Isolating segments, fixed point index and symbolic dynamics II. Homoclinic solutions.

Klaudiusz Wójcik ^{1} Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland. E-mail: wojcik@im.uj.edu.pl

Piotr Zgliczyński ^{2} ^{3} Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, Georgia 30332, USA E-mail: piotrz@math.gatech.edu

March 14, 2000

Abstract

We present a topological method for detecting homoclinic solutions in nonautonomous ODE’s.

As an application we show that there exist infinitely many (geometrically distinct) homoclinic solutions to the trivial 0 solution in the planar system

ż = z̄(1 + e^{iκt}|z|^{2})

for 0 < κ ≤ 0.495.

Keywords: isolating segments, fixed point index, homoclinic solutions

1 Introduction

This paper is the sequel to the paper [WZ] and is devoted to a study of chaotic properties of the equation

ż = z̄(1 + e^{iκt}|z|^{2}),  z ∈ ℂ, 0 < κ ≤ 0.495  (1)

Before we state our main result we need a few notation. Let Σ_{3} = {−1, 0, 1}ℤ, σ : Σ_{3} → Σ_{3} be a shift map. For α ∈ Σ_{3} by p(α) we will denote its principal period. For α non-periodic we set p(α) = ∞.

Let T = \frac{2π}{κ} be a period of Equation (1). Let φ be a local process induced by (1). Let P = φ_{(0,T)} be a T -time shift along a trajectory of (1) starting at t = 0.

The results of this paper and the paper [WZ] can be summarized as

Theorem 1 There exists a compact set S, such that P (S) = S and a continuous map g : S → Σ_{3}, such that 1. σ ∘ g = g ∘ P

1supported by Polish KBN grant 2 P 03A 028 17

2supported by Polish KBN grant 2P03A 021 15, 2P03A 029 12 and NSF–NATO grant DGE– 98–04459

3on leave from: Institute of Mathematics, Jagiellonian University, Cracow, Poland; e-mail:

zgliczyn@im.uj.edu.pl

2. g(S) = Σ_{3}

3. if α ∈ Σ_{3} is a periodic sequence, then g^{−1}(α) contains a point with a period equal to p(α)

4. For any α ∈ Σ_{3} such that α_{i} = 0 for i > i_{0} , g^{−1}(α) contains a point x such that lim_{t→∞} φ(0, t, x) = (0, 0)

5. For any α ∈ Σ_{3} such that α_{i} = 0 for i < i_{0} , g^{−1}(α) contains a point x such that lim_{t→−∞} φ(0, t, x) = (0, 0)

6. For any α ∈ Σ_{3} such that α_{i} = 0 for i < i_{0} or i > j_{0} g^{−1}(α) contains a point x such that lim_{t→±∞} φ(0, t, x) = (0, 0)

Assertions 1–3 are shown in [WZ] already. The new element in this paper are the statements about existence of complicated orbits (α in 4–6 can be an arbitrary finite sequence of −1, 0, 1) which are asymptotic to the hyperbolic periodic orbits passing through the origin. Assertion 6 is a statement about an existence of plenty of orbits homoclinic to the origin, in literature (see [CT] and references cited there) such orbits are often called multibump solutions.

We believe that the most interesting aspect of our paper and [WZ] is the way we prove our main result. We construct a topological model for a semiprocess induced by (1), for which Assertions 1–6 hold. Then we show by a continuation argument, that these assertions also hold for φ.

The method is based on the concept of an periodic isolating segment introduced by Srzednicki (see [S1]) to study periodic points for a time periodic non–autonomous ordinary differential equation. In all practical applications isolating segments are manifolds with corners in the extended phase space, such that at any point at the boundary of the segment the vector field is directed either outward or inward with respect of the segment (compare the notion of the isolating block in the Conley index theory [C]).

In applications of topological methods in the field of chaotic dynamics like the Conley index (see for example [MM]), isolating segments (see [WZ]) or the TSmaps method (see [Z1, Z2, GZ]) the obtained results are similar to Assertions 1–3 in our main theorem. Apparently, no information concerning the sensitive dependence on the initial conditions or the existence of orbits asymptotic to known hyperbolic periodic orbits can be obtained by these methods.

In this paper we show how, by using isolating segments, one can extract the existence of homoclinic solutions for Equation (1). The basic idea can be explained as follows. To obtain a symbolic dynamics (Assertions 1–3) we have built in [WZ] ’large’ isolating segments. To deal with solutions which are asymptotic to the origin we shrink the isolating segments almost to a line. Obviously the actual proof is much more complicated.

