F . Let t u , t s , N u , N s > 0 be fixed. Let S Γ Nu,Ns be the transition map corresponding to Γ for the flow Φ, and let S ΓΣ Nu,Ns be the transition map corresponding to Γ Σ for the return map F . Then The conclusion of the proposition now follows from the definition of the transition map in the flow case and the definition of the transition map in the map case.

5.
Topological method of correctly aligned windows. We review briefly the topological method of correctly aligned windows. We follow [22]. See also [14,13].
We adopt the following notation: When the local parametrization χ is evident from context, we suppress the subscript χ from the notation.
Definition 5.2. Let R 1 and R 2 be (m 1 , m 2 )-windows, and let χ 1 and χ 2 be the corresponding local parametrizations. Let F be a continuous map on M with F (im(χ 1 )) ⊆ im(χ 2 ). We say that R 1 is correctly aligned with R 2 under F if the following conditions are satisfied: (i) There exists a continuous homotopy h : [0, 1] × (R 1 )χ 1 → R m1 × R m2 , such that the following conditions hold true where π m1 : R m1 × R m2 → R m1 is the projection onto the first component, and deg is the Brouwer degree of the map A y0 at 0.
The following result allows the detection of orbits with prescribed itineraries.
Theorem 5.3. Let {R i } i∈Z , be a collection of (m 1 , m 2 )-windows in M , and let F i be a collection of continuous maps on M . If for each i ∈ Z, R i is correctly aligned with R i+1 under F i , then there exists a point p ∈ R 0 such that Moreover, under the above conditions, and assuming that for some k > 0 we have R i = R (i mod k) and F i = F (i mod k) for all i ∈ Z, then there exists a point p as above that is periodic in the sense Often, the maps F i represent different powers of the return map associated to a certain surface of section. The orbit of the point p found above is not necessarily unique.
The correct alignment of windows is robust, in the sense that if two windows are correctly aligned under a map, then they remain correctly aligned under a sufficiently small perturbation of the map.
Proposition 5. Assume R 1 , R 2 are (m 1 , m 2 )-windows in M . Let G be a continuous maps on M . Assume that R 1 is correctly aligned with R 2 under G. Then there exists > 0, depending on the windows R 1 , R 2 and G, such that, for every continuous map F on M with F (x) − G(x) < for all x ∈ R 1 , we have that R 1 is correctly aligned with R 2 under F . Also, the correct alignment satisfies a natural product property. Given two windows and a map, if each window can be written as a product of window components, and if the components of the first window are correctly aligned with the corresponding components of the second window under the appropriate components of the map, then the first window is correctly aligned with the second window under the given map. For example, if we consider a pair of windows in a neighborhood of a normally invariant normally hyperbolic invariant manifold, if the center components of the windows are correctly aligned and the hyperbolic components of the windows are also correctly aligned, then the windows are correctly aligned. Although the product property is quite intuitive, its rigorous statement is rather technical, so we will omit it here. The details can be found in [13]. 6. A shadowing lemma for normally hyperbolic invariant manifolds. In this section we present a shadowing lemma-type of result. It is assumed the existence of a sequence of windows in the normally hyperbolic invariant manifold Λ, with the windows of the same dimension l as Λ. It is assumed that the sequence of windows is made up of pairs of windows D − i , D + i+1 that are correctly aligned under the transition map S Γ , alternately with pairs of windows D + i+1 , D − i+1 that are correctly aligned under some power F N 0 i of the restriction of F to Λ. Here, the superscript ± for the windows D ± i suggest that D ± i is typically obtained by taking some positive (negative) iteration of some other window that lies in the codomain (domain) of the scattering map.
It is required that the numbers N 0 i , N − i , N + i can be chosen arbitrarily large, but uniformly bounded relative to j. The conclusion is that there exists a true orbit in the full space dynamics that follows these windows arbitrarily closely. The resulting orbit is not necessarily unique.
