Continuation and collapse of homoclinic tangles
aa r X i v : . [ m a t h . D S ] D ec Continuation and collapse of homoclinictangles
Wolf-J¨urgen Beyn ∗ Thorsten H¨uls ∗ Department of Mathematics, Bielefeld UniversityPOB 100131, 33501 Bielefeld, Germany [email protected] [email protected]
October 31, 2018
Abstract
By a classical theorem transversal homoclinic points of maps lead toshift dynamics on a maximal invariant set, also referred to as a homoclinictangle. In this paper we study the fate of homoclinic tangles in parameter-ized systems from the viewpoint of numerical continuation and bifurcationtheory. The bifurcation result shows that the maximal invariant set near ahomoclinic tangency, where two homoclinic tangles collide, can be charac-terized by a system of bifurcation equations that is indexed by a symbolicsequence. For the H´enon family we investigate in detail the bifurcationstructure of multi-humped orbits originating from several tangencies. Thehomoclinic network found by numerical continuation is explained by com-bining our bifurcation result with graph-theoretical arguments.
Keywords:
Homoclinic tangency, symbolic dynamics, numerical continuation,bifurcation of homoclinic orbits.
AMS Subject Classification:
We consider parameter dependent, discrete time dynamical systems of the form x n +1 = f ( x n , λ ) , n ∈ Z , (1)where f ( · , λ ) , λ ∈ R are smooth diffeomorphisms in R k . We assume that the sys-tem (1) has a smooth branch of hyperbolic fixed points and our main interest is ∗ Supported by CRC 701 ’Spectral Structures and Topological Methods in Mathematics’.
1n branches of homoclinic orbits that return to these fixed points. Generically onefinds turning points on these branches which correspond to homoclinic tangencieswhere stable and unstable manifolds of the fixed point intersect nontransversally,see [15], [3] for the precise relation. While the dynamics near transversal in-tersections are well understood through the celebrated Smale-Shilnikov-BirkhoffTheorem (see [26],[25],[9],[21]), the picture near homoclinic tangencies still seemsto be far from being complete. Among the many references, we mention themonograph [20] which contains a detailed geometrical study of the bifurcationsthat occur near tangencies, the work [7] which supports the generic occurrenceof homoclinic tangencies of all orders, and the paper [17] which proves shift dy-namics in arbitrarily small neighborhoods of the tangency. We also mentionthat homoclinic orbits with a tangency can be computed numerically in a robustway by solving boundary value problems on a finite interval, and that the errorscaused by this approximation have been completely analyzed, see [16, 15, 3].In this paper, we consider bifurcations of tangential homoclinic orbits from alocal as well as from a global viewpoint.The local study determines the elements of the maximal invariant set in aneighborhood of the tangent orbit and of the critical parameter from a set ofbifurcation equations. Using the same shift space as in the transversal case,we associate with any sequence of symbols a bifurcation equation that describesthose branches of orbits which have the return pattern of the symbolic sequence.Such a result does not fully resolve the dynamics near tangencies, but reduces theproblem to a set of perturbed bifurcation equations for which the unperturbedform is known (similar to Liapunov-Schmidt reduction). For example, multi-humped homoclinic orbits that enter and leave a neighborhood of the fixed pointseveral times, relate to a perturbed system of hilltop bifurcations, cf. [6]. Ourmain results will be stated in Section 2 with the proofs deferred to Sections 5, 6.The global viewpoint asks for possible bifurcations of multi-humped orbitsthat are known to emerge from the tangencies. We take the H´enon family as amodel equation for a detailed numerical study of the homoclinic network thatarises from a total of 4 primary homoclinic tangencies. It turns out that theconnected components of this network are by no means arbitrary. Rather, theyfollow certain rules governing the bifurcations of multi-humped orbits. Combiningthese rules with graph-theoretical and combinatorial arguments allows to predictthe structure to a large extent. Only some fine details are left to numericalcomputations as will be demonstrated in Sections 3 and 4.
The aim of this section is to state our main result on bifurcation equations nearhomoclinic tangencies. We first describe the setting and state our assumptions: A1 f ∈ C ∞ ( R k × Λ , R k ) for some open set Λ ⊂ R and f ( · , λ ) is a diffeomor-2hism for all λ ∈ Λ , A2 f ( ξ ( λ ) , λ ) = ξ ( λ ) for some smooth branch ξ ( λ ) ∈ R k , λ ∈ Λ , A3 f x ( ξ ( λ ) , λ ) ∈ R k,k is hyperbolic for all λ ∈ Λ .Clearly, if ξ is a hyperbolic fixed point of f ( · , λ ) for some λ ∈ R then A2 and A3 follow for some neighborhood Λ of λ . Replacing f by g ( x, λ ) = f ( x + ξ ( λ + λ ) , λ + λ ) − ξ ( λ + λ ) shows that A2 , A3 can be assumed to hold for the trivialbranch ξ ( λ ) = 0 and for a neighborhood Λ of zero. This will be our standingassumption throughout Sections 5 and 6.It is well known that transversal homoclinic orbits lead to chaotic dynamicson a nearby invariant set commonly referred to as a homoclinic tangle. Let usfirst assume that this situation occurs at some parameter value ˜ λ ∈ Λ . A4 For some ˜ λ ∈ Λ there exists a nontrivial homoclinic orbit ˜ x Z = (˜ x n ) n ∈ Z ,i.e. lim n →±∞ ˜ x n = ˜ ξ := ξ (˜ λ ) and ˜ x n = ˜ ξ for some n ∈ Z . This orbit is transversal in the sense that the variational equation y n +1 = f x (˜ x n , ˜ λ ) y n , n ∈ Z (2)has no nontrivial bounded solution on Z .In this case the stable and the unstable manifold of ˜ ξ intersect transversallyat each ˜ x n and the set ˜ H = { ˜ x n } n ∈ Z ∪ { ˜ ξ } is hyperbolic, cf. [22]. Moreover, thereexists an open neighborhood U of ˜ H such that the dynamics on the maximalinvariant set M ( U, ˜ λ ) = { x ∈ U : f n ( x, ˜ λ ) ∈ U ∀ n ∈ Z } (3)is conjugate to a subshift of finite type (see the Smale-Shilnikov-Birkhoff Homo-clinic Theorem in [9] and [21, Chapter 5] for a proof). To be precise, let N ≥ S N = { , , . . . , N − } Z be the shift space with N symbols which is compact w.r.t. the metric d ( s, t ) = X j ∈ Z −| j | | s j − t j | , s = ( s j ) j ∈ Z ∈ S N . (4)Let β be the Bernoulli shift β ( s ) i = s i +1 , i ∈ Z , s ∈ S N . Consider a special subshift of finite type, see [18]Ω N = { s ∈ S N : A ( N ) s i ,s i +1 = 1 ∀ i ∈ Z } N × N binary matrix A ( N ) = · · ·
00 0 1 . . . ...... . . . 1 00 . . . 11 0 · · · · · · ∈ { , } N × N . Then there exists a neighborhood U of ˜ H , an integer N ≥ h : Ω N → M ( U, ˜ λ ) such that f ( · , ˜ λ ) ◦ h = h ◦ β in Ω N . (5)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. Asan example we refer to Figure 1 for the H´enon map. Parts of the branch that canbe parametrized by λ belong to transversal homoclinic orbits while (quadratic)turning points correspond to homoclinic orbits with a (quadratic) tangency, seeTheorem 3 for a precise statement. In this case we replace Assumption A4 by B4 For some ¯ λ ∈ Λ there exists a nontrivial homoclinic orbit ¯ x Z = (¯ x n ) n ∈ Z converging towards ¯ ξ = ξ (¯ λ ). The orbit is tangential in the sense that thevariational equation y n +1 = f x (¯ x n , ¯ λ ) y n , n ∈ Z (6)has a non-trivial solution u Z = ( u n ) n ∈ Z that is unique up to constant mul-tiples.Since the fixed point stays hyperbolic we have exponential decay for both theorbit and the solution of (6), i.e. for some α, C e > k ¯ x n − ¯ ξ k + k u n k ≤ C e e − α | n | , n ∈ Z . (7)Therefore, we may normalize k u Z k ℓ = h u Z , u Z i ℓ = X n ∈ Z u Tn u n = 1 . (8)In the following we use h· , ·i to denote the inner product in ℓ . The assumptionon (6) in B4 holds if and only if the tangent spaces of the stable and unstablemanifold have a one-dimensional intersection, i.e. T ¯ x n W s ( ¯ ξ ) ∩ T ¯ x n W u ( ¯ ξ ) = span(¯ u n ) , n ∈ Z . We refer to Theorem 3 and to [15, Appendix] for a more general statement.4onsider open neighborhoods U ⊂ R k of H = { ¯ x n } n ∈ Z ∪ { ¯ ξ } and Λ ⊂ Λ of¯ λ , respectively. Our main interest is in the dynamics on the maximal