Continuation and collapse of homoclinic tangles

By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a system of bifurcation equations that is indexed by a symbolic sequence. For the H\'{e}non family we investigate in detail the bifurcation structure of multi-humped orbits originating from several tangencies. The homoclinic network found by numerical continuation is explained by combining our bifurcation result with graph-theoretical arguments.

1. Introduction. We consider parameter dependent, discrete time dynamical systems of the form x n+1 = f (x n , λ), n ∈ , where f is smooth and f (·, λ), λ ∈ Ê are diffeomorphisms in Ê k . We assume that the system (1) has a smooth branch of hyperbolic fixed points and our main interest is in branches of homoclinic orbits that return to these fixed points. Generically one finds turning points on these branches which correspond to homoclinic tangencies where stable and unstable manifolds of the fixed point intersect nontransversally, see [26,4] for the precise relation. While the dynamics near transversal intersections are well understood through the celebrated Smale-Shilnikov-Birkhoff Theorem (see [44,43,18,36]), the picture near homoclinic tangencies still seems to be far from being complete. There is a wealth of references that deal with phenomena occurring near homoclinic tangencies, mostly for planar diffeomorphisms. We mention the two monographs [35,5] which deal with various aspects of nonuniformly hyperbolic behavior, such as the Newhouse phenomenon, measures of sets with secondary tangencies etc. Another source showing the richness of phenomena is [6] where the authors study the socalled fattened Arnold map as a model for tangencies of planar diffeomorphisms.
We also refer to the work [15] which supports the generic occurrence of homoclinic tangencies of all orders. Among the further references, we mention the papers [7,9,8] which relate homoclinic tangencies to global structural changes, such as boundary crisis or basin boundary metamorphosis. Moreover, a theory of trellises (sets of pieces of stable and unstable manifolds) is developed that allows to study global changes in two-dimensional phase space by symbolic coding and graph representations. In [28] it is shown that shift dynamics occurs in arbitrarily small neighborhoods of the tangency.
Homoclinic (or heteroclinic) tangencies also occur in a natural way for Poincaré maps of continuous time dynamical systems. These maps are either generated by classical Poincaré sections of periodic orbits or by special cross-sections taken in the vicinity of continuous homoclinic orbits. We refer to [22,Ch.4,5] for a recent overview, to [39] for a Lyapunov-Schmidt reduction of heteroclinic networks with tangencies, and to [31,29,34] for numerical approaches utilizing such sections. In particular, these references study codimension-two homoclinic orbits ('orbit flip' and 'inclination flip') which lead to multi-humped orbits in a fan of the two-parameter region. More details on the relation to continuous systems will be discussed in Section 2.
Finally, we mention that transversal and tangential homoclinic orbits of maps can be computed numerically in a robust way by solving boundary value problems on a finite interval, cf. [3,13,16,26]. Errors caused by this approximation have been completely analyzed, see [27,26,4]. The works [26,16] also study curves of homoclinic tangencies in two-parameter systems.
To the best of our knowledge the maximal invariant set near a homoclinic tangency has not been completely characterized in the literature. Nor are we aware of any study -particularly for Hénon's map -of global connected components of multi-humped homoclinic orbits containing tangencies. Both topics will be central to this paper. We consider bifurcations of multi-humped homoclinic orbits from tangential orbits in arbitrary space dimensions and investigate in detail the global behavior of branches of multi-humped orbits for the classical Hénon example.
Our main local result (see Theorem 2.1) determines the elements of the maximal invariant set in a neighborhood of the tangent orbit and of the critical parameter from a set of bifurcation equations. Using the same shift space as in the transversal case, we associate with any sequence of symbols a bifurcation equation that describes those branches of orbits that have the return pattern of the symbolic sequence. Such a result does not fully resolve the dynamics near tangencies, but reduces the problem to a set of perturbed bifurcation equations for which the unperturbed form is known (similar to Liapunov-Schmidt reduction, cf. [14,Ch.I]). The theorem covers multihumped homoclinic orbits that enter and leave a neighborhood of the fixed point several times, by relating it to a perturbed system of hilltop bifurcations (see [14] for the hilltop normal form). Our main results will be stated in Section 2 (Theorems 2.1 and 2.5) with the rather involved proofs deferred to Sections 5 and 6.
The global approach asks for possible bifurcations of multi-humped orbits that are known to emerge from the tangencies. We take the Hénon family as a model equation for a detailed numerical study of the homoclinic network that arises from a total of 4 primary homoclinic tangencies. We show that the connected components of this network are by no means arbitrary. Rather, they follow certain rules governing the bifurcations of multi-humped orbits. Combining these rules with graph-theoretical and combinatorial arguments allows to predict the structure to a large extent, see Theorem 4.2. Only some fine details are left to numerical computations as will be demonstrated in Sections 3 and 4.