The result below provides a method to reduce the problem of the existence of orbits in the full dimensional phase space to a lower dimensional problem of the existence of pseudo-orbits in the normally hyperbolic invariant manifold. Theorem 6.1. Assume that there exists a bi-infinite sequence {D + i , D − i } i∈Z of ldimensional windows contained in a compact subset of Λ such that, for any integers n 0 1 , n − 1 , n + 1 > 0, there exist integers n 0 2 > n 0 1 , n − 2 > n − 1 , n + 2 > n + 1 and sequences of integers such that the following properties hold for all i ∈ Z: Then, for every ε > 0, there exist an orbit {F N (z)} N ∈Z of F for some z ∈ M , and an increasing sequence of integers such that, for all i: Proof. The idea of this proof is to 'thicken' the windows D + i , D − i in Λ to fulldimensional windows R − i , R + i in M , so that the successive windows in the sequence {R − i , R + i } i are correctly aligned under some appropriate iterations of the map F . The argument is done in several steps. In the first three steps, we only specify the relative sizes of the windows involved in each step. In the fourth step, we explain how to make the choices of the sizes of the windows uniform.
Step 1. At this step, we take a pair of l-dimensional windows D − i and D + i+1 as in the statement of the theorem and, through applying iteration and the wave maps we construct two (l + 2n)-dimensional windowsR − i andR + i+1 near Γ such thatR − i is correctly aligned withR + i+1 under the identity map.

Note that conditions (i) and (ii) imply thatD
) be the copies ofD − i and D + i+1 , respectively, in the homoclinic channel Γ. By making some arbitrarily small changes in the sizes of their exit and entry directions, we can alter the windowsD − i andD + i+1 such thatD − i is correctly aligned withD − i under (Ω − Γ ) −1 ,D − i is correctly aligned withD + i+1 under the identity mapping, andD + i+1 is correctly aligned witĥ D + i+1 under Ω + Γ .

AMADEU DELSHAMS, MARIAN GIDEA AND PABLO ROLDÁN
We 'thicken' the l-dimensional windowsD − i andD + i+1 in Γ, which are correctly aligned under the identity mapping, to (l + 2n)-dimensional windows that are correctly aligned under the identity map. We now explain the 'thickening' procedure.
First, we describe how to thickenD − i to a full dimensional windowR − i . We choose some 0 <δ − i < ε and 0 <η and is homeomorphic to an (l+n)-dimensional rectangle. We define the exit set and the entry set of this rectangle as follows: We consider the normal bundle N + to W u (Λ). At each point y ∈∆ − i , we choose an n-dimensional closed ballB + η − i (y) centered at y and contained in the . By the Tubular Neighborhood Theorem (see, for example [2]), we have that forη − i > 0 sufficiently small, the setR − i is a homeomorphic copy of an (l + 2n)-rectangle. We now define the exit set and the entry set ofR − i as follows: Second, we describe in a similar fashion how to thickenD + i+1 to a full dimensional windowR + i+1 . We choose 0 <δ + i+1 < ε and 0 <η + i+1 < ε. We consider the (l + n)- is the n-dimensional closed ball of radiusη + i+1 centered at x and contained in W s (x + ), with x + = Ω Γ + (x). The exit set and entry set of this window are defined as follows: (y) is the n-dimensional closed ball centered at y and contained in the image of N − y ⊆ T y M under the exponential map exp y : N − y → M , and N − is the normal bundle to W s (Λ). The Tubular Neighborhood Theorem implies that forδ + i+1 > 0 sufficiently small the setR + i+1 is a homeomorphic copy of a (l + 2n)-rectangle. The exit set and the entry set ofR + i+1 TRANSITION MAP AND SHADOWING LEMMA 1103 are defined by: This completes the description of the thickening of the l-dimensional windowD − i into an (l + 2n)-dimensional windowR − i , and of the thickening of the l-dimensional windowD + i+1 into an (l+2n)-dimensional windowR + i+1 . Note that, by construction, R − i andR + i+1 are both contained in an ε-neighborhood of Γ. Now we want to makeR − i correctly aligned withR + i+1 under the identity map. This is achieved by choosingδ + i+1 sufficiently small relative toδ − i , and by choosinḡ η − i sufficiently small relative toη + i+1 . Thus, we haveδ (we stress that these inequalities alone may not suffice for the correct alignment). Choosingδ + i+1 andη − i small enough agrees with the constraints imposed by the Tubular Neighborhood Theorem.