invariantset M ( U, Λ) = { ( x, λ ) ∈ U × Λ : f n ( x, λ ) ∈ U ∀ n ∈ Z } . As in the transversal case, we will still work with the subshift (Ω N , β ) but theconjugacy (5) will be replaced by a set of bifurcation equations. For any s ∈ Ω N define the index set I ( s ) = { n ∈ Z : s n = 1 } , (9)and note that I : Ω N → Z ( N ) ⊂ Z is bijective, where Z ( N ) = { J ⊂ Z : | j − k | ≥ N ∀ j, k ∈ J, j = k } . (10)With any s ∈ Ω N we associate the Banach space ℓ ∞ ( s ) = { τ ∈ R I ( s ) : k τ k ∞ < ∞} , k τ k ∞ = sup ℓ ∈ I ( s ) | τ ( ℓ ) | , B ρ = { τ ∈ ℓ ∞ ( s ) : k τ k ∞ ≤ ρ } . Our aim is to determine the elements of M ( U, Λ) from a set of bifurcationequations g s ( τ, λ ) = 0 , τ ∈ B ρ τ , λ ∈ Λ , (11)where s ∈ Ω N , ρ τ > s and g s : B ρ τ × Λ → ℓ ∞ ( s )( τ, λ ) g s ( τ, λ )is a sufficiently smooth map. Note that (11) constitutes a finite or an infinitesystem of equations depending on the cardinality of I ( s ).In order to formulate the precise statement we define the pseudo orbits p n ( s ) = ¯ ξ + X ℓ ∈ I ( s ) (¯ x n − ℓ − ¯ ξ ) , n ∈ Z . (12)Equation (7) shows that p Z ( s ) is a bounded sequence, in particular k p n ( s ) − ¯ ξ k ≤ ¯ C = C e − α − e − α , n ∈ Z . (13)Setting ¯ ξ n = ¯ ξ for all n ∈ Z we write (12) more formally as p Z ( s ) = ¯ ξ Z + X ℓ ∈ I ( s ) β − ℓ (¯ x Z − ¯ ξ Z ) . Here and it what follows we use the symbol β to denote the shift of sequences in R k . Thus β acts as an operator in sequence spaces such as ℓ p ( R k ) , ≤ p ≤ ∞ .5imilarly, for every τ ∈ ℓ ∞ ( s ) we define the bounded sequence v Z ( s, τ ) = X ℓ ∈ I ( s ) τ ℓ β − ℓ u Z . (14)Note that the sequence p Z ( s ) has humps at the positions defined by I ( s ) and that p Z ( s ) is a pseudo orbit of f ( · , ¯ λ ) with a small error, see Lemma 12. The term v Z ( s, τ ) shifts the solution of the variational equation to the positions defined by I ( s ) and combines them linearly. Theorem 1
Let assumptions
A1 - A3 and B4 hold. Then there exist constants < r τ ≤ ρ τ , N ∈ N and neighborhoods U of H , Λ of ¯ λ and for any s ∈ Ω N smooth functions g s : B ρ τ × Λ → ℓ ∞ ( s ) ,x Z ,s : B ρ τ × Λ → ℓ ∞ ( R k ) with the following properties.(i) For any point ( y , λ ) ∈ M ( U, Λ) with orbit y n = f n ( y , λ ) , n ∈ Z thereexists an index ν ∈ Z and elements s ∈ Ω N , τ ∈ B ρ τ ⊂ ℓ ∞ ( s ) such that β ν y Z = x Z ,s ( τ, λ ) + p Z ( s ) + v Z ( s, τ ) , (15) g s ( τ, λ ) = 0 . (16) (ii) Conversely, if s ∈ Ω N , τ ∈ B r τ ⊂ ℓ ∞ ( s ) , λ ∈ Λ satisfy (16) , then thereexists ν ∈ Z such that ( y n , λ ) n ∈ Z , with y Z given by (15) , belongs to M ( U, Λ) . Remark 2
Theorem 1 reduces the study of M ( U, Λ) to the set of bifurcationequations (16) with a symbolic index s ∈ Ω N . It may be regarded as a type ofLiapunov-Schmidt reduction though we have not formally put it into this frame-work. The construction of the neighborhood U × Λ uses some features from thetransversal case [21, Theorem 5.1], but is considerably more involved, see Sections5 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 thefunctions x Z ,s and g s on the symbolic sequence s with respect to the metric (4) . The functions g s and x Z ,s have several properties that we discuss next.Due to B4 the adjoint equation y Tn +1 f x (¯ x n +1 , ¯ λ ) = y Tn , n ∈ Z (17)has a non-trivial solution w Z that is unique up to constant multiples, cf. [22,Section 2]. It decays exponentially as in (7) and can thus be normalized suchthat k w Z k ℓ = 1. Without loss of generality we take C e in (7) such that k w n k ≤ C e e − α | n | , n ∈ Z . (18)6s is shown in [15] the quantities c λ = h w Z , ( f λ (¯ x n , ¯ λ )) n ∈ Z i , c x = 12 h w Z , ( f xx (¯ x n , ¯ λ ) u n ) n ∈ Z i (19)characterize the behavior of the branch of homoclinic orbits which passes through(¯ x Z , ¯ λ ). Theorem 3
The operator F : ℓ ∞ ( R k ) × R → ℓ ∞ ( R k ) defined by F ( x Z , λ ) = (cid:0) ( x n +1 − f ( x n , λ )) n ∈ Z (cid:1) (20) has a limit point at (¯ x Z , ¯ λ ) in the sense that F (¯ x Z , ¯ λ ) = 0 and N ( D x F (¯ x Z , ¯ λ )) = span { u Z } . The limit point is transversal, i.e. D λ F (¯ x Z , ¯ λ ) / ∈ R ( D x F (¯ x Z , ¯ λ )) if and only if c λ = 0 . Moreover, it is a quadratic turning point, i.e. D x F (¯ x Z , ¯ λ ) u Z / ∈ R ( D x F (¯ x Z , ¯ λ )) if and only if c x = 0 . Remark 4
For transversal homoclinic orbits ( λ = ¯ λ ), the Sacker-Sell spectrum,cf. [24], of the variational equation (2) is a pure point spectrum Σ ED ( λ ) = {| µ | : µ ∈ σ ( f x ( ξ ( λ ) , λ )) } . At a turning point, we find a spectral explosion to a continuous Sacker-Sell spec-trum Σ ED (¯ λ ) = {| µ | : µ ∈ σ ( f x ( ¯ ξ, ¯ λ )) } ∪ [ µ s , µ u ] , where µ s = max {| µ | < µ ∈ σ ( f x ( ¯ ξ, ¯ λ )) } , µ u = min {| µ | > µ ∈ σ ( f x ( ¯ ξ, ¯ λ )) } . Our second result shows that the constants c λ , c x play an important role inthe behavior of the bifurcation function g s ( τ, λ ). Theorem 5
Let the assumptions of Theorem 1 hold. Then the functions x Z ,s , g s have the following properties(i) x Z ,βs ( βτ, λ ) = βx Z ,s ( τ, λ ) , where I ( βs ) = I ( s ) − , (21) g βs ( βτ, λ ) = βg s ( τ, λ ) . (22)7 ii) For some constants C > , α > , independent of s, N and ℓ ∈ I ( s ) (cid:12)(cid:12) g s ( τ, λ ) ℓ − ( c λ ( λ − ¯ λ ) + c x τ ℓ ) (cid:12)(cid:12) ≤ C (cid:0) ( λ − ¯ λ ) + | λ − ¯ λ |k τ k ∞ + k τ k ∞ + e − αN/ (cid:1) . (23) Remark 6
In order to apply unfolding theory to the bifurcation equations (16) one also needs estimates of derivatives in (23) . Though the functions g s ( · , · ) come out smoothly from Theorem 10, estimating their derivatives seems to bequite 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 the theorem shows that the bifurcation equations are small perturbationsof a set of K identical turning point equations0 = c λ ( λ − ¯ λ ) + c x τ ℓ , ℓ ∈ I ( s ) . If c λ , c x = 0 one can shift ¯ λ to zero and scale λ and τ ℓ such that one obtains aset of hilltop bifurcations of order K , cf. [6, Ch. IX, § λ − τ ℓ , ℓ ∈ I ( s ) . (24)For two-humped orbits the set I ( s ) contains two elements and the solution curvesof (24) are shown in Figure 7. This case will be crucial for understanding theglobal behavior of homoclinic curves in the next sections. A typical example, which plays the role of a normal form for quadratic two-dimensional mappings, is the famous H´enon map, cf. [11, 19, 5, 10] which isdefined as f ( x, λ ) = (cid:18) x − λx . x (cid:19) . This map has fixed points ξ ± ( λ ) = (cid:18) ν ( λ )1 . ν ( λ ) (cid:19) , where ν ( λ ) = 15 λ (cid:16) ± √ λ (cid:17) , and for ˜ λ = 0 .
35 a transversal homoclinic orbit x Z (˜ λ ) w.r.t. the fixed point ξ + (˜ λ )exists, satisfying Assumption A4 .For numerical computations, we approximate an infinite homoclinic orbit x Z (˜ λ ) by a finite orbit segment x J , where J = [ n − , n + ] ∩ Z . The segment isdetermined as a zero of the boundary value operatorΓ J ( x J , ˜ λ ) = (cid:18) x n +1 − f ( x n , ˜ λ ) , n = n − , . . . , n + − b ( x n − , x n + ) (cid:19) . b : R k → R k defines a boundary condition, for example b per ( x, y ) = x − y, or b proj ( x, y ) = (cid:18) B s ( x − ¯ ξ ) B u ( x − ¯ ξ ) (cid:19) , in case of periodic and projection boundary conditions, where B s and B u yieldlinear approximations of the stable and the unstable manifold. Due to our hy-perbolicity assumption, Γ J ( · , ˜ λ ) has for J sufficiently large a unique zero in aneighborhood of the exact solution. Moreover approximation errors decay at anexponential rate that depends on the type of boundary condition, cf. [4].For H´enon’s map, we solve the corresponding boundary value problem, obtainin this way an approximation of x Z (˜ λ ) and continue this orbit w.r.t. the parameter λ , using the method of pseudo arclength continuation, cf. [14, 1, 8]. In Figure 1,we plot the amplitude of these orbits amp( x J ( λ )) := (cid:0)P n ∈ J k x n ( λ ) − ξ + ( λ ) k (cid:1) versus the parameter. PSfrag replacements 0123 ℓ , = ( x , Z , λ , ) ℓ , = ( x , Z , λ , ) r , = ( x , Z , λ , ) r , = ( x , Z , λ , ) λ ampFigure 1: Continuation of homoclinic H´enon orbits. At the parameter ˜ λ = 0 . four distinct orbits x i Z , i ∈ { , . . . , } exist that turn into each other via left orright turning points. At the value ˜ λ = 0 .