For the convenience of the reader we will comment on the mutual dependencies between [WZ] and the present paper. The full proof of Theorem 1 is contained in the present paper. The parts of [WZ] one needs to read to follow fully the proof are sections 2,5 and 6, which contains definitions of periodic isolating segments , semiprocesses and methods of constructing them (section 2), a construction of U, W (section 5) and a construction of the model semiprocess (section 6). Section 4 from [WZ] containing the continuation theorem have been rewritten here as section 3 to include a more general continuation theorem, which we have found suitable to deal with the homoclinc solutions.

2 Basic notions

Some notation: ℝ_{+} = [0, ∞), ρ is a distance function. In the context of ℝ^{n} ρ will be always a distance induced by the max norm i.e. ∥(x_{1}, …, x_{n})∥ = max_{i=1,...,n} |x_{i}|. B(Z, δ) is a ball of size δ around the set Z, ind (F, D) - fixed point index of the map F relatively to the set D (see [D]). We assume that reader is familiar with the content of section 2 of [WZ], where the notions of semiprocess and periodic isolating segment are defined.

2.1 Isolating segments

As in [WZ] we use the following notation: by π_{1} : ℝ×X → ℝ and π_{2} : ℝ×X → X we denote the projections and for the subset Z ⊂ ℝ × X and t ∈ ℝ , [t_{1}, t_{2}] we put

Z_{t} = {x ∈ X : (t, x) ∈ Z}
Z_{[t_{1},t_{2}]} = Z ∩ ([t_{1}, t_{2}] × X)

Let φ be a global semiprocess and Φ be a semiflow associated to φ.

Definition 1 Let (W, W^{−}) ⊂ ℝ × X be a pair of subsets. We call W an isolating segment for the global semiprocess φ if:

(ii) (W_{|[−A,A]}, W_{|[−A,A]}^{−}) is a pair of compact sets for any A ∈ ℝ

(iii) for every σ ∈ ℝ, x ∈ ∂W_{σ} there exists δ > 0 such that either φ_{(σ,t)}(x) ∉ W_{σ+t} for all t ∈ (0, δ) or φ_{(σ,t)}(x) ∈ intW_{σ+t} for all t ∈ (0, δ),

(iv)

W^{−} = {(σ, x) ∈ W : ∃δ > 0 ∀t ∈ (0, δ) φ_{(σ,t)}(x) ∉ W_{σ+t}},
W^{+} := cl (∂W \ W^{−})

(v) for all z ∈ W^{+} and all v : (a, 0] → ℝ × X a left solution through z there is a ≤ b < 0 such that for all t ∈ (b, 0) v(t) ∉ W

(vi) there exists η > 0 such that for all x ∈ W^{−} there exists t > 0 such that for all τ ∈ (0, t] Φ_{τ} (x) ∉ W and ρ(Φ_{t}(x), W) > η

Roughly speaking, W^{−} and W^{+} are sections for the semiflows, through which trajectories leave and enter the segment W, respectively.

It is easy to see that a T –periodic isolating segment for the semiprocess φ defined in [WZ] (def. 3) is an isolating segment in the sense of the above definition. The only essential difference between them is a requirement of the T –periodicity of pair (W, W^{−}), which constituted condition (i) in [WZ] and which is dropped here. There is also a change in condition (ii), but it is easy to see that both forms of (ii) are equivalent for a T –periodic semiprocess.

The following lemma corresponds to Lemma 4 in [WZ] and has essentially the same proof.

Lemma 2 Let (W, W^{−}) be an isolating segment. For every A > 0 there exists an ϵ > 0 such that for every x ∈ W_{[−A,A]}^{−} if for some t > 0 Φ_{t}(x) ∈ W then t > ϵ.

The two definitions below are the natural extensions of the analogous definitions in [WZ], where they are stated for W_{0} only.

Definition 2 For the isolating segment W we define an exit time function τ_{W,φ}

τ_{W,φ} : W ∋ x ↦ sup{t ≥ 0 : ∀s ∈ [0, t] Φ_{s}(x) ∈ W } ∈ [0, ∞]

By the Ważewski Retract Theorem the map τ_{W,φ} is continuous (compare [C]).

Definition 3 Let W be an isolating segment for φ and x ∈ W. Let C ⊂ W^{−}. We will say that x leaves W at the time t through C, iff t = τ_{W,φ}(x) and Φ_{t}(x) ∈ C.

When x ∈ W_{0} we say that x leaves W at the time t through C, iff a point (0, x) leaves W at the time t through C.