Step 2. At this step, we expand the given l-dimensional window D − i to an (l + 2n)-dimensional window R − i such that R − i is correctly aligned withR − i under some positive iterate, and we also expand the given l-dimensional window D + i+1 to an (l + 2n)-dimensional window R + i+1 such thatR + i+1 is correctly aligned withR + i+1 under some positive iterate. We take a negative iterate We have that F −M (Γ) is ε-close to Λ on a neighborhood in the C 1 -topology, for all M sufficiently large. The vectors tangent to the fibers W u (x − ) inR − i are contracted, and the vectors transverse to W u (Λ) alongR − i ∩ W u (Λ) are expanded by the derivative of F −M . We choose and fix M = N − i sufficiently large. We obtain that, in particular, In other words, D − i is correctly aligned under the identity mapping with the projection of F −N − i (D − i ) onto Λ along the unstable fibres. Let us consider 0 < δ − i < ε and 0 < η − i < ε. To define the window R − i we use a local linearization of the normally hyperbolic invariant manifold.
For Λ normally hyperbolic, let N Λ = (E u ⊕ E s ) |Λ = p∈Λ {p} × E u p × E s p be the normal bundle to Λ, and N F = T F |N Λ , where By Theorem 1 in [20], there exists a homeomorphisms h from an open neighborhood of the zero section of N Λ to a neighborhood of Λ in M such that Since D − i is contractible the bundles are trivial on D − i and we can identify is the closed ball centered at 0 of radius η − i in the stable space E s x . We set the exit and entry sets of Then we define the window R − i as follows: In order to ensure the correct alignment of In a similar fashion, we construct a window R + i+1 contained in an ε-neighborhood of Λ such thatR + i+1 is correctly aligned with R + i+1 under F N + i+1 . The window R + i+1 , and its entry and exit sets, are defined by: (0)), (H + i+1 (x)) exit , and (H + i+1 (x)) entry are defined as before for some appropriate choices of radii δ + i+1 , η + i+1 > 0.
Step 3. At this step, we take the (l + 2n)-dimensional window R + i+1 andR − i+1 as constructed in the previous step, and we make R + i+1 correctly aligned withR − i+1 under some positive iterate.
Suppose that we have constructed the window R + i+1 about the l-dimensional exponentially expanded in the unstable direction and exponentially contracted in the stable direction by DF . ThusB u (0)) under the identity map (both rectangles are Step 4. At this step we will use the previous steps to construct a bi-infinite sequences of windows {R ± i ,R ± i } i∈Z such that, for each i, the windows {R ± i } are obtained by thickening the rectangles {D ± i } ⊆ Λ, the windows {R ± i+1 } are obtained by thickening some rectangles {D ± i } ⊆ Γ, and, moreover, We can assume without loss of generality that Λ and Γ are compact. We fix an ε-neighborhood V of Λ. Using the compactness of Λ and Γ and the uniform boundedness of the iterates N − i , N + i , N 0 i , we now show how to choose the sizes of the stable and unstable components of the windows {R ± i ,R ± i } i∈Z constructed in the previous steps in a uniformly bounded manner. For each point x in Λ we consider a (2n)-dimensional window h({x} ×B u δ ×B s η ), for some 0 < δ, η < ε, where h is the local conjugacy between F and DF near Λ.
), for all n 0 1 ≤ N 0 i ≤ n 0 2 , provided that n 0 1 is chosen sufficiently large. For each i, we thicken D + i and D − i into full dimensional windows R + i and of R − i respectively, as described in Step 2, where for the sizes of the components of these windows we choose δ ± i = δ and η ± i = η for all i. Since D + i is correctly aligned with D − i under F N 0 i , then, as in Step 3, it follows that R + i is correctly aligned with R − i under F N 0 i . We also define the set This set cannot be realized as a window since it does not have exit/entry directions associated to the Λ components. However, for each x ∈ Λ, the set h({x} ×B s η (0) × B u δ (0)) is a well defined window, with the exit given by the hyperbolic unstable directions. Note that Υ 0 (x) ⊆ h({x} ×W u (x) ×W s (x)) for each x ∈ Γ.