35 four distinct homoclinic orbits occur that we denoteby x i Z , i ∈ { , . . . , } . We choose their index by following the order given by thecontinuation routine. The orbit x Z is shown in Figure 2 together with parts ofthe stable and the unstable manifold of the fixed point ξ + (˜ λ ). The enlargementin this figure shows where the four orbits lie in the intersection of manifolds.At each turning point, two orbits collide; with r and ℓ , we distinguish rightand left turning points. Figure 3 illustrates intersections of stable and unstablemanifolds at these four turning points.Errors of turning point calculations for finite approximations of homoclinic9 PSfrag replacements 0 123 ξ + x Z Figure 2:
Primary homoclinic orbit x Z w.r.t. the fixed point ξ + , and parts of thestable manifold (green) and the unstable manifold (red). The enlargement showsthe intersections of manifolds that lead to the four homoclinic orbits in Figure 1. orbits decay exponentially fast w.r.t. the length of the computed orbit segment,cf. [16, Theorem 5.1.1]. For H´enon’s map, we find four distinct transversal homoclinic orbits x s Z , s ∈{ , . . . , } at ˜ λ = 0 .
35 and we identify x Z ( s ) with its symbol s . Note that theorbits 0 , , , , r , , ℓ , , r , , ℓ , , see Figure 3. The graph in Figure 4 gives an alternative illustrationof these transitions.PSfrag replacements 0 1 2 3 RR LL
Figure 4:
Transition graph for one-humped orbits .For the construction of an n -humped orbit, we choose a sufficiently long in-terval J = [ n − , n + ] around zero and a sequence s ∈ S n := { , . . . , } n . We define10Sfrag replacements ℓ , ℓ , r , r ,
01 233 0 , , , , Intersections of stable and unstable manifolds at the four turning pointsin the cutout region from Figure 2. the pseudo orbit ˜ x Z [ s ] := x s ( −∞ ,n + ] x s J . . . x s n − J x s n [ n − , ∞ ) , (25)see Figure 5. Since the collection of single orbits x r Z , r ∈ { , . . . , } forms ahyperbolic set, the Shadowing-Lemma, cf. [23, 21] shows that the pseudo orbit˜ x Z ( s ) lies close to a true n -humped f -orbit which we denote by x Z ( s ). In S n there are 4 n different symbols and thus we expect to find 4 n different n -humpedorbits x Z ( s ). We identify these orbits with their symbol.Note that the construction of pseudo orbits in (25) slightly differs from (12),where we add up shifted orbits. With both approaches, we expect to find thesame shadowing orbit for sufficiently large intervals J .11Sfrag replacements x Z x Z x Z x Z x Z [01] x Z [123]Figure 5: Construction of multi-humped orbits.
Given two symbols s, ¯ s ∈ S n , we analyze whether the n -humped orbits x Z ( s )and x Z (¯ s ) can turn into each other via continuation.Let us first look at the two-humped case. The continuation of two-humped orbits exhibits three closed curves of homoclinicorbits, cf. Figure 6, and at the parameter value ˜ λ , 16 different homoclinic orbits x Z ( s ), s ∈ S exist. PSfrag replacements λ ampFigure 6: Continuation of two-humped orbits of length n − = − , n + = 21 . One observes that at each turning point in Figure 6, exactly one componentof the symbol changes. For example, the symbol (1 ,
1) changes at a left turningpoint into the symbol (2 , decides , whether (1 ,
1) bifurcates into the symbol(2 ,
1) or into (1 , λ, τ , τ , called the hilltop normal form, cf. [6] λ = τ , λ = τ . (26)Figure 7 (left) shows the solution curves of (26) while the red curves in Figure7 (right) indicate the generic solution picture of a perturbed equation. Herewe neglect more detailed bifurcation diagrams which take into account smallhysteresis effects w.r.t. the parameter λ , see [6, Ch. IX, §
3] for the unfoldingtheory.PSfrag replacements21 (2 , , ,
1) (1 , ,
2) (1 , ,
2) (2 , τ + τ τ + τ τ − τ τ − τ λλ Figure 7:
Unperturbed (left) and perturbed hilltop-bifurcation (right) at the turn-ing point ℓ , . Homoclinic orbits that lie on a common closed curve define a connected compo-nent of H := (cid:8) ( y Z , λ ) ∈ ℓ ∞ ( R k ) × R : y n +1 = f ( y n , λ ) ∀ n ∈ Z , lim n →±∞ y n = ξ ( λ ) (cid:9) . More precisely, let s ∈ S n and denote by C ( s ) ⊂ H the connected componentthat satisfies ( x Z ( 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. s ∼ = ¯ s ⇔ ( x Z (¯ s ) , ˜ λ ) ∈ C ( s ) . (27)In the following, we discuss how to find these equivalence classes. Particularly,we show under some generic assumptions that each equivalence class has at least13our elements, and for n -humped orbits it turns out that one class has at least4 n elements.For this task, we introduce a labeled graph G with vertices s ∈ S n . Twovertices s and ¯ s ∈ S n are connected with an L or R -edge, if x Z ( s ) bifurcates into x Z (¯ s ) via a left or right turning point. Since we do not know the effect of theperturbed hilltop bifurcation a priori, we put an edge, if the transition is possiblefor at least one perturbation. For example, the vertices (1 ,
1) and (2 ,
1) as wellas (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. Con-sequently, all symbols lie in the same equivalence class, which matches the factthat all one-humped orbits lie on the same closed curve and thus, in the sameconnected component of H . In this section, we give precise rules for defining the labeled graph G which weidentify with its adjacency tensor with entries R and L .First, we assume that only one of the n humps can turn into a neighboringhump at a turning point. R1 There is no edge from s ∈ S n to ¯ s ∈ S n if s = ¯ s or k s − ¯ s k = n X i =1 d ( s i , ¯ s i ) ≥ , where d is the distance on the cycle 01230.Now let s, ¯ s ∈ S n and assume k s − ¯ s k = 1, then there exists a unique j suchthat s j = ¯ s j .From λ , > λ , we conclude that the right transition at r , can only occurif the orbit contains no 0 and no 1 hump. Therefore, we define R -edges in G according to the following rule. R2 G ( s, ¯ s ) = R if • { s j , ¯ s j } = { , } , • { s j , ¯ s j } = { , } and s i ∈ { , } for all i = 1 , . . . , n .Similarly from λ , > λ , we conclude that the left transition at ℓ , can onlyoccur if the orbit contains no 1 and no 2 hump. Our rules for L -edges are: R3 G ( s, ¯ s ) = L if 14 { s j , ¯ s j } = { , } , • { s j , ¯ s j } = { , } and s i ∈ { , } for all i = 1 , . . . , n .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 hy-pothesis is as follows. Hypothesis 7
The connected components of n -humped orbits and thus the equiv-alence classes (27) are in one to one correspondence to a partition of G intodisjoint LR -cycles. In case n = 2, the L and R -edges are shown in the left and center picture ofFigure 8. The right diagram additionally shows the LR -cycles that correspondto the connected components from Figure 6.Similar diagrams for n = 3 are shown in Figure 9. −0.5 0 0.5 1 1.5 2 2.5 3 3.5−0.500.511.522.533.5−0.5 0 0.5 1 1.5 2 2.5 3 3.5−0.500.511.522.533.5 −0.5 0 0.5 1 1.5 2 2.5 3 3.5−0.500.511.522.533.5 PSfrag replacements 0000 00 1111 11 2222 22 3333 33Figure 8: L -edges of G (left) and R -edges (center) for two-humped orbits. Thecycles in the right figure correspond to the closed curves that are computed nu-merically in Figure 6. PSfrag replacements 000130 430 203Figure 9: L -edges of G (left) and R -edges (center) for three-humped orbits. Thegreen lines in the right figure show the cycles that are computed numerically.