It is well known that transversal homoclinic orbits lead to chaotic dynamics on a nearby invariant set commonly referred to as a homoclinic tangle. Let us first assume that this situation occurs at some parameter valueλ ∈ Λ 0 .
In this case the stable and the unstable manifold ofξ intersect transversally at eachx n and the setH = {x n } n∈ ∪{ξ} is hyperbolic, cf. [37]. Moreover, there exists an open neighborhood U ofH such that the dynamics on the maximal invariant set is conjugate to a subshift of finite type (see the Smale-Shilnikov-Birkhoff Homoclinic Theorem in [18] and [36, Chapter 5] for a proof). To be precise, let N ≥ 2 and let Let β be the Bernoulli shift Consider a special subshift of finite type, see [32] Ω N = {s ∈ S N : A (N ) si,si+1 = 1 ∀i ∈ } 74 WOLF-JÜRGEN BEYN AND THORSTEN HÜLS generated by the N × N binary matrix Then there exists a neighborhood U ofH, an integer N ≥ 2 and a homeomorphism h : A continuation of the transversal homoclinic orbit w.r.t. the parameter λ leads to a curve of homoclinic orbits that typically exhibits turning points. As an example we refer to Figure 1 for the Hénon map. Parts of the branch that can be parametrized by λ belong to transversal homoclinic orbits while (quadratic) turning points correspond to homoclinic orbits with a (quadratic) tangency, see Theorem 2.3 for a precise statement. In this case we replace Assumption A4 by B4 For someλ ∈ Λ 0 there exists a nontrivial homoclinic orbitx = (x n ) n∈ converging towardsξ = ξ(λ). The orbit is tangential in the sense that the variational equation has a non-trivial bounded solution u = (u n ) n∈ in Ê k that is unique up to constant multiples. Since the fixed point stays hyperbolic we have exponential decay for both the orbit and the solution of (6), i.e. for some α, C e > 0 x n −ξ + u n ≤ C e e −α|n| , n ∈ .
Therefore, we may normalize In the following we use ·, · to denote the inner product in 2 . The assumption on (6) in B4 holds if and only if the tangent spaces of the stable and unstable manifold have a one-dimensional intersection, i.e.
We refer to Theorem 2.3 and to [ As in the transversal case, we will still work with the subshift (Ω N , β) but the conjugacy (5) will be replaced by a set of bifurcation equations. For any s ∈ Ω N define the index set I(s) = {n ∈ : s n = 1}, (9) and note that I : With any s ∈ Ω N we associate the Banach space Our aim is to determine the elements of M (U, Λ) from a set of bifurcation equations where s ∈ Ω N , ρ τ > 0 is independent of s and g s : is a sufficiently smooth map. Note that (11) constitutes a finite or an infinite system of equations depending on the cardinality of I(s).
In order to formulate the precise statement we define the pseudo orbits Equation (7) shows that p (s) is a bounded sequence, in particular Settingξ n =ξ for all n ∈ we write (12) more formally as Here and in what follows we use the symbol β to denote the shift of sequences in Ê k . Thus β acts as an operator in sequence spaces such as p (Ê k ), 1 ≤ p ≤ ∞.
Similarly, for every τ ∈ ∞ (s) we define the bounded sequence v (s, τ ) = Note that the sequence p (s) has humps at the positions defined by I(s) and that p (s) is a pseudo orbit of f (·,λ) with a small error, see Lemma 5.3. The term v (s, τ ) shifts the solution of the variational equation to the positions defined by I(s) and combines them linearly.
Theorem 2.1. Let assumptions A1 -A3 and B4 hold. Then there exist constants 0 < r τ ≤ ρ τ , N ∈ AE and neighborhoods U of H, Λ ofλ and for any s ∈ Ω N smooth functions with the following properties.
Remark 2.2. Theorem 2.1 reduces the study of M (U, Λ) to the set of bifurcation equations (16) with a symbolic index s ∈ Ω N . It may be regarded as a type of Liapunov-Schmidt reduction though we have not formally put it into this framework. The construction of the neighborhood U ×Λ uses some features from the transversal case [36,Theorem 5.1], but is considerably more involved, see Sections 5 and 6. We also note that we were not able to prove that one can take r τ = ρ τ which would give a complete characterization of M (U, Λ) in terms of (15), (16). Another issue which has not yet been resolved, is continuous dependence of the functions x ,s and g s on the symbolic sequence s with respect to the metric (4).
The functions g s and x ,s have several properties that we discuss next. Due to B4 the adjoint equation has a non-trivial solution w that is unique up to constant multiples, cf. [37,Section 2]. It decays exponentially as in (7) and can thus be normalized such that w 2 = 1. Without loss of generality we take C e in (7) such that w n ≤ C e e −α|n| , n ∈ .