We let∆ − = x∈ΓB ū δ − (x), withB ū δ − (x) being the closed ball centered at x of radiusδ − in W u (x − ). For each point y ∈∆ − we consider the closed ballB s η − (y) centered at y of radiusη − in the image under exp y of the normal subspace N y to W u (Λ) at y. Similarly, we let∆ + = x∈ΓB s η + (x), whereB ū η + (x) ⊆ W s (x + ), and for each y ∈∆ + we consider the closed ballB ū δ + (y) in the image under exp y of the normal subspace N y to W s (Λ) at y. We define the sets These sets cannot be realized as windows as there are no well defined exit/entry directions associated to their Γ components. However, for each x ∈ Γ, the set Υ − (x) = y∈B ū δ − (x)B s η − (y) is a well defined (2n)-dimensional window, with the exit given by the hyperbolic unstable directions. Note that Υ − (x) ⊆ y∈W u (x−) exp y (N y ). The intersection of y∈W u (x−) exp y (N y ) with Υ + defines a window Υ + (x) with the exit given by the hyperbolic unstable directions. Due to the compactness of Γ, there exist δ ± , η ± such that Υ − (x) is correctly aligned with Υ + (x) for all x ∈ Γ. We choose and fix such δ ± , η ± . We define the windowsR ± i at Step 1 with the choices of δ ± i = δ ± , and η ± i = η ± , for all i. It follows thatR − i is correctly aligned withR + i under the identity map for all i.
Due to the compactness of Γ and the uniform expansion and contraction of the hyperbolic directions, there exist n − 1 , n + 1 such that, for all where V is the neighborhood of Λ where the local linearization is defined. For any such n − 1 , n + 1 , the assumptions of Lemma 6.1 provide us with some i (x + )))] under the identity map. From these choices, it follows that the windows The windows R ± i are contained in ε-neighborhoods of the given rectangles D ± i , respectively, and the windows R ± i are contained in ε-neighborhoods of some rectanglesD ± i , respectively. By Theorem 5.3, there exits an orbit F N (z) that visits the windows The existence of the shadowing orbit concludes the proof. describe two models for Hamiltonian instability and we explain how the transition map and the shadowing lemma are utilized.
In the sequel, we show that the arguments in [6,13,8,15] can be combined with the techniques developed in this paper to prove the existence of diffusing trajectories as above.
(1) We can make the Hamiltonian H µ into an autonomous Hamiltonian by adding an extra variable A symplectically conjugate with the time t. The variable A has no dynamical role. We then restrict to an energy level, which can be parametrized by the variables (p, q, I, φ, t) with A an implicit function of these variables. When µ = 0 the setΛ 0 = {(p, q, I, φ, t), p = q = 0, I ∈ [I − , I + ]} is a 3-dimensional normally hyperbolic invariant manifold with boundary for the flow of H 0 . It is foliated by 2dimensional invariant toriT I = {(p, q, I, φ, t), p = q = 0, I = I } with I ∈ [I − , I + ]. The stable and unstable manifolds ofΛ 0 coincide, i.e. W u (Λ 0 ) = W s (Λ 0 ). Also W u (T I ) = W s (T I ) for all I ∈ [I − , I + ]. This follows from (ii).
(2) There exists µ 0 > 0 small such that for each 0 < µ < µ 0 the manifoldΛ 0 has a continuationΛ µ that is a normally hyperbolic invariant manifold for the flow of H µ . Inside Λ µ there are finitely many resonant regions, due to (iv). Outside the resonant regions, the conditions (i), (iii) allow one to apply the KAM theorem, and obtain KAM tori that are at a distance of order O(µ 3/2 ) from one another. The resonant regions yield gaps of size O(µ j/2 ) between the KAM tori, where j is the order of the resonance. Only the resonances of order 1 and 2 are of interest, as they produce gaps of size O(µ) and O(µ 1/2 ) respectively. Inside each resonant region, there exist primary KAM tori and secondary KAM tori (homotopically trivial relative to Λ ε ). They can be chosen to be O(µ 3/2 ) from one another.