15e continued n -humped orbits numerically for H´enon’s map up to n = 5.Table 1 summarizes the number of cycles and their lengths found in the compu-tation. length of cycle n Continuation of n -humped orbits – length of occurring cycles in numer-ical experiments. These experiments for n -humped orbits of H´enon’s map suggest that thelength of occurring cycles is a multiple of 4. Furthermore one cycle exists ofat least length 4 n . Theorem 8
Fix n ∈ N and assume that Hypothesis 7 holds true. Then, all n -humped orbits lie on cycles whose length is a multiple of .For the specific symbols s = (0 , . . . , and s = (2 , . . . , ∈ S n , the corre-sponding orbits x Z ( s ) and x Z ( s ) lie on a common cycle of at least length n . Remark 9
Table 1 shows that there is exactly one orbit of length n and allother orbits are shorter. Hence, the orbit of length n contains s and s . Proof:
By assuming Hypothesis 7 we see that it suffices to analyze LR -cycles ofthe graph G . More precisely, we prove Theorem 8 along the following steps.(i) Each LR -cycle in G has length 4 m , m ≥ m ∈ N .(ii) There exists an LR -cycle from s to s of length 4 n and each LR -cycle thatcontains s and s has at least length 4 n .(iii) Each LR -cycle that contains s also contains s .(i) Let the vertices v , . . . , v m , v m +1 = v form an LR -cycle in G . Fix j ∈{ , . . . , n } and let l j = (cid:8) i ∈ { , . . . , m } : v ij = v i +1 j , G ( v i , v i +1 ) = L (cid:9) bethe number of L -edges for which the corresponding vertices only differ inthe j th component.From R1 it follows that l j is an even number and max j =1 ,...,n { l j } ≥ j th component. Furthermore,16n LR -cycle has the same number of L and R -edges. Thus, the cycle haslength m = n X i =1 l i = 4 n X i =1 l i p with p = n X i =1 l i ≥ . (ii) We explicitly construct an LR -cycle in G from s to s : (0 · · · (10 · · · (20 · · · (210 · · · (220 · · ·
7→ · · · 7→ (2 · · · (32 · · · (312 · · · (302 · · · (3012 · · · (3002 · · · (30 · · · (0 · · ·
0) which has length 4 n . Note that the distance from s to s onthe full quadratic grid is 2 n and consequently, each cycle containing thesetwo points has at least length 4 n .(iii) For proving that each LR -cycle with s also contains s , assume that an LR -cycle exists that contains s but not s . This cycle lies in the subgraphthat we obtain by deleting the vertex s with its corresponding edges.Note that the LR -cycles start with an L and end with an R -edge, and itfollows from R2 that all R -edges that start at s end in V := { s ∈ S n : ∃ j ∈ { , . . . , n } : s j = 3 } . We define the graph ˜ G by deleting the R -edges of s from the remaininggraph. Figure 10 illustrates this construction in case n = 2.PSfrag replacements 0 01 12 23 3 s s Figure 10:
Transition graph in case n = 2 . L -edges and R -edges are plotted inred and black, respectively. Vertices and edges that are deleted in the proof ofTheorem 8 are marked by green crosses. If ˜ G breaks into two components V and ˜ G \ V , then we get a contradictionto the above assumption and an LR -cycle in the original graph G thatcontains s but not s cannot exist.17o finish the proof, we show that an LR -path in ˜ G from s to V does notexist.From s we cannot go directly via an R -edge to V , since the correspondingedges are deleted in ˜ G . Thus without loss of generality, we get: (2 · · · (12 · · · (02 · · · v m the m th vertex on this path. A 3-component can only be achieved by the left transition ℓ , or by the righttransition r , , see R2 and R3 .If a j exists with v mj ∈ { , } , then the r , transition is impossible by R2 .If a j exists with v mj ∈ { , } , then the ℓ , transition is impossible by R3 .Thus, we only obtain a 3 component via the vertex s = (0 . . .
0) which isdeleted in ˜ G . (cid:4) In this section we prove the main Theorem 1 and Theorem 5(i) by using anexistence and uniqueness result for a suitable operator equation in spaces ofbounded 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. First recall the operator F : ℓ ∞ ( R k ) × R → ℓ ∞ ( R k ) from (20) and the normal-ization ¯ λ = 0 and ξ ( λ ) = 0 for λ close to ¯ λ , see A2 , A3 in Section 2. Then forany s ∈ Ω N define the operator G s : ℓ ∞ ( R k ) × ℓ ∞ ( s ) × ℓ ∞ ( s ) × R → ℓ ∞ ( R k ) × ℓ ∞ ( s )by G s ( x Z , g, τ, λ ) = (cid:18) F ( p Z ( s ) + x Z + v Z ( s, τ ) , λ ) + w ( s, g ) h β − ℓ u Z , x Z i , ℓ ∈ I ( s ) (cid:19) . (28)Here p Z , v Z are defined in (12), (14) and w ( s, g ) is given by (recall w Z from (17)) w ( s, g ) = X ℓ ∈ I ( s ) g ℓ β − ℓ w Z , g ∈ ℓ ∞ ( s ) . Our aim is to derive the functions x Z ,s , g s in Theorem 1 by solving G s ( x Z , g, τ, λ ) = 0 (29)for k τ k ∞ , | λ | sufficiently small and for all s ∈ Ω N . More precisely, we prove inSection 6 the following Reduction Theorem .18 heorem 10
There exist constants C , ρ x , ρ g , ρ τ , ρ λ > and a number N ∈ N such that for all N ≥ N and for all s ∈ Ω N the following statements hold. Forall τ ∈ B ρ τ , λ ∈ B ρ λ the system (29) has a unique solution g = g s ( τ, λ ) ∈ B ρ g ⊂ ℓ ∞ ( s ) , x Z = x Z ,s ( τ, λ ) ∈ B ρ x ⊂ ℓ ∞ ( R k ) . Moreover, the following estimate is satisfied: max( k g − ˜ g k ∞ , k x Z − ˜ x Z k ∞ ) ≤ C k G s ( x Z , g, τ, λ ) − G s (˜ x Z , ˜ g, τ, λ ) k (30) for all g, ˜ g ∈ B ρ g , x Z , ˜ x Z ∈ B ρ x , τ ∈ B ρ τ , λ ∈ B ρ λ . In order to construct the neighborhoods U and Λ in Theorem 1 we need severallemmata. Lemma 11
There exists N ∈ N , such that for all N ≥ N , s ∈ Ω N the linearsystem X k ∈ I ( s ) h β − ℓ u Z , β − k u Z i τ k = r ℓ , ℓ ∈ I ( s ) , r ∈ ℓ ∞ ( s ) (31) has a unique solution τ ∈ ℓ ∞ ( s ) and k τ k ∞ ≤ k r k ∞ . (32) Proof:
Rewrite (31) as fixed point equation τ = P τ + r, ( P τ ) ℓ = − X k ∈ I ( s ) ,k = ℓ h β − ℓ u Z , β − k u Z i τ k and note k P τ k ∞ ≤ k τ k ∞ sup ℓ ∈ I ( s ) X k ∈ I ( s ) ,k = ℓ |h u Z , β ℓ − k u Z i|≤ k τ k ∞ C X j ≥ e − αjN = C e − αN − e − αN k τ k ∞ . Thus, we choose N such that C e − αN − e − αN ≤ . Then P is contractive and (32)follows. (cid:4) Lemma 12
There exist N ∈ N , C > such that k F ( p Z ( s ) + v Z ( s, τ ) , λ ) k ∞ ≤ C ( | λ | + k τ k ∞ + e − αN/ ) (33) for all N ≥ N , s ∈ Ω N , | λ | ≤ , τ ∈ B ⊂ ℓ ∞ ( s ) . roof: We estimate p n +1 ( s ) + v n +1 ( s, τ ) − f ( p n ( s ) + v n ( s, τ ) , λ )= X ℓ ∈ I ( s ) (¯ x n +1 − ℓ + τ ℓ u n +1 − ℓ ) − f ( p n ( s ) + v n ( s, τ ) ,
0) + O ( | λ | )= X ℓ ∈ I ( s ) (cid:0) f (¯ x n − ℓ ,
0) + τ ℓ f x (¯ x n − ℓ , u n − ℓ (cid:1) − f (cid:0) X ℓ ∈ I ( s ) ¯ x n − ℓ + X ℓ ∈ I ( s ) τ ℓ u n − ℓ , (cid:1) + O ( | λ | )= X ℓ ∈ I ( s ) f (¯ x n − ℓ ,
0) + X ℓ ∈ I ( s ) τ ℓ f x (¯ x n − ℓ , u n − ℓ − f (cid:0) X ℓ ∈ I ( s ) ¯ x n − ℓ , (cid:1) − f x (cid:0) X j ∈ I ( s ) ¯ x n − j , (cid:1) X ℓ ∈ I ( s ) τ ℓ u n − ℓ + O ( | λ | + k τ k ∞ ) . For n ∈ Z , choose ˜ ℓ ∈ I ( s ) such that | n − ˜ ℓ | ≤ | n − ℓ | for all ℓ ∈ I ( s ). Then | n − ℓ | ≥ N for all ℓ = ˜ ℓ and, therefore, by (7), (cid:13)(cid:13)(cid:13) X ℓ ∈ I ( s ) f (¯ x n − ℓ , − f (cid:0) X ℓ ∈ I ( s ) ¯ x n − ℓ , (cid:1)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) X ℓ ∈ I ( s ) ,ℓ =˜ ℓ f (¯ x n − ℓ , (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) f (¯ x n − ˜ ℓ , − f (cid:0) X ℓ ∈ I ( s ) ¯ x n − ℓ , (cid:1)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) X ℓ ∈ I ( s ) ,ℓ =˜ ℓ ¯ x n +1 − ℓ (cid:13)(cid:13)(cid:13) + L (cid:13)(cid:13)(cid:13) X ℓ ∈ I ( s ) ,ℓ =˜ ℓ ¯ x n − ℓ (cid:13)(cid:13)(cid:13) ≤ C e − αN/ . (34)In a similar way, X ℓ ∈ I ( s ) τ ℓ f x (cid:0) X j ∈ I ( s ) ¯ x n − j , (cid:1) u n − ℓ = X ℓ ∈ I ( s ) τ ℓ f x (¯ x n − ℓ , u n − ℓ + O (e − αN/ k τ k ∞ ) . (35)Combining these estimates, we obtain (33). (cid:4) Lemma 13
Assume A1 , A2 and let ¯ x Z be a homoclinic f ( · , -orbit with respectto the hyperbolic fixed point . Then, there exist zero neighborhoods U ⊂ R k , Λ ⊂ Λ and constants N , n , α > , C ≥ such that the following statementholds for all K ≥ N , − n − , n + ≥ n :If x n +1 = f ( x n , λ ) for n ∈ ˜ J := [0 , K − , λ ∈ Λ , and if x n ∈ U for all n ∈ J := [0 , K ] then we have the estimate sup j ∈ J k x j − ¯ x n − +1 − K + j − ¯ x n + − j k ≤ C (cid:16) k x K − ¯ x n − +1 k + k x − ¯ x n + − k + | λ | + e − α K (cid:17) . Furthermore we obtain in case K = ∞ : sup j ≥ k x j − ¯ x n + − j k ≤ C ( k x − ¯ x n + − k + | λ | ) . (36)20 roof: Consider the pseudo-orbit p j := ¯ x n + − j + ¯ x n − +1 − K + j , j ∈ J which is azero of the boundary value operatorΓ J ( y J , λ ) := (cid:18) y n +1 − f ( y n , λ ) − ρ n , n ∈ ˜ Jb K ( y , y K ) (cid:19) , at λ = 0, where ρ n := f (¯ x n + − n ,
0) + f (¯ x n − +1 − K + n , − f ( p n , , n ∈ J,b K ( y , y K ) := (cid:18) P s ( y − p ) P u ( y K − p K ) (cid:19) . Here P s and P u are the stable and unstable projectors of the fixed point 0.Let b be the bound from Theorem 21 for the difference equation u n +1 = ( f x (0 ,
0) + B n ) u n , B n = f x ( p n , λ ) − f x (0 , , n ∈ ˜ J .