As is shown in [26] the quantities characterize the behavior of the branch of homoclinic orbits that passes through (x ,λ).
has a limit point at (x ,λ) in the sense that F (x ,λ) = 0 and N (D x F (x ,λ)) = span{u }.
The limit point is transversal, i.e.
Moreover, it is a quadratic turning point, i.e.
Our second result shows that the constants c λ , c x play an important role in the behavior of the bifurcation function g s (τ, λ).
(ii) For some constants C > 0, α > 0, independent of s, N and ∈ I(s) Remark 2.6. Note that one can prove existence of n-humped orbits in the transversal case from the well-known shadowing principle, see the remarks following Eq. (25). To obtain an analogous result for the tangential case is much more difficult. Although Theorem 2.1 completely characterizes the maximal invariant set near a tangency, such a result requires to prove existence of smooth solution branches for the reduced system (16). In principle, this task is solved by the unfolding theory for singularities, see [14]. However, in order to make the theory applicable to the bifurcation equations (16) one needs estimates of (all) derivatives of g s in (23). In fact, the function g s (·, ·) comes out smoothly from Theorem 5.1, but estimating their derivatives is quite involved (cf. the proof of (23) in Section 6) and has not yet been done.
If we consider a homoclinic symbol s ∈ Ω N with K = card(I(s)) < ∞ humps, then Theorem 2.5 shows that the bifurcation equations are small perturbations of a set of K identical turning point equations If c λ , c x = 0 one can shiftλ to zero and scale λ and τ such that one obtains a set of hilltop bifurcations of order K, cf. [14, Ch. IX, §3], For two-humped orbits the set I(s) contains two elements and the solution curves of (24) are shown in Figure 8. This case will be crucial for understanding the global behavior of homoclinic curves in the next sections.
Remark 2.7. The emergence of multi-humped homoclinic orbits near tangencies is reminiscent of the fan of multi-humped orbits created near codimension-two homoclinics in continuous systems, see [22,Sect. 5.1] and [41,21]. However, the underlying mechanisms seem to be different. This is discussed in the survey paper [22,Sect. 4.3] which also suggests to further compare both phenomena. For continuous systems the occurrence of a fan of homoclinics near an inclination flip or orbit flip bifurcation is explained by a so-called singular horse-shoe appearing in suitable Poincaré sections. On the other hand multi-humped orbits in discrete systems are created by maps of horse-shoe type with tangencies. Though it is natural that the codimensions of both phenomena differ by one, there is another distinguishing feature that is relevant for numerical continuation: the curves of multi-humped homoclinics in the fan of a continuous system all accumulate at a single pair of parameters, while the corresponding curves in the discrete system have their own turning points which relate to secondary tangencies close to but not at the critical parameter value. This will be illustrated in Section 4.

3.
Homoclinic orbits and their continuation. A typical example, which plays the role of a normal form for quadratic two-dimensional mappings, is the famous Hénon map, cf. [20,33,11,19] which is defined as This map has fixed points and forλ = 0.35 a transversal homoclinic orbit x (λ) w.r.t. the fixed point ξ + (λ) exists, satisfying Assumption A4.
For numerical computations, we approximate an infinite homoclinic orbit x (λ) by a finite orbit segment x J , where J = [n − , n + ] ∩ . The segment is determined as a zero of the boundary value operator Here b : Ê 2k → Ê k defines a boundary condition, for example in case of periodic and projection boundary conditions, where B s and B u yield linear approximations of the stable and the unstable manifold. Due to our hyperbolicity assumption, Γ J (·,λ) has for J sufficiently large a unique zero in a neighborhood of the exact solution. Moreover approximation errors decay at an exponential rate that depends on the type of boundary condition, cf. [3]. For Hénon's map, we solve the corresponding boundary value problem, obtain in this way an approximation of x (λ) and continue this orbit w.r.t. the parameter λ, using the method of pseudo arclength continuation, cf. [25,1,17,13]. In Figure   1, we plot the amplitude of these orbits amp(x J (λ)) := n∈J x n (λ) − ξ + (λ) 2 1 2 versus the parameter. This figure is well known, cf. [3], but is reproduced here to illustrate and introduce our labeling of left and right turning points. At the valueλ = 0.35 four distinct homoclinic orbits occur that we denote by x i , i ∈ {0, . . . , 3}. We choose their index by following the order given by the continuation routine. The orbit x 0 is shown in Figure 2 together with parts of the stable and the unstable manifold of the fixed point ξ + (λ). The enlargements in the figure show which intersection of the manifolds leads to the four homoclinic orbits.