(3) The conditions (v) on the Melnikov potential imply that if µ 0 is chosen small enough then W u (Λ µ ) intersects W s (Λ µ ) transversally at an angle O(µ). Choose and fix a homoclinic channelΓ µ as in (1) and (2), and let SΓ µ denote the corresponding scattering map for the flow of H µ , as in Definition 3.1. It turns out that the scattering map has the property that there exists a constant C > 0 such that for each I ∈ [I − , I + ] there exists I ∈ [I − , I + ] with |I − I | > Cµ and x , x ∈Λ µ with I(x ) = I , I(x ) = I , such that SΓ µ (x ) = x . In the above, one can always choose I > I or I > I , as one wishes. That is, there are always points whose I-coordinates in increased or decreased by the scattering map by O(µ).
(4) Fix a Poincaré surface of section Σ = {(p, q, I, φ, t), t = 0 (mod 2π)}, and let F µ be the first return map to Σ. By Theorem 2.3 Λ 0 =Λ 0 ∩ Σ is a normally hyperbolic invariant manifold with boundary for F µ . The map F µ restricted to Λ µ satisfies a uniform twist condition, due to (iii). Each 2-dimensional torusT I inΛ µ intersects Σ in a 1-dimensional T I in Λ µ . From Subsection 3.2 it follows that Γ µ =Γ µ ∩ Σ is a homoclinic channel. The scattering map S Γµ : U − ε → U + ε associated to Γ µ for F µ is related to SΓ µ by the relation given in Proposition 3. (5) Choose a sequence {T Ij } j∈Z of KAM tori (primary or secondary) in Λ µ with the following properties: (a) Each leave T Ij is within O(µ), relative to the I-variable, from the next leave T Ij+1 , (b) S Γµ takes each T Ij transversally across T Ij+1 , for all n. From Proposition 2, W u (T Ij ) intersects transversally W s (T Ij+1 ) at some point z j ∈ Γ µ . Thus, the set {T Ij } j∈Z , together with the transverse homoclinic connections among consecutive tori, forms a transition chain.
(6) Now we explain how Theorem 6.1 is applied to this problem. We have to show that the assumptions of the theorem are met for some sequence of 2-dimensional windows {D − j , D + j } in Λ ε . Following the arguments in [13], for each j we can construct a pair of windowsD − j ⊆ U − ε ,D + j+1 ⊆ U + ε , withD − j in some neighborhood of T Ij andD + j+1 in some neighborhood of T Ij+1 such thatD − j is correctly aligned witĥ D + j+1 under S Γµ . The exit and entry directions of each window are 1-dimensional. The windowD − j is chosen to have its entry set components on some pair of tori T Ij , T Ij neighboring T Ij on both its sides in Λ ε , and the windowD + j+1 is chosen to have its exit components set on some pair of tori T Ij+1 , T Ij+1 neighboring T Ij+1 on both its sides in Λ ε .
where r 2 1 = (x − µ) 2 + y 2 + z 2 and r 2 2 = (x − µ + 1) 2 + y 2 + z 2 . The system has five equilibrium points; one of them, denoted L 1 , lies between the two primaries. Its stability is of saddle × center × center type. We focus our attention on the dynamics near L 1 . At a given energy level, near L 1 we can distinguish a family of quasi-periodic orbits that are uniquely defined by their out-of-plane amplitude of the motion. The objective is to show that there exits trajectories that move from near one such a quasi-periodic orbit to near another, in the prescribed order, and so to change the out-of-plane amplitude of the motion near L 1 with zero cost. The mechanism to achieve such trajectories involves starting about a quasi-periodic orbit about L 1 of some out-of-plane amplitude, moving the infinitesimal mass around one of the primaries, returning to L 1 , and moving about some other quasi-periodic orbit about L 1 of a different out-of-plane amplitude, and so on. Since this problem is not close to integrable, the methods from perturbation theory do not apply. The approach discussed below is semi-numerical.