3 Continuation theorem

Let φ : ℝ × ℝ_{+} × ℝ^{d} → ℝ^{d} be a T -periodic global semiprocess.

By Φ : ℝ_{+} × ℝ^{d+1} → ℝ^{d+1} we will denote a T -periodic semidynamical system induced by φ (see [WZ]).

Let U, W be two T -periodic isolating segments for φ, let {t_{i}}_{i=0}^{n} , n ≥ 1 such that

t_{0} = 0 < t_{1} < … < t_{n} = T  (2)
U ⊂ W, U_{t_{i}} = W_{t_{i}}, U_{t_{i}}^{−} = W_{t_{i}}^{−},  i = 0, …, n − 1  (3)

Let

U^{−} = ⋃_{l=1}^{K} U_{l}^{−}  (4)

be a decomposition of the exit set U^{−} into a disjoint union of closed T –periodic sets U_{l}^{−}.

Observe that

ρ(U_{l}^{−}, U_{j}^{−}) > 0, when l ≠ j  (5)

In the sequel we will often work with sequences of sets of length n, where n is as in (2).

Definition 4 Let

C = (C_{0}, C_{1}, …, C_{n−1}), where C_{i} ⊂ ℝ^{d}

We will call such sequence an n–set. We can extend the n–set C to n–periodic sequence of sets by setting

C_{kn+l} = C_{l}

Definition 5 Let C, B be n–sets. Then

int C = (int C_{0}, …, int C_{n−1})
\overline{C} = (\overline{C}_{0}, …, \overline{C}_{n−1})
∂C = (∂C_{0}, …, ∂C_{n−1})
C ⊂ B iff C_{i} ⊂ B_{i}, for i = 0, …, n − 1

C is open (closed), iff C = int C (C = \overline{C})

With the periodic isolating segment W we associate the following n–set W

W = (W_{0}, W_{t_{1}}, …, W_{t_{n−1}})

Definition 6 Let α = (α_{0}, α_{1}, …, α_{n−1}) ∈ {0, 1, …, K}^{n} and D be an n-set, D ⊂ W. We define D_{α}(φ) as a set of points fulfilling the following conditions

φ(0, t_{l}, x) ∈ D_{l}, for l = 0, …, n  (6)
Φ_{t}(0, x) = (t, φ(0, t, x)) ∈ int W, for t ∈ (0, T)  (7)
if α_{l} = 0, then Φ_{t_{l}+t}(0, x) = (t_{l} + t, φ(0, t_{l} + t, x)) ∈ int U
for t ∈ (0, t_{l+1} − t_{l})  (8)
if α_{l} ≠ 0, then Φ_{t_{l}}(0, x) = (t_{l}, φ(0, t_{l}, x)) leaves U in time less than
(t_{l+1} − t_{l}) through U_{α_{l}}^{−}  (9)

Lemma 3 If D is open, then D_{α}(φ) is open. If D is closed, then D_{α}(φ) is closed.

Proof:

First suppose that D is open. From the continuity of φ(0, t_{l}, ·) and φ(t_{l}, t, ·) it follows that conditions (6–8) define an open subset of D_{0}. It remains to show that (9) holds for an open subset of D_{0}. Suppose that x ∈ D_{0} is such that (6) and (9) hold. Since φ(0, t_{l}, x) ∈ int U_{t_{l}} and φ(0, t_{l+1}, x) ∈ int U_{t_{l+1}} the exit time t = τ_{U,φ}((t_{l}, φ(0, t_{l}, x))) belongs to the open segment (0, t_{l+1} − t_{l}). From the continuity of the exit time function it follows that

τ_{U,φ}((t_{l}, φ(0, t_{l}, y)) ∈ (0, t_{l+1} − t_{l}).

for y sufficiently close to x . Hence the function

y ↦ (τ_{U,φ}(t_{l}, φ(0, t_{l}, y)), φ(t_{U,φ}((t_{l}, φ(0, t_{l}, y), t_{l}, y)) ∈ U^{−}  (10)

is continuous. Observe that each set U_{s}^{−} is open in U^{−}. Therefore, all points sufficiently close to x are mapped by this function into the same component U_{α_{l}}^{−}.

Now suppose that D is closed. D_{0} ⊂ W_{0}. Suppose that x_{m} → x and x_{m} ∈ D_{α}(φ). We have to show that conditions (6–9) hold for x. Condition (6) is obvious. Suppose that (7) does not hold for x, then for some t ∈ (0, T) we have (t, φ(0, t, x)) ∉ W. However, from the continuity of φ(0, t, ·) it follows that (t, φ(0, t, x_{m})) ∉ W for all but a finite number of the x_{m}’s. This contradicts (7) for x_{m}. The proof of (8) is analogous.