At each turning point, two orbits collide; with r and , we distinguish right and left turning points. Figure 3 illustrates intersections of stable and unstable manifolds at these four turning points.
Errors of turning point calculations for finite approximations of homoclinic orbits decay exponentially fast w.r.t. the length of the computed orbit segment, cf. [27, Theorem 5.1.1]. We refer to [12], for algorithms that compute approximations of one-dimensional and two-dimensional stable and unstable manifolds and to [30] for a comparison of competing methods. 4. Connected components of multi-humped orbits. For Hénon's map, we find four distinct transversal homoclinic orbits x s , s ∈ {0, . . . , 3} atλ = 0.35 and we identify x (s) with its symbol s. Note that the orbits 0, 1, 2, 3, 0 pass into each   other via left (L) and right (R) turning points r 0,1 , 1,2 , r 2,3 , 3,0 , see Figure 3. The graph in Figure 4 gives an alternative illustration of these transitions.
The transition graph for one-humped orbits in Figure 4 consists of exactly one LR-cycle. Note that all one-humped orbits lie on a single closed curve and thus are found via numerical continuation.
The corresponding analysis of closed curves of n-humped orbits, with n ≥ 2, is more involved and is the main goal of this section. Indeed, several disjoint curves of  n-humped orbits occur. We introduce a symbolic coding for n-humped orbits and put two orbits into the same equivalence class, if they lie on the same closed curve. For the construction of an n-humped orbit, we choose a sufficiently long interval J = [n − , n + ] around zero and a sequence s ∈ S n := {0, . . . , 3} n . We define the pseudo orbitx see Figure 5. Since the collection of single orbits x r , r ∈ {0, . . . , 3} together with the fixed point forms a hyperbolic set, the Shadowing-Lemma, cf. [38,36] shows that the pseudo orbitx (s) lies close to a true n-humped f -orbit which we denote by x (s). In S n there are 4 n different symbols and thus we expect to find 4 n different n-humped orbits x (s). We identify these orbits with their symbol. Note that composing initial approximations of multi-humped orbits from pieces of single orbits is a very common approach that appears in discrete as well as continuous time dynamical systems, see [8,31,29,34]. We also note that our construction of pseudo orbits in (25) slightly differs from (12), where we add up shifted orbits. With both approaches, we expect to find the same shadowing orbit for sufficiently large intervals J.
x 0 x [123] Given two symbols s,s ∈ S n , we analyze whether the n-humped orbits x (s) and x (s) can turn into each other via continuation.
Let us first look at the two-humped case.

4.1.
Bifurcation of two-humped orbits. The continuation of two-humped orbits exhibits three closed curves of homoclinic orbits, see Figure 6. At the parameter valueλ, there exist 16 different homoclinic orbits x (s), indexed by s ∈ S 2 . These orbits lie on different closed curves, namely 8 on the first, and 4 on the second and the third curve.  We note that it is challenging to compute the continuation pictures in Figure 6 due to the sharp turns that occur, see Figure 7.
An accurate step size control is essential for achieving these diagrams. With too large continuation steps, the algorithm directly jumps from the upper light green to the lower dark green curve.
For the continuation of n-humped orbits, this problem also shows up with available software like Matcont, cf. [13], when using the standard setup of parameters. We therefore apply our own implementation of an Euler-Newton method which controls the step size by cond −α , where α ∈ [0.7, 0.81] and cond denotes the condition number of the Jacobian, occurring in the Newton process of the corrector step. The difficulty of continuing n-humped orbits across the sharp turns is illustrated by the minimal step sizes that we find during the computation, see Table 1. h min 9 · 10 −4 8 · 10 −6 9 · 10 −7 2 · 10 −7 Table 1. Minimal step sizes h min of our algorithm for the continuation of n-humped orbits.
The following discussion explains why a large condition number close to the sharp turns is to be expected.
One observes that at each turning point in Figure 6, exactly one component of the symbol changes. For example, the symbol (1, 1) changes at a left turning point into the symbol (2, 1). But why does the symbol (1, 1) not change into the symbol (1, 2)? The answer, which transition occurs depends on small terms that perturb a hilltop bifurcation, see the discussion below. These small terms are caused by the finite distance between the single humps.
For two-humped orbits, the system (24) is a set of two equations in three variables λ, τ 1 , τ 2 , called the hilltop normal form, cf. [14]  Note that the Jacobian of the Newton process becomes singular at a simple bifurcation point, cf. [10], and hence is ill-conditioned at a perturbed bifurcation.

4.2.
Connected components and equivalent symbols. Homoclinic orbits that lie on a common closed curve define a connected component of More precisely, let s ∈ S n and denote by C(s) ⊂ H the connected component that satisfies (x (s),λ) ∈ C(s).