(1) Fix an energy level h slightly above the energy level of L 1 , so that the dynamical channel between the m 1 -region and the m 2 -region is open. Restrict the study of the dynamics to the 5-dimensional energy manifold M h = {H = h}.
(2) Compute numerically the 3-dimensional normally hyperbolic invariant mani-foldΛ ⊆ M h by writing the Hamiltonian in Birkhoff normal form in a neighborhood of L 1 . By taking a high-order truncation of the normal form we obtain a system of action-angle coordinates (I 1 , I 2 , φ 1 , φ 2 ) onΛ, where we can choose I 1 as the out-of-plane amplitude of a quasi-periodic orbit inΛ, and I 2 implicitly defined by the restriction to the energy level. As the truncated normal form is integrable, the numerically computedΛ ⊆ M h is filled with 2-dimensional invariant toriT I1 parametrized by the out-of-plane amplitude I 1 . Dynamically, these invariant tori are only almost invariant.
(3) The Birkhoff normal form is also used to compute the stable and unstable manifolds W s (Λ), W u (Λ) of the normally hyperbolic invariant manifoldΛ. One shows numerically that W s (Λ), W u (Λ) intersect transversally along some homoclinic channelΓ. In particular, one can compute the scattering map SΓ for the flow as in [9]. We choose some ε > 0 sufficiently small, and choose the t u , t s to be the first times when φ −tu (Γ), φ −ts (Γ), respectively, land in the ε-neighborhood ofΛ. We then compute the corresponding transition map SΓ t u ,t s as in Definition 4.1.
(4) We reduce the dimension of the problem by choosing a suitable Poincaré section, e.g. Σ = {φ 2 = 0}. Let F denote the first return map to Σ. We have that Λ = Σ ∩Λ is normally hyperbolic for F , and Γ = Σ ∩Γ is a homoclinic channel. The intersections of the 2-dimensional toriT I1 with the Poincaré section yields 1dimensional tori T I1 in Λ, each torus corresponding to a fixed value of I 1 . The inner dynamics induced on the Poincaré section is a monotone twist map. Using the relationship between the transition map for a flow and the transition map for the return map from Proposition 4, we compute the transition map S Γ N u ,N s for F . The image of each 1-dimensional torus T I 1 under S Γ N u ,N s is a curve along which the value of I 1 variable sometimes goes above and sometimes below the original value I 1 . See Figure 3.
(5) To design trajectories that visit some finite collection of I 1 -level sets in the prescribed order, we first use the transition map S Γ N u ,N s to move between level sets. We obtain pairs of points x j , x j+1 with I(x j ) = (I 1 ) j , I(x ) = (I 1 ) j+1 , and S Γ N u ,N s (x j ) = x j+1 , moving from a level set (I 1 ) j to a level set (I 1 ) j+1 . Using continuation, we construct a window D − j about the level set (I 1 ) j , and a window D + j+1 about the level set (I 1 ) j+1 , such that D − j is correctly aligned with D + j+1 under S Γ N u ,N s . Similarly, we construct a window D − j+1 about the level set (I 1 ) j+1 , and a window D + j+2 about a level set (I 1 ) j+2 , such that D − j+1 is correctly aligned with D + j+2 under S Γ N u ,N s . To align D + j+1 with D − j+1 we take some large number of iterates N 0 j such that D + j+1 is correctly aligned with D − j+1 under F N 0 j . We obtain the situation described in Theorem 6.1. In particular, we obtain diffusing orbits for which the action variable I 1 increases as much as possible.
We should note that, since the correct alignment of windows is robust, the fact that the tori T I1 are only almost-invariant does not matter. This makes the method suitable for a computer assisted proof.
In both examples, the main advantage of using the transition map and the Theorem 6.1 is that, via these tools, one only needs to construct windows of lower dimension (the dimension of the normally hyperbolic invariant manifold) that are correctly aligned either under the transition map, or under some power of the inner map. Both these maps are defined on the lower dimensional normally hyperbolic invariant manifold. The result from Theorem 6.1 is the existence of a certain trajectory that lives in the full dimensional phase space.