It remains to show (9). Since for all m φ(0, t_{l+1}, x_{m}) ∈ D_{l+1} ⊂ W_{t_{l+1}} it follows that the exit times of (t_{l}, φ(0, t_{l}, x_{m})) are less than (t_{l+1}−t_{l})−ϵ, where ϵ is obtained from lemma 4 in [WZ] applied to the segment (U, U^{−}). From this and condition (vi) it follows that (t_{l}, φ(0, t_{l}, x)) leaves U before time t_{l+1} − t_{l}. Suppose now that (t_{l}, φ(0, t_{l}, x) does leave U through U_{s}^{−}, where s ≠ α_{l}. Then the trajectories of all points sufficiently close to x will remain near the trajectory of x. Hence all but a finite number of the (t_{l}, φ(0, t_{l}, x_{m}))’s will leave U through U_{s}^{−}, which contradicts our assumption. Therefore, condition (9) holds for x. □

Definition 7 Let τ ≥ τ_{0} and A ⊂ W_{τ_{0}}. We define

Im (A, φ, τ_{0}, τ) = {φ(τ_{0}, τ, x) | x ∈ A, φ(τ_{0}, t, x) ∈ W_{τ_{0}+t} for 0 ≤ t ≤ τ − τ_{0}} (11)

For τ_{0} = 0 and A ⊂ W_{0} we will often drop τ_{0} in the above notation, hence

Im (A, φ, τ) = Im (A, φ, 0, τ)  (12)

This notation agrees with the one used in [WZ].

Consider now a continuous family of semiprocesses H : [0, 1] × ℝ × ℝ_{+} × ℝ^{d} → ℝ^{d}. We will use the notation H_{λ} for the map with the first parameter fixed, H_{λ}(t_{0}, τ, x) = H(λ, t_{0}, τ, x).

The main theorem in this section is the following

Theorem 4 Let H_{λ} be a continuous family of T -periodic semiprocesses and t_{0} = 0 < t_{1} < … < t_{n} = T and U, W - T -periodic isolating segments for the semiprocess H_{λ} for every λ ∈ [0, 1], such that (3) holds.

Assume that condition (vi) holds uniformly for H_{λ}, where λ ∈ [0, 1] and isolating segments (W, W^{−}) and (U, U^{−}). By this we mean the following statement:

Let Φ^{λ} be a semiflow induced by H_{λ}, then there exists η > 0 such that for every λ ∈ [0, 1] and for every x ∈ W^{−} (x ∈ U^{−}) there exists t > 0 such that for 0 < τ ≤ t holds Φ_{τ}^{λ}(x) ∉ W and ρ(Φ_{t}^{λ}(x), W) > η (resp. Φ_{τ}^{λ}(x) ∉ U and ρ(Φ_{t}^{λ}(x), U) > η).

Then for every α = (α_{0}, α_{1}, …, α_{n−1}) ∈ {0, 1, …, K}^{n} the fixed point indices ind (H_{λ}(0, T), (int W)_{α}(H_{λ})) are well defined and equal (i.e. do not depend on λ).

Before we prove this theorem we first show the following

Lemma 5 Under hypotheses of Theorem 4 there exists δ > 0 such that for every λ ∈ [0, 1] and l = 0, …, n − 1 holds

Im (W_{t_{l}}, H_{λ}, t_{l}, t_{l+1}) ∩ \overline{B(W_{t_{l+1}}^{+}, δ)} = ∅  (13)
every point from \{t_{l}\} × \overline{B(W_{t_{l}}^{−}, δ)} ∩ W_{t_{l}} leaves W in time τ < t_{l+1} − t_{l} (14)

Proof:

Let S_{l} = {x ∈ W_{t_{l}} : τ_{W,H_{λ}}(t_{l}, x) ≥ t_{l+1} −t_{l}}. It is easy to check that S_{l} is compact and

Im (W_{t_{l}}, H_{λ}, t_{l}, t_{l+1}) = H_{λ}(t_{l}, t_{l+1} − t_{l}, S_{l}).

If z ∈ H_{λ}(t_{l}, t_{l+1} − t_{l}, S_{l}) then Φ_{t}(t_{l+1}, z) ∈ W for t ∈ (t_{l} − t_{l+1}, 0]. Therefore, from condition (v) in the definition of the isolating segment it follows (t_{l}, z) ∉ W^{+}. Hence

H_{λ}(t_{l}, t_{l+1} − t_{l}, S) ∩ W_{t_{l+1}}^{+} = ∅,

hence ρ(H_{λ}(t_{l}, t_{l+1} − t_{l}, S), W_{t_{l=1}}^{+}) > 0 and

min_{l} inf_{λ∈[0,1]} ρ(H_{λ}(t_{l}, t_{l+1} − t_{l}, S_{l}), W_{t_{l+1}}^{+}) > 0.