Then, we obtain an equivalence relation by identifying two sequences s,s ∈ S n , if the corresponding orbits lie in the same component i.e.
In the following, we discuss how to find these equivalence classes. In particular, we show under some generic assumptions that each equivalence class has at least four elements, and for n-humped orbits it turns out that one class has at least 4n elements.
For this task, we introduce a labeled graph G with vertices s ∈ S n . Two vertices s ands ∈ S n are connected with an L or R-edge, if x (s) bifurcates into x (s) via a left or right turning point. Since we do not know the effect of the perturbed hilltop bifurcation a priori, we put an edge, if the transition is possible for at least one perturbation. For example, the vertices (1, 1) and (2, 1) as well as (1, 1) and (1,2) are connected with L-edges in case n = 2, see Section 4.1. Precise rules for constructing this graph are stated in Section 4.3.
Our hypothesis is that the desired equivalence classes correspond to a special decomposition of this graph into disjoint LR-cycles.
In case of one-humped orbits, the only LR-cycle is 01230, see Figure 4. Consequently, all symbols lie in the same equivalence class, which matches the fact that all one-humped orbits lie on the same closed curve and thus, in the same connected component of H.

4.3.
Graph structure of homoclinic network. In this section, we give precise rules for defining the labeled graph G which we identify with its adjacency tensor with entries R and L.
First, we assume that only one of the n humps can turn into a neighboring hump at a turning point.

R1
There is no edge from s ∈ S n tos ∈ S n if s =s or where d is the distance on the cycle 01230. Now let s,s ∈ S n and assume s −s 1 = 1, then there exists a unique j such that s j =s j .
From λ 2,3 > λ 0,1 we conclude that the right transition at r 2,3 can only occur if the orbit contains no 0 and no 1 hump. Therefore, we define R-edges in G according to the following rule.
. . , n. Similarly from λ 1,2 > λ 3,0 we conclude that the left transition at 3,0 can only occur if the orbit contains no 1 and no 2 hump. Our rules for L-edges are: We expect a connected component to correspond to an LR-cycle in this graph, i.e. a cycle on which L and R-edges alternate. A precise statement of our hypothesis is as follows.
Hypothesis 4.1. The connected components of n-humped orbits and thus the equivalence classes (27) are in one to one correspondence to partitions of G into disjoint LR-cycles.
In case n = 2, the L and R-edges are shown in the left and center picture of Figure 9, respectively. The right diagram additionally shows the LR-cycles that correspond to the connected components from Figure 6.  Figure 9. L-edges of G (left) and R-edges (center) for twohumped orbits. The cycles in the right figure correspond to the closed curves that are computed numerically in Figure 6.
A partition of the graph into disjoint LR-cycles is not unique and consequently, the decomposition in the right of Figure 9 is not the only possible candidate, satisfying Hypothesis 4.1. Figure 10 illustrates all possible partitions (up to reflections at the diagonal).
Our numerical results show that the first partition in the second row of Figure  10 actually occurs in the Hénon system. This indicates the particular paths (red or magenta in Figure 8) taken on each perturbed hilltop bifurcation. Corresponding diagrams for n = 3 are given in Figure 11. We continued n-humped orbits numerically for Hénon's map up to n = 5. Table  2 Table 2. Continuation of n-humped orbits -length of occurring cycles in numerical experiments.
Hypothesis 4.1 relates connected components of H to LR-cycles in the graph G, given by our rules R1-R3. Even though we cannot decompose G uniquely into disjoint LR-cycles, our experiments for n-humped orbits of Hénon's map suggest that the length of occurring cycles is always a multiple of 4. Furthermore, there exists one cycle of length at least 4n. Indeed, we prove that these observations follow from Hypothesis 4.1.
Theorem 4.2. Fix n ∈ AE and assume that Hypothesis 4.1 holds true. Then, all n-humped orbits lie on cycles whose length is a multiple of 4.
For the specific symbols s 0 = (0, . . . , 0) and s 2 = (2, . . . , 2) ∈ S n , the corresponding orbits x (s 0 ) and x (s 2 ) lie on a common cycle of at least length 4n.  Table 2 shows that there is exactly one orbit of length 4n and all other orbits are shorter. Hence, the orbit of length 4n contains s 0 and s 2 .
Proof. By assuming Hypothesis 4.1 we see that it suffices to analyze LR-cycles of the graph G. More precisely, we prove Theorem 4.2 along the following steps.
We define the graphG by deleting the R-edges of s 2 from the remaining graph. Figure 12 illustrates this construction in case n = 2.
IfG breaks into two components V 3 andG \ V 3 , then we get a contradiction to the above assumption and an LR-cycle in the original graph G that contains s 2 but not s 0 cannot exist.