For sufficiently large − n − , n + ≥ n and λ ∈ Λ sufficiently small, we get k B n k ≤ 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 sn , P un and an exponentialrate α that is independent of n − , n + , λ and K .As in the proof of [12, Theorem 4], we show that for λ ∈ Λ , n − , n + ≥ n and K ≥ N we have a uniform bound k D Γ J ( p J , λ ) − k ∞ ≤ σ − . (37)In order to see this, consider the inhomogeneous difference equation u n +1 − f x ( p n , λ ) u n = r n , n = 0 , . . . K − , (38) P s u + P u u K = γ. (39)Denote by Φ the solution operator of the homogeneous equation and let G be thecorresponding Green’s function, cf. (87). The general solution of (38) is given by u n = Φ( n, v + X m ∈ ˜ J G ( n, m + 1) r m , (40)where v = v − + Φ(0 , K ) v + , v − ∈ R ( P s ) , v + ∈ R ( P uK ) . Inserting (40) into (39), it remains to solve P s (cid:0) v − + Φ(0 , K ) v + (cid:1) + P u (cid:0) Φ( K, v − + v + (cid:1) = R, R = γ − P s X m ∈ ˜ J G (0 , m + 1) r m − P u X m ∈ ˜ J G ( K, m + 1) r m . This finite-dimensional system has a unique solution for K ≥ N sufficiently largesince k P s − P s k → k P u − P uK k → K → ∞ . Therefore, the system (38),(39) also has a unique solution u J for K large and the dichotomy estimates leadto a bound k u J k ∞ ≤ σ − ( | γ | + k r J k ∞ ) , i.e. (37) holds.We apply Theorem 19 with the space Y = ℓ ∞ J = { ( y n ) n ∈ J : y n ∈ R k } of finitesequences and with Z = ℓ ∞ ˜ J × R k , both endowed with the sup-norm. We take y = p J and use uniform data for all λ ∈ Λ . For δ sufficiently small we have k D Γ J ( x J , λ ) − D Γ J ( p J , λ ) k ∞ ≤ σ x J ∈ B δ ( p J ) , λ ∈ Λ , and by choosing the neighborhood Λ sufficiently small we get k Γ J ( p J , λ ) k ∞ = sup n ∈ ˜ J k p n +1 − f ( p n , λ ) − ρ n k + k b K ( p , p K ) k = sup n ∈ ˜ J k p n +1 − f ( p n , λ ) + f ( p n , − f (¯ x n + − n , − f (¯ x n − +1 − K + n , k = sup n ∈ ˜ J k f ( p n , − f ( p n , λ ) k ≤ σ δ for λ ∈ Λ . Theorem 19 applies to λ ∈ Λ with uniform data, and it follows from (119) withsome constant C > k x J − y J k ∞ ≤ C k Γ J ( x J , λ ) − Γ J ( y J , λ ) k ∞ for all x J , y J ∈ B δ ( p J ) , λ ∈ Λ . (41)From (7) we find a number n such that ¯ x n ∈ B δ (0) for all | n | ≥ n and also p n ∈ B δ (0) , n ∈ J for all − n − , n + ≥ n . Then we take U := B δ (0) as ourneighborhood and note that (41) holds for any two sequences x J , y J in U . For n ∈ ˜ J and λ ∈ Λ it follows that f ( p n , λ ) − p n +1 = f (¯ x n + − n + ¯ x n − +1 − K + n , λ ) − ¯ x n + − n +1 − ¯ x n − +1 − K + n +1 = f (¯ x n + − n + ¯ x n − +1 − K + n , − f (¯ x n + − n , − f (¯ x n − +1 − K + n ,
0) + O ( | λ | )= ( − f (¯ x n − +1 − K + n ,
0) + O ( k ¯ x n − +1 − K + n k ) for 0 ≤ n ≤ K − f (¯ x n + − n ,
0) + O ( k ¯ x n + − n k ) for K < n ≤ K ) + O ( | λ | )= O (e − α K + | λ | ) . x J be a sequence in U such that x n +1 = f ( x n , λ ) for all n ∈ ˜ J , andsome λ ∈ Λ . Then k x J − p J k ∞ ≤ C k Γ J ( x J , λ ) − Γ J ( p J , λ ) k ∞ ≤ C (cid:16) sup n ∈ ˜ J k f ( p n , λ ) − p n +1 k + k b K ( x , x K ) k (cid:17) ≤ C (cid:18) e − α K + | λ | + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) P s ( x − p ) P u ( x K − p K ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:19) ≤ C (cid:16) e − α K + k x − p k + k x K − p K k + | λ | (cid:17) ≤ C (cid:16) e − α K + k x − ¯ x n + − k + k x K − ¯ x n − +1 k + | λ | (cid:17) . In case K = ∞ , one uses the operator˜Γ N ( y N , λ ) = (cid:18) y n +1 − f ( y n ) , n ∈ N P s ( y − ¯ x n + − ) (cid:19) , and it turns out that D ˜Γ N ( p N , λ ) with p j = ¯ x n + − j has a uniformly boundedinverse for λ ∈ Λ and sufficiently large − n − , n + . Then the estimate (36) followsimmediately. (cid:4) Let us first prove assertion (i) in Theorem 1.
Step 1: (Construction of neighborhoods U, Λ)In the following Λ ⊃ Λ ⊃ . . . will denote shrinking neighborhoods of 0. Let ρ g , ρ x > ρ τ , ρ λ with-out changing the assertion of Theorem 10. Introduce the constants (cf. (7) andLemma 12, 13) α ∗ = e − α , C = C + 2 C e − α ∗ , C = C e − α ∗ − α ∗ , C = C C + 2 C e − α ∗ . (42)Let ρ τ > C ρ τ ≤ ρ x . (43)By Lemma 13 we can choose a ball B ε ⊂ U and numbers n + , − n − ≥ n suchthat ¯ x n ∈ B ε for all n ≥ n + − n ≤ n − + 1. It is well known that the onlyfull orbit in a small neighborhood of a hyperbolic fixed point is the fixed pointitself. That is, we can assume w.l.o.g. that U , Λ satisfy y n +1 = f ( y n , λ ) , y n ∈ U ( n ∈ Z ) , λ ∈ Λ ⇒ y n = 0 for all n ∈ Z . (44)23he set K = { } ∪ { ¯ x n : n ≤ n − or n ≥ n + } is compact and satisfies f ( K , ⊂ K ∪ { ¯ x n − +1 } . Thus we find an ε ≤ ε and Λ ⊂ Λ such that the following properties hold V := B ε ∪ [ n ≤ n − ,n ≥ n + B ε (¯ x n ) ⊂ B ε ⊂ U , (45)the balls B j := B ε (¯ x n − + j ) , j = 1 , . . . , κ := n + − − n − (46)are mutually disjoint, f ( V , Λ ) ∩ B j = ∅ for j = 2 , . . . , κ , (47)2 C (2 C + 1) ε ≤ ρ τ C + 1) ε ≤ ρ x , (48)see Figure 11.PSfrag replacements 0 ¯ x n − ¯ x n + ¯ x n − +1 ¯ x n + − V V V V κ − V κ B B B κ B κ − Figure 11:
Construction of neighborhoods.