For the proof of (14) let us consider a continuous function

G : [0, 1] × W_{t_{l}} ∋ (λ, x) → τ_{W,H_{λ}}(t_{l}, x) ∈ [0, +∞].

Since for all λ ∈ [0, 1] and x ∈ W_{t_{l}}^{−} G(λ, x) = 0, it follows by the standard argument based on the compactness of [0, 1] and W_{t_{l}}^{−} that there is an open neighborhood V of W_{t_{l}}^{−} such that for all (λ, x) ∈ [0, 1] × V G(λ, x) < \frac{t_{l+1}−t_{l}}{2} . This completes the proof. □

Proof of Theorem 4 :

Let us choose β > 0 such that

β < η, and β < dist(U_{l}^{−}, U_{j}^{−}), for l ≠ j  (15)

Let us take δ obtained from Lemma 5. By eventually decreasing δ we can assume that

β/2 > δ  (16)

Let us define n–sets D and C by

D_{l} := W_{t_{l}} \ \overline{B(∂W_{t_{l}}, δ)}, C_{l} := W_{t_{l}} \ \overline{B(∂W_{t_{l}}, δ/2)}  (17)

The n–sets D and C are open and

B(D_{l}, δ/2) ⊂ C_{l}, B(C_{l}, δ/2) ⊂ int W_{t_{l}}  (18)

Let λ_{0} ∈ [0, 1]. There exists a set Λ open in [0, 1], such that λ_{0} ∈ Λ and for every λ_{1}, λ_{2} ∈ Λ

ρ(H_{λ_{1}}(0, t, x), H_{λ_{2}}(0, t, x)) ≤ δ/2, for 0 ≤ t ≤ T, x ∈ W_{0}  (19)

We show now the following

Lemma 6

D_{α}(H_{λ}) ⊂ C_{α}(H_{λ_{0}}) ⊂ (int W)_{α}(H_{λ}), for λ ∈ Λ  (20)

Proof: We will show only the inclusion D_{α}(H_{λ}) ⊂ C_{α}(H_{λ_{0}}). The proof for the second one is analogous.

Let x ∈ D_{α}(H_{λ}). From (6) we have H_{λ}(0, t_{l}, x) ∈ D_{l} for l = 0, …, n. Hence by (18) and (19) we obtain immediately that

H_{λ_{0}}(0, t_{l}, x) ∈ C_{l}, for l = 0, …, n

To prove (7) for the semiprocess H_{λ_{0}} we suppose that it is not fulfilled. x leaves W in time less than T, so we know that it has to move away from W to a distance greater than β before it enters W again (it must do it before T from just proved (6)). Hence there exists t < T such that

ρ((t, H_{λ_{0}}(0, t, x)), W) ≥ β  (21)

From Equations (19) and (16) it follows that

ρ((t, H_{λ}(0, t, x), W)) > β − δ/2 > 0  (22)

This is in a contradiction with the assumption that x does not leave W in time less than T for the semiprocess H_{λ}.

The proof of (8) is analogous, one has to replace W by U in the above reasoning.

It remains to show (9) for the semiprocess H_{λ_{0}}. Suppose that (t_{l}, H_{λ}(0, t_{l}, x)) leaves U in time less than t_{l+1} − t_{l}. By repeating the reasoning which led to a contradiction in the proof of (7) we see that (t_{l}, H_{λ_{0}}(0, t_{l}, x)) leaves U in time less than t_{l+1} − t_{l}.

It remains to show that both points (t_{l}, H_{λ_{0}}(0, t_{l}, x)) and (t_{l}, H_{λ}(0, t_{l}, x)) leave U through the same set U_{k}^{−}.

Let us define the exit times t_{λ}, t_{λ_{0}} by

t_{λ} = τ_{U,H_{λ}}(Φ_{t_{l}}^{λ}(0, x))  (23)
t_{λ_{0}} = τ_{U,H_{λ_{0}}} (Φ_{t_{l}}^{λ_{0}}(0, x))  (24)

Without any loss of generality we can assume that t_{λ} ≤ t_{λ_{0}}.

Let s ∈ {1, …, K} be such that Φ_{t_{l}+t_{λ}}^{λ}(0, x) ∈ U