To finish the proof, we show that an LR-path inG from s 2 to V 3 does not exist.
From s 2 we cannot go directly via an R-edge to V 3 , since the corresponding edges are deleted inG. Thus without loss of generality, we get: (2 · · · 2) →  In summary, 4 n different n-humped orbits for Hénons map exist at the parameter valueλ = 0.35. These orbits share the same connected component, if they turn into each other via parameter-continuation. We have introduced a symbolic coding and proposed rules to decide a-priori, if two orbits lie on a common component. Via a graph theoretical argument, it turns out that the minimal component size is 4. Furthermore, the orbits coded by (0 · · · 0) and (2 · · · 2) lie on a common curve and this curve connects at least 4n orbits via continuation.
A complete classification of the 4 n orbits into connected components requires more knowledge on perturbations of hilltop bifurcations that we have not yet analyzed. We presented numerical data found for the cases of 2 − 5-humped orbits.
5. Bifurcation analysis near homoclinic tangencies. In this section we prove the main Theorem 2.1 and Theorem 2.5(i) by using an existence and uniqueness result for a suitable operator equation in spaces of bounded sequences. In the following we use the notation B ρ (x) and B ρ = B ρ (0) to denote closed balls of radius ρ in some Banach space.
5.1. The operator equation. First recall the operator F : ∞ (Ê k )×Ê → ∞ (Ê k ) from (20) and the normalizationλ = 0 and ξ(λ) = 0 for λ close toλ, see A2, A3 in Section 2. Then for any s ∈ Ω N define the operator Here p , v are defined in (12), (14) and w(s, g) is given by (recall w from (17)) Our aim is to derive the functions x ,s , g s in Theorem 2.1 by solving for τ ∞ , |λ| sufficiently small and for all s ∈ Ω N . More precisely, we prove in Section 6 the following Reduction Theorem.
Theorem 5.1. There exist constants C 0 , ρ x , ρ g , ρ τ , ρ λ > 0 and a number N 0 ∈ AE such that for all N ≥ N 0 and for all s ∈ Ω N the following statements hold. For all τ ∈ B ρτ , λ ∈ B ρ λ the system (29) has a unique solution Moreover, the following estimate is satisfied:

Preparatory lemmata.
In order to construct the neighborhoods U and Λ in Theorem 2.1 we need several lemmata.

WOLF-JÜRGEN BEYN AND THORSTEN HÜLS
at λ = 0, where Here P s and P u are the stable and unstable projectors of the fixed point 0. Let b be the bound from Theorem A.4 for the difference equation For sufficiently large −n − , n + ≥ n 0 and λ ∈ Λ 3 sufficiently small, we get B n ≤ b for all n ∈J. Consequently u n+1 = f n (p n , λ)u n , n ∈J has an exponential dichotomy on J with projectors P s n , P u n and an exponential rate α that is independent of n − , n + , λ and K.
As in the proof of [23, Theorem 4], we show that for λ ∈ Λ 3 , n − , n + ≥ n 0 and K ≥ N 3 we have a uniform bound In order to see this, consider the inhomogeneous difference equation Denote by Φ the solution operator of the homogeneous equation and let G be the corresponding Green's function, cf. (87). The general solution of (38) is given by where , v + ∈ R(P u K ). Inserting (40) into (39), it remains to solve This finite-dimensional system has a unique solution for K ≥ N 3 sufficiently large since P s − P s 0 → 0 and P u − P u K → 0 as K → ∞. Therefore, the system (38), (39) also has a unique solution u J for K large and the dichotomy estimates lead to a bound u J ∞ ≤ σ −1 (|γ| + r J ∞ ), i.e. (37) holds. We apply Theorem A.2 with the space Y = ∞ J = {(y n ) n∈J : y n ∈ Ê k } of finite sequences and with Z = ∞ J × Ê k , both endowed with the sup-norm. We take y 0 = p J and use uniform data for all λ ∈ Λ 3 . For δ sufficiently small we have and by choosing the neighborhood Λ 3 sufficiently small we get Theorem A.2 applies to λ ∈ Λ 3 with uniform data, and it follows from (119) with some constant C 3 > 0 that From (7) we find a number n 0 such thatx n ∈ B δ 2 (0) for all |n| ≥ n 0 and also p n ∈ B δ 2 (0), n ∈ J for all −n − , n + ≥ n 0 . Then we take U 3 := B δ 2 (0) as our neighborhood and note that (41) holds for any two sequences x J , y J in U 3 . For n ∈J and λ ∈ Λ 3 it follows that Now let x J be a sequence in U 3 such that x n+1 = f (x n , λ) for all n ∈J, and some λ ∈ Λ 3 . Then In case K = ∞, one uses the operator and it turns out that D 1ΓAE (p AE , λ) with p j =x n+−1+j has a uniformly bounded inverse for λ ∈ Λ 3 and sufficiently large −n − , n + . Then the estimate (36) follows immediately.