Next we take N ≥ max( N , N , N ) (see Lemmata 11, 12, 13) such that2 C C e − αN ≤ ρ τ , C e − αN ≤ ρ x , (2( C e + C ) + 18 C C )e − αN ≤ ε . (49)We can also find Λ ⊂ Λ such that for all λ ∈ Λ C C | λ | ≤ ρ τ , C | λ | ≤ ρ x , | λ | ≤ ρ λ , C | λ | (3 C C + 1) ≤ ε. (50)Finally, we define N = κ + N ∗ , where N ∗ = 2 N . (51)24hen we choose Λ ⊂ Λ such that the following settings define neighborhoods V j of ¯ x n − + j , j = κ , . . . , V κ = B κ ∩ N ∗ +1 \ n =1 f − n ( V , Λ ) ,V j = B j ∩ f − ( V j +1 , Λ ) , for j = κ − , . . . , . (52)Here we use the notation f − n ( V j , Λ ) = { x : f n ( x, λ ) ∈ V j for all λ ∈ Λ } .With these settings we consider the maximal invariant set M ( U, Λ), cf. (3),that belongs to U = κ [ j =0 V j , Λ = Λ . (53)Let us note that our construction (46), (47), (52), (53) implies the following threeassertions for any f ( · , λ )-orbit y Z ⊂ U , λ ∈ Λ y n ∈ V , y n +1 / ∈ V ⇒ y n +1 ∈ V , (54) y n ∈ V j for some 1 ≤ j ≤ κ − ⇒ y n +1 ∈ V j +1 , (55) y n ∈ V κ ⇒ y n + ℓ ∈ V for ℓ = 1 , . . . , N ∗ + 1 . (56) Step 2: (Construction of symbolic sequence s )For some λ ∈ Λ consider an orbit y Z of (1) that lies in U . If it lies in V then y n = 0 , n ∈ Z by (44) and we set s = 0 ∈ Ω N . Otherwise we have y ˜ n / ∈ V forsome ˜ n ∈ Z . We show that˜ I ( y Z ) := { n ∈ Z : y n ∈ V } is nonempty and that there is a unique s ∈ Ω N such that ˜ I ( y Z ) = I ( s ), see (9).By our assumptions we have y ˜ n ∈ V j for some j ∈ { , . . . , κ } . If j = 1 then˜ I ( y Z ) = ∅ whereas in case j ≥ y ˜ n − m ∈ V j − m , m = 0 , . . . , j − V j are mutually disjoint. Therefore, wehave ˜ ℓ := ˜ n − j + 1 ∈ ˜ I ( y Z ). Moreover, from (54) and (56) we obtain y ˜ ℓ + j ∈ V j +1 , j = 0 , . . . κ − , y ˜ ℓ + j ∈ V , j = κ , . . . , κ + N ∗ . (57)This shows that the difference of two consecutive indices ˜ ℓ < ℓ in ˜ I ( y Z ) is at least N = κ + N ∗ . Therefore ˜ I ( y Z ) belongs to Z ( N ) (cf. (10)) and there is a uniquesequence s ∈ Ω N such that ˜ I ( y Z ) = I ( s ).The relations (57) hold whenever ℓ ∈ I ( s ). By (46) and (52) this gives us theestimates k y p + ℓ + ν − ¯ x p k ≤ ε for ℓ ∈ I ( s ) , p = n − + 1 , . . . , n + − ν = − n − − T n := k y n + ν − X m ∈ I ( s ) ¯ x n − m k ≤ C e − αN + C | λ | + (2 C + 1) ε for n ∈ Z . (59)Consider first indices n = p + ℓ with ℓ ∈ I ( s ) and p = n − + 1 , . . . , n + −
1. Thenwith (51) we find T n ≤ k y p + ℓ + ν − ¯ x p k + X m ∈ I ( s ) ,m = ℓ k ¯ x p + ℓ − m k≤ ε + C e X I ( s ) ∋ m<ℓ α p + ℓ − m ∗ + X I ( s ) ∋ m>ℓ α m − ℓ − p ∗ ≤ ε + C e X µ ≥ α µN + n − +1 ∗ + X µ ≥ α µN − n + +1 ∗ ! = ε + C e − α N ∗ (cid:0) α N + n − +1 ∗ + α N − n + +1 ∗ (cid:1) ≤ ε + 2 C e − α ∗ α N ∗ ∗ . (60)Next consider two consecutive indices ℓ < ˜ ℓ in I ( s ) and n = p + ℓ for p = n + − , . . . , ˜ p = ˜ ℓ − ℓ + n − + 1. For these indices we get y n + ν ∈ V κ , for p = n + − ,V , for p = ˜ p,V , otherwise , and we can apply Lemma 13 to this sequence in place of ˜ x , . . . , ˜ x K , where K =˜ ℓ − ℓ + 2 − ( n + − n − ) = ˜ ℓ − ℓ + 1 − κ ≥ N ∗ + 1 = 2 N + 1. With (58) this yieldsthe estimatesup j =0 ,...,K k y ℓ + ν + n + − j − ¯ x n + + j − − ¯ x n − + j − K +1 k≤ C (cid:0) k y ℓ + ν + n + − − ¯ x n + − k + k y ℓ + ν + n + − K − ¯ x n − +1 k + | λ | + α − K/ ∗ (cid:1) ≤ C (cid:0) ε + | λ | + α N ∗ (cid:1) . (61)Finally, we use this to estimate for n = p + ℓ and p = n + − , . . . , ˜ pT n ≤ k y n + ν − ¯ x n − ℓ − ¯ x n − ˜ ℓ k + X m ∈ I ( s ) ,m = ℓ, ˜ ℓ k ¯ x n − m k . The first term is handled by (61). Further note that X ˜ ℓ
1. Since y Z is an f ( · , λ ) orbit weconclude by induction from the definition (52) y p + ℓ + ν ∈ V p − n − , for p = n + − , . . . , n − + 1 . Therefore the sequence y Z lies in U which proves our assertion. Step 5: (Proof of Theorem 5 (i)) The proof of (21),(22) is easily accomplishedby noting the equivariance relations I ( βs ) = I ( s ) − , p Z ( βs ) = βp Z ( s ) , v Z ( βs, βτ ) = βv Z ( s, τ ) ,w ( βs, βg ) = βw ( s, g ) , F ( βx Z , λ ) = βF ( x Z , λ ) , and h β ℓ u Z , x Z i = h β ℓ +1 u Z , βx Z i for ℓ ∈ I ( βs ) = I ( s ) − . The assertion then follows by uniqueness from Theorem 10 since neighborhoodsare shift invariant as well.The proof of Theorem 5(ii) will be deferred to the next section.29
Proof of Reduction Theorem
According to (28) the Frechet derivative of G s w.r.t. x Z , g is given by D ( x,g ) G s ( x Z , g, τ, λ )( y Z , h ) = (cid:18) D x F ( p Z ( s ) + x Z + v Z ( s, τ ) , λ ) y Z + w ( s, h ) h β − ℓ u Z , y Z i , ℓ ∈ I ( s ) (cid:19) ,D x F ( x Z , λ ) y Z =( y n +1 − f x ( x n , λ ) y n ) n ∈ Z . (72)The key step in the proof of Theorem 10 will be a uniform bound for the inverseof D ( x,g ) G s (0 , , , Lemma 14
There exist constants ˆ C, ˆ N > such that for all N ≥ ˆ N , s ∈ Ω N the operator D ( x,g ) G s (0 , , , is invertible and satisfies k y Z k ∞ + k h k ∞ ≤ ˆ C k D ( x,g ) G s (0 , , , y Z , h ) k , y Z ∈ ℓ ∞ ( R k ) , h ∈ ℓ ∞ ( s ) . Before proving Lemma 14 we finish the proof of Theorem 10.
Proof:
Let L x and L λ be Lipschitz constants of the Jacobian f x ( x, λ ) with respectto x and λ in a compact ball that contains the homoclinic orbit in its interior.Then formula (72) and the bound (65) directly lead to the Lipschitz estimate k D ( x,g ) G s ( x Z , g , τ , λ ) − D ( x,g ) G s ( x Z , g , τ , λ ) k≤ L x ( k x Z − x Z k ∞ + C k τ − τ k ∞ ) + L λ | λ − λ | (73)for all x Z , x Z ∈ B ρ x , g , g ∈ ℓ ∞ , τ , τ ∈ B ρ τ , and λ , λ ∈ Λ with ρ x , ρ τ takensufficiently small. From Lemma 14 and Lemma 18 we obtain that the operators D ( x,g ) G s (0 , , τ, λ ) are invertible for τ ∈ B ρ τ , λ ∈ B ρ λ provided we choose N ≥ ˆ N and L x C ρ τ + L λ ρ λ ≤
12 ˆ
C .
Then we have k D ( x,g ) G s (0 , , τ, λ ) − k ≤ C, τ ∈ B ρ τ , λ ∈ B ρ λ . Now we apply Theorem 19 to every operator F = G s ( · , · , τ, λ ) in the spaces Y = Z = ℓ ∞ ( R k ) × ℓ ∞ ( s ). Setting σ =
12 ˆ C , y = 0 and taking L x ρ x ≤
14 ˆ C we findthat condition (116) is satisfied with κ =
14 ˆ C and δ = ρ x . Finally, we obtain fromLemma 12 for all N ≥ N , k G s (0 , , τ, λ ) k = k F ( p Z ( s ) + v Z ( s, τ ) , λ ) k ∞ ≤ C ( | λ | + k τ k ∞ + e − αN/ ) ≤ C ( ρ λ + ρ τ + e − αN/ ) . N ≥ max( ˆ N , N ) and ρ λ , ρ τ such that C ( ρ λ + ρ τ + e − αN ) ≤ ρ x C .Then we find k G s (0 , , τ, λ ) k ≤ ( σ − κ ) δ = ρ x C , for all N ≥ N , i.e. condition(117) is satisfied. An application of Theorem 19 finishes the proof. (cid:4) Remark 15
If instead of Theorem 19 we use a Lipschitz inverse mapping the-orem with smooth parameters (cf. [13, Appendix]) then it is easily seen that thesolutions g s , x Z ,s are smooth functions of τ and λ . Proof: (Theorem 5 (ii))
With c λ , c x from (19) define h = h ( τ, λ ) ∈ ℓ ∞ ( s ) by h ℓ = c λ λ + c x τ ℓ , ℓ ∈ I ( s ) . The idea is to construct elements y Z = y Z ( τ, λ ) ∈ ℓ ∞ ( R k ) and γ = γ ( τ, λ ) ∈ ℓ ∞ ( s )such that the residual G s ( p Z ( s ) + y Z + v Z ( s, τ ) , h + γ, τ, λ )and γ are of higher order than O ( | λ | + k τ k ∞ ). Then the assertion follows from(30) by comparing them to x Z ,s ( τ, λ ) , g s ( τ, λ ). We find y Z , γ by Taylor expansionof F (we abbreviate F x = F x ( p Z ( s ) ,