5.3.
Proof of main theorem. Let us first prove assertion (i) in Theorem 2.1.
Step 1. (Construction of neighborhoods U, Λ) In the following Λ 1 ⊃ Λ 2 ⊃ . . . will denote shrinking neighborhoods of 0. Let ρ g , ρ x > 0 be given by Theorem 5.1 and note that we can decrease ρ τ , ρ λ without changing the assertion of Theorem 5.1. Introduce the constants (cf. (7) and Lemma 5.3, 5.4) Let ρ τ > 0 be such that By Lemma 5.4 we can choose a ball B 3ε0 ⊂ U 3 and numbers n + , −n − ≥ n 0 such thatx n ∈ B ε0 for all n ≥ n + − 1 and n ≤ n − + 1. It is well known that the only full orbit in a small neighborhood of a hyperbolic fixed point is the fixed point itself (this follows from [42,Theorem III.7]). That is, we can assume w.l.o.g. that U 3 , Λ 3 satisfy The set K = {0} ∪ {x n : n ≤ n − or n ≥ n + } is compact and satisfies Thus we find an ε ≤ ε 0 and Λ 4 ⊂ Λ 3 such that the following properties hold the balls are mutually disjoint, see Figure 13. Figure 13. Construction of neighborhoods.
Step 4. (Proof of Theorem 2.1 (ii))The radius r τ will be taken such that Let τ ∈ B rτ ⊂ ∞ (s), λ ∈ Λ satisfy g s (τ, λ) = 0 for some s ∈ Ω N and let x ,s be given as in Theorem 5.1. Then clearly, the sequence is an orbit of (1). It remains to show that y n ∈ U for all n ∈ . Application of (30) in Theorem 5.1 and of Lemma 5.3 yields the estimate We estimate the distance of y to the centers of the balls B j in U by showing for ν = −n − − 1 y p+ +ν −x p ≤ ε 6C 3 for p = n − + 1, . . . , n + − 1, ∈ I(s).
Therefore the sequence y lies in U which proves our assertion.
The assertion then follows by uniqueness from Theorem 5.1 since neighborhoods are shift invariant as well.
The proof of Theorem 2.5(ii) will be deferred to the next section.
Before proving Lemma 6.1 we finish the proof of Theorem 5.1.
The idea is to construct elements y = y (τ, λ) ∈ ∞ (Ê k ) and γ = γ(τ, λ) ∈ ∞ (s) such that the residual and γ are of higher order than O(|λ| + τ 2 ∞ ). Then the assertion follows from (30) by comparing them to x ,s (τ, λ), g s (τ, λ). We find y , γ by Taylor expansion of F (we abbreviate F 0 From the estimates (34), (35) in the proof of Lemma 5.3 we have . In a similar way, using Lipschitz constants for f λ and f xx we find Therefore, Taylor expansion of G s yields where This suggests to define (y , γ) by From this equation and Lemma 6.1 we have the estimate Taking the inner product of the first coordinate in (75) with β − w , ∈ I(s) and using (19), (84) leads to the improved estimate By (76), (75) the Taylor expansion (74) of G s assumes the form (30) and (77) this leads us to the final result 6.2. Linear estimates. In this subsection we prove Lemma 6.1. For any two integers n l ≤ n r and for any number a ≥ 0 consider the weight function ω n = ω n (a, n l , n r ) = min e a(n−n l ) , 1, e a(nr−n) , n ∈ .
Note that ω has a constant plateau of arbitrary width with exponentially decaying tails on both sides. We also allow n l = −∞ and n r = ∞ (but neither n l = n r = −∞ nor n l = n r = ∞), in which case ω has only one-sided decay or degenerates to the maximum norm if n l = −∞ and n r = ∞. In the following we will suppress the dependence of the norm on the data n l , n r , a, but all our estimates will be uniform with respect to − ∞ ≤ n l ≤ n r ≤ ∞ 0 ≤ a ≤ a 0 < α, (80) where 0 < a 0 < α is fixed. The following lemma shows that exponentially decaying kernels preserve the weight. Taking the exponent α from (7), (18) we have the following result for the variational equation (6).