0) etc.) F ( p Z ( s ) + y Z + v Z ( s, τ ) , λ ) = F + F x ( y Z + v Z ( s, τ )) + F λ λ + 12 F xx ( y Z + v Z ( s, τ )) + F x,λ ( y Z + v Z ( s, τ )) λ + 12 F λλ λ + O (( | λ | + k τ k ∞ + k y Z k ∞ ) ) . From the estimates (34), (35) in the proof of Lemma 12 we have k F k ∞ = O (e − αN/ ) , k F x v Z ( s, τ ) k = O (e − αN/ k τ k ∞ ) . In a similar way, using Lipschitz constants for f λ and f xx we find( F λ ) n = − X ℓ ∈ I ( s ) f λ (¯ x n − ℓ ,
0) + O (e − αN/ )( F xx ( v Z ( s, τ )) ) n = − X ℓ ∈ I ( s ) τ ℓ f xx (¯ x n − ℓ , u n − ℓ ) + O (e − αN/ k τ k ∞ ) . Therefore, Taylor expansion of G s yields G s ( p Z ( s ) + y Z + v Z ( s, τ ) , h + γ, τ, λ ) = D ( x,g ) G s ( y Z , γ ) + ( ϕ Z , O (e − αN/ + k τ k ∞ | λ | + k τ k ∞ k y Z k ∞ + λ + k y Z k ∞ + k τ k ∞ ) , (74)where ϕ n = X ℓ ∈ I ( s ) λ ( − f λ (¯ x n − ℓ ,
0) + c λ w n − ℓ ) + τ ℓ ( − f xx (¯ x n − ℓ , u n − ℓ ) + c x w n − ℓ ) . y Z , γ ) by D ( x,g ) G s ( y Z , γ ) = − ( ϕ Z , . (75)From this equation and Lemma 14 we have the estimate k y Z k ∞ + k γ k ∞ ≤ ˜ C ( | λ | + k τ k ∞ ) . (76)Taking the inner product of the first coordinate in (75) with β − ℓ w Z , ℓ ∈ I ( s ) andusing (19), (84) leads to the improved estimate k γ k ∞ ≤ ˜ C e − αN/ ( | λ | + k τ k ∞ ) . (77)By (76), (75) the Taylor expansion (74) of G s assumes the form k G s ( p Z ( s ) + y Z + v Z ( s, τ ) , h + γ, τ, λ ) k = O (e − αN/ + k τ k ∞ | λ | + λ + k τ k ∞ ) . (78)With (30) and (77) this leads us to the final result k x Z ,s ( τ, λ ) − y Z ( τ, λ ) k ∞ + k g s ( τ, λ ) − h ( τ, λ ) k ∞ = O (e − αN/ + k τ k ∞ | λ | + λ + k τ k ∞ ) . (cid:4) In this subsection we prove Lemma 14. For any two integers n l ≤ n r and for anynumber a ≥ ω n = ω n ( a, n l , n r ) = min (cid:0) e a ( n − n l ) , , e a ( n r − n ) (cid:1) , n ∈ Z . (79)Note that ω Z has a constant plateau of arbitrary width with exponentially de-caying 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 Z has only one-sided decay ordegenerates to the maximum norm if n l = −∞ and n r = ∞ . In the followingwe will suppress the dependence of the norm on the data n l , n r , a , but all ourestimates will be uniform with respect to − ∞ ≤ n l ≤ n r ≤ ∞ ≤ a ≤ a < α, (80)where 0 < a < α is fixed. The following lemma shows that exponentiallydecaying kernels preserve the weight. Lemma 16
There exists a constant K , depending only on a , α , such that X m ∈ Z e − α | n − m − | ω m ≤ K ω n , for all n ∈ Z and for all weight functions satisfying (80) . roof: We consider δ n = X m ∈ Z ω − n e − α | n − m − | ω m only for n ≤ n l and leave cases n l + 1 ≤ n ≤ n r and n r + 1 ≤ n to the reader. δ n = X m ≤ n − e a ( n l − n ) − α ( n − − m )+ a ( m − n l ) + n l − X m = n e a ( n l − n ) − α ( m +1 − n )+ a ( m − n l ) + n r − X m = n l e a ( n l − n ) − α ( m +1 − n ) + X m ≥ n r e a ( n l − n ) − α ( m +1 − n )+ a ( n r − m ) =e − a X m ≤ n − e − ( α + a )( n − − m ) + e − α n l − X m = n e − ( α − a )( m − n ) +e − ( α − a )( n l − n ) − α n r − X m = n l e − α ( m − n l ) + e − ( α − a )( n l − n )+ α ( n l − n r − X m ≥ n r e − ( α + a )( m − n r ) ≤ e − a + e − α − e − ( α + a ) + e − α (cid:18) − e − ( α − a ) + 11 − e − α (cid:19) . A suitable constant for all cases is K = 2 − α − e − α + e a + e a − e − ( α − a . (cid:4) With the weights from above we consider the Banach space ℓ ∞ ω = { y Z ∈ ( R k ) Z : k y Z k ω = sup n ∈ Z k y n ω − n k < ∞} . Taking the exponent α from (7), (18) we have the following result for the varia-tional equation (6). Lemma 17
Let conditions
A1 - A3 and B4 hold. Then the linear system y n +1 − f x (¯ x n , y n + hw n = r n , n ∈ Z h u Z , y Z i = γ (81) has a unique solution ( y Z , h ) ∈ ℓ ∞ ω × R for every ( r Z , γ ) ∈ ℓ ∞ ω × R . Moreoverthere is a constant C ∗ such that for all weights (79) with (80) , k y Z k ω + | h | ≤ C ∗ ( k r Z k ω + | γ |k u Z k ω ) . (82) If n l ≤ ≤ n r , then (82) simplifies to k y Z k ω + | h | ≤ C ∗ ( k r Z k ω + | γ | ) . Proof:
Let us abbreviate A n = f x (¯ x n , L y Z = ( y n +1 − A n y n ) n ∈ Z and denoteby Φ the solution operator (121) of (120). From [22, Section 2] (see also [15])we obtain the following facts. Equation (120) has an exponential dichotomy for33 ≥ K, α, P + sn , P + un ) and for n ≤ K, α, P − sn , P − un ) (seeDefinition 20). Due to B4 we have R ( P + s ) ∩ R ( P − u ) = span { u } =: Y andthere exist decompositions R ( P + s ) = Y ⊕ Y + , R ( P − u ) = Y ⊕ Y − , Y + ∩ Y − = { } , (83)where k s = rank( P + sn ) = dim( Y + ) + 1 , ( n ≥ ,k u = rank( P − un ) = dim( Y − ) + 1 , ( n ≤ ,k u = k − k s , P + sn + P + un = I ( n ≥ , P − sn + P − un = I ( n ≤ . The operator L : ℓ ∞ ( R k ) → ℓ ∞ ( R k ) is Fredholm of index 0 with N ( L ) = span { u Z } , R ( L ) = { r Z ∈ ℓ ∞ ( R k ) : h w Z , r Z i = 0 } . (84)One can also choose the ranges of P + u and P − s such that, in addition to (83), R k = Y ⊕ Y + ⊕ Y − ⊕ Y , R ( P + u ) = Y − ⊕ Y , R ( P − s ) = Y + ⊕ Y . (85)Here dim Y = 1 and one can take Y = span { w − } . Following [22, Lemma 2.7]the general bounded solution of y n +1 − A n y n = r n , n ≥ y + n = Φ( n, η + + X m ≥ G + ( n, m + 1) r m , n ≥ , η + ∈ R ( P + s ) (86)with the Green’s function defined as follows G + ( n, m ) = (cid:26) Φ( n, m ) P + sm , ≤ m ≤ n, − Φ( n, m ) P + um , ≤ n < m. (87)Similarly, all bounded solutions of y n +1 − A n y n = r n , n ≤ − y − n = Φ( n, η − + X m ≤− G − ( n, m + 1) r m , n ≤ , η − ∈ R ( P − u ) , (88)where G − ( n, m ) = (cid:26) Φ( n, m ) P − sm , m ≤ n ≤ , − Φ( n, m ) P − um , n < m ≤ . (89)By the exponential dichotomies the Green’s functions satisfy k G ± ( n, m ) k ≤ K e − α | n − m | , n, m ∈ Z ± . (90)With u T Z y Z := h u Z , y Z i we may write (81) in block operator form as (cid:18) L w Z u T Z (cid:19) (cid:18) y Z h (cid:19) = (cid:18) r Z γ (cid:19) . (91)34y the bordering lemma [2, Appendix] the block operator is Fredholm of the sameindex 0 as L and, using (84), it is a linear homeomorphism in ℓ ∞ ( R k ) × R . Since ℓ ∞ ω is a closed subspace of ℓ ∞ ( R k ) it suffices to prove that the unique solution( y Z , h ) of (91) in ℓ ∞ ( R k ) × R satisfies the estimate (82) in case r Z ∈ ℓ ∞ ω .Take the inner product of the first equation of (81) with w Z . Then (84) andthe normalization k w Z k ℓ = 1 show h = h w Z , r Z i . Therefore, by (18), | h | ≤ C e k r Z k ω X n ∈ Z e − α | n | ω n ≤ C e − α − e − α k r Z k ω . (92)With this h we have z Z := r Z − hw Z ∈ R ( L ) by (84). Below we will construct aspecial solution ˆ y Z ∈ ℓ ∞ ( R k ) of L ˆ y Z = z Z such that for some constant C > k ˆ y Z k ω ≤ C k z Z k ω . (93)By (84) and (8) the solution of (91) is given by( y Z , h ) = (ˆ y Z + cu Z , h ) , c = γ − h u Z , ˆ y Z i , h = h w Z , r Z i . (94)From the exponential decay (7), (18) and Lemma 16, applied to the kernele − α ( | n | + | m | ) ≤ e α e − α | n − m − | , we obtain with C ′ = C e e α K , kh u Z , ˆ y Z i u Z k ω ≤ C ′ k ˆ y Z k ω , kh w Z , r Z i w Z k ω ≤ C ′ k r Z k ω . Then (92)-(94) yield the assertion k y Z k ω ≤ (1 + C ′ ) k ˆ y Z k ω + | γ |k u Z k ω ≤ C (1 + C ′ ) k r Z k ω + | γ |k u Z k ω . It remains to construct ˆ y Z with L ˆ y Z = z Z and (93). We determine η + ∈ R ( P + s )and η − ∈ R ( P − u ) such that ˆ y n = y + n , n ≥ y + n from (86), and ˆ y n = y − n , n ≤ y − n from (88), and such that the definitions coincide at n = 0. The lastcondition holds if and only if η + − η − = X m ≤− G − (0 , m + 1) z m − X m ≥ G + (0 , m + 1) z m =: ∆ . (95)By (87), (89), (85) the first sum on the right is in Y + ⊕ Y and the second sumis in Y − ⊕ Y while the left-hand side is in Y ⊕ Y − ⊕ Y + . Since z Z ∈ R ( L ) holds,equation (95) has a solution and thus ∆ ∈ Y + ⊕ Y − . We conclude from (85) that η + = P + s ∆ and η − = − P − u ∆ are the unique solutions of (95). With (90) andLemma 16 we estimate for n ≥ k ˆ y n k ω − n ≤ K k z Z k ω X m ≥ ω − n e − α | n − m − | ω m ≤ KK k z Z k ω . An analogous estimate holds for n ≤ (cid:4) roof: (Lemma 14) We use Lemma 18 and construct an approximate rightinverse B + of D ( x,g ) G s = D ( x,g ) G s (0 , , , ℓ ∈ I ( s ) we define theinterval J ( ℓ ) = { ℓ − , . . . , ℓ, . . . , ℓ + } where right and left neighbors are given by ℓ + = (cid:26) ∞ , if ℓ = max I ( s ) ,ℓ + N ∗ , otherwise , (96) ℓ − = (cid:26) −∞ , if ℓ = min I ( s ) , max { ˜ ℓ + : ˜ ℓ ∈ I ( s ) , ˜ ℓ < ℓ } + 1 , otherwise . (97)Note that the sets J ( ℓ ) , ℓ ∈ I ( s ) define a partitioning of Z . In the following weconsider N ≥ ˆ N = 2 N ∗ + 1 which implies ℓ − ℓ − ≥ ℓ + − ℓ ≥ N ∗ . During the proof N ∗ will be taken sufficiently large. Given an element ( r Z , γ ) ∈ ℓ ∞ ( R k ) × ℓ ∞ ( s ),we decompose r Z = X ℓ ∈ I ( s ) J ( ℓ ) r Z , where ( J ( ℓ ) ) n = (cid:26) , n ∈ J ( ℓ ) , , elseis the characteristic function of J ( ℓ ). Let B denote the solution operator of (91),then we set ( y ℓ Z , h ℓ ) = B ( β ℓ J ( ℓ ) r Z , γ ℓ ) for ℓ ∈ I ( s ) , (98)and define B + as a blockwise inverse via B + ( r Z , γ ) = ( y Z , h ) = ( X ℓ ∈ I ( s ) β − ℓ y ℓ Z , ( h ℓ ) ℓ ∈ I ( s ) ) . (99)Using Lemma 17 with the settings n l = ℓ − − ℓ , n r = ℓ + − ℓ we obtain a bound k y ℓ Z k ω + | h ℓ | ≤ C ∗ (cid:0) k β ℓ J ( ℓ ) r Z k ω + | γ ℓ | (cid:1) ≤ C ∗ ( k r Z k ∞ + | γ ℓ | ) . (100)Let us abbreviate the weights from (79), ω n,ℓ = ω n ( a, ℓ − , ℓ + ) , n ∈ Z , ℓ ∈ I ( s ) . Then equation (99) and (100) lead to the estimate k y n k ≤ k X ℓ ∈ I ( s ) y ℓn − ℓ k ≤ C ∗ ( k r Z k ∞ + k γ k ∞ ) X ℓ ∈ I ( s ) ω n,ℓ , For 2e − aN ∗ ≤ − a and (100) yields k B + ( r Z , γ ) k = k y Z k ∞ + k h k ∞ ≤ (2 C ∗ + 1 + 4e − a )( k r Z k ∞ + k γ k ∞ ) . (101)In the next step we show for N ∗ sufficiently large, k ( r Z , γ ) − D ( x,g ) G s B + ( r Z , γ ) k ≤
12 ( k r Z k ∞ + k γ k ∞ ) , (102)36hich by (115), (101) gives the desired estimate k ( D ( x,g ) G s ) − k ≤ C ∗ + 1 + 4e − a ) . We introduce (˜ r Z , ˜ γ ) = ( r Z , γ ) − D ( x,g ) G s ( y Z , h ). From (99) and the variationalequation in (98) we have˜ r n = r n − (cid:16) X m ∈ I ( s ) y mn +1 − m − f x ( X p ∈ I ( s ) ¯ x n − p , y n + X m ∈ I ( s ) h m w n − m (cid:17) = X m ∈ I ( s ) h f x ( X p ∈ I ( s ) ¯ x n − p , − f x (¯ x n − m , i y mn − m . For n ∈ Z there exists a unique ℓ ∈ I ( s ) such that n ∈ J ( ℓ ). Define the neighbor-hood U ( ℓ ) of ℓ by U ( ℓ ) = { ˆ ℓ, ℓ, ˜ ℓ } with left neighbor ˆ ℓ = sup { m ∈ I ( s ) : m < ℓ } and right neighbor ˜ ℓ = inf { m ∈ I ( s ) : m > ℓ } (as usual let ˆ ℓ = −∞ and ˜ ℓ = ∞ if the sets are empty). Using the Lipschitz constant L x of f x and (13), (100) weobtain k ˜ r n k ≤ L x (cid:16) X m ∈ I ( s ) \U ( ℓ ) k y mn − m k X m = p ∈ I ( s ) k ¯ x n − p k + X m ∈U ( ℓ ) k y mn − m k X m = p ∈ I ( s ) k ¯ x n − p k (cid:17) ≤ L x C ∗ ( k r Z k ∞ + k γ k ∞ ) n ¯ C X m ∈ I ( s ) \U ( ℓ ) ω n,m + X ℓ = p ∈ I ( s ) k ¯ x n − p k + ω n, ˜ ℓ (cid:16) k ¯ x n − ℓ k + X p ∈ I ( s ) \{ ℓ, ˜ ℓ } k ¯ x n − p k (cid:17) + ω n, ˆ ℓ (cid:16) k ¯ x n − ℓ k + X p ∈ I ( s ) \{ ℓ, ˆ ℓ } k ¯ x n − p k (cid:17)o . We show that the terms in { . . . } are of order O (e − aN ∗ ) so that the contractionestimate (102) holds for the first component if N ∗ is sufficiently large. A criticalterm on the right-hand side is ω n, ˜ ℓ k ¯ x n − ℓ k ≤ C e e − α | n − ℓ | + a ( n − ˜ ℓ − ) ≤ C e e a ( ℓ − ˜ ℓ − ) ≤ C e e − aN ∗ , ℓ − ≤ n ≤ ℓ + . (103)The term ω n, ˆ ℓ k ¯ x n − ℓ k is handled analogously. Further, X ℓ = p ∈ I ( s ) k ¯ x n − p k ≤ C e (cid:16) X ℓ>p ∈ I ( s ) e − α ( n − p ) + X ℓ
p ∈ I ( s ) e − α ( ℓ − − p ) + X ℓ
37e estimate the remaining sum by using (103) X n ∈ Z e − α | n − ℓ | X ℓ = m ∈ I ( s ) ω n,m ≤ ℓ + X n = ℓ − e − α | n − ℓ | (cid:16) ω n, ˜ ℓ + ω n, ˆ ℓ + X m ∈ I ( s ) \U ( ℓ ) ω n,m (cid:17) + C (cid:16) ∞ X n = ℓ + e − α ( n − ℓ ) + ℓ − − X n = −∞ e − α ( ℓ − n ) (cid:17) . The last two terms are O (e − αN ∗ ) since ℓ + − ℓ ≥ N ∗ and ℓ − ℓ − ≥ N ∗ . Furthermore,we have for ℓ − ≤ n ≤ ℓ + X m ∈ I ( s ) \U ( ℓ ) ω n,m ≤ X ˜ ℓ
Lemma 18 (Banach Lemma) Let
X, Y be Banach spaces and let A ∈ L [ X, Y ] , B − , B + ∈ L [ Y, X ] be bounded linear operators such that k I Y − AB + k < , k I X − B − A k < . Then A is a homeomorphism with k A − k ≤ min (cid:18) k B + k − k I Y − AB + k , k B − k − k I X − B − A k (cid:19) . (115) Proof:
Note that y = ( I Y − AB + ) y + r has a unique solution y for every r ∈ Y .Then y satisfies k y k ≤ k r k − k I Y − AB + k 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 thesolution is unique and the estimates follow. (cid:4) A key tool in the proofs of Lemma 13 and Theorem 10 is the following quan-titative version of the Lipschitz inverse mapping theorem, cf. [13].
Theorem 19
Assume Y and Z are Banach spaces, F ∈ C ( Y, Z ) and F ′ ( y ) isfor y ∈ Y a homeomorphism. Let κ, σ, δ > be three constants, such that thefollowing estimates hold: (cid:13)(cid:13) F ′ ( y ) − F ′ ( y ) (cid:13)(cid:13) ≤ κ < σ ≤ (cid:13)(cid:13) F ′ ( y ) − (cid:13)(cid:13) ∀ y ∈ B δ ( y ) , (116) (cid:13)(cid:13) F ( y ) (cid:13)(cid:13) ≤ ( σ − κ ) δ. (117) Then F has a unique zero ¯ y ∈ B δ ( y ) and the following inequalities are satisfied (cid:13)(cid:13) F ′ ( y ) − (cid:13)(cid:13) ≤ σ − κ ∀ y ∈ B δ ( y ) , (118) k y − y k ≤ σ − κ (cid:13)(cid:13) F ( y ) − F ( y ) (cid:13)(cid:13) ∀ y , y ∈ B δ ( y ) . (119)We collect some well known results on exponential dichotomies from [22].Denote by Φ the solution operator of the linear difference equation y n +1 = A n y n , n ∈ Z , (120)which is defined asΦ( n, m ) := A n − . . . A m , for n > m,I, for n = m,A − n . . . A − m − , for n < m. (121) Definition 20
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