By the exponential dichotomies the Green's functions satisfy With u T y := u , y we may write (81) in block operator form as By the bordering lemma [2, Appendix] the block operator is Fredholm of the same index 0 as L and, using (84), it is a linear homeomorphism in ∞ (Ê k )×Ê. Since ∞ ω is a closed subspace of ∞ (Ê k ) it suffices to prove that the unique solution (y , h) of (91) in ∞ (Ê k ) × Ê satisfies the estimate (82) in case r ∈ ∞ ω . Take the inner product of the first equation of (81) with w . Then (84) and the normalization w 2 = 1 show h = w , r . Therefore, by (18), With this h we have z := r − hw ∈ R(L) by (84). Below we will construct a special solutionŷ ∈ ∞ (Ê k ) of Lŷ = z such that for some constant C > 0 By (84) and (8) the solution of (91) is given by (y , h) = (ŷ + cu , h), c = γ − u ,ŷ , h = w , r .
Then (92)-(94) yield the assertion It remains to constructŷ with Lŷ = z and (93). We determine η + ∈ R(P +s 0 ) and η − ∈ R(P −u 0 ) such thatŷ n = y + n , n ≥ 0 with y + n from (86), andŷ n = y − n , n ≤ 0 with y − n from (88), and such that the definitions coincide at n = 0. The last condition holds if and only if By (87), (89), (85) the first sum on the right is in Y + ⊕ Y 1 and the second sum is Since z ∈ R(L) holds, equation (95) has a solution and thus ∆ 0 ∈ Y + ⊕ Y − . We conclude from (85) that η + = P +s 0 ∆ 0 and η − = −P −u 0 ∆ 0 are the unique solutions of (95). With (90) and Lemma 6.3 we estimate for n ≥ 0 An analogous estimate holds for n ≤ 0 and this completes the proof.

0, else
is the characteristic function of J( ). Let B 0 denote the solution operator of (91), then we set and define B + as a blockwise inverse via Using Lemma 6.4 with the settings n l = − − , n r = + − we obtain a bound Let us abbreviate the weights from (79), ω n, = ω n (a, − , + ), n ∈ , ∈ I(s).
Then equation (99) and (100) For 2e −aN * ≤ 1 the last sum is bounded by 1 + 4e −a and (100) yields In the next step we show for N * sufficiently large, We show that the terms in {. . .} are of order O(e −aN * ) so that the contraction estimate (102) holds for the first component if N * is sufficiently large. A critical term on the right-hand side is The term ω n,ˆ x n− is handled analogously. Further, The remaining terms allow similar estimates since n always lies in an exponential decaying tail of the weights and of the shifted homoclinic orbits. Finally, we use (7) and (98), (100) for ∈ I(s), We estimate the remaining sum by using (103)

Our final estimate is
The term + n= − e −α|n− | ω n,ˆ is estimated in a similar way. This finishes the proof of (102).
Using the exponential weights of t m in (106) and (108) leads to the same estimate for h − t ∞ and (104) is proved.
In a similar manner, holds for the weightsω * n = ω n (2a,ˆ + −ˆ ,ˆ + −ˆ ). The t-values satisfy In particular, this proves the t-estimates in (108). Next we estimate the difference d = β − y + β −ˆ yˆ − y by using the exponential dichotomy on for the constant coefficient operator L 0 y = (y n+1 − Ay n ) n∈ , A = f x (0, 0). From (109) and the definition of y , yˆ we find Then A is a homeomorphism with Proof. Note that y = (I Y − AB + )y + r has a unique solution y for every r ∈ Y . Then y satisfies y ≤ r 1 − I Y − AB + and x = B + y solves Ax = r. To prove uniqueness, note that any solution x of Ax = r solves x = (I X − B − A)x + B − r. Since I X − B − A is also contractive the solution is unique and the estimates follow.
A key tool in the proofs of Lemma 5.4 and Theorem 5.1 is the following quantitative version of the Lipschitz inverse mapping theorem, cf. [24].
Theorem A.2. Assume Y and Z are Banach spaces, F ∈ C 1 (Y, Z) and F (y 0 ) is for y 0 ∈ Y a homeomorphism. Let κ, σ, δ > 0 be three constants, such that the following estimates hold: Then F has a unique zeroȳ ∈ B δ (y 0 ) and the following inequalities are satisfied We collect some well known results on exponential dichotomies from [37]. Denote by Φ the solution operator of the linear difference equation Definition A.3. The linear difference equation (120) with invertible matrices A n ∈ Ê k,k has an exponential dichotomy with data (K, α, P s n , P u n ) on an interval J ⊂ , if there exist two families of projectors P s n and P u n = I − P s n and constants K, α > 0, such that the following statements hold: with an interval J ⊆ , has an exponential dichotomy with data (K, α, P s n , P u n ).
Suppose B n ∈ Ê k,k satisfies B n ≤ b for all n ∈ J with a sufficiently small b.
Then A n + B n is invertible and the perturbed difference equation y n+1 = (A n + B n )y n has an exponential dichotomy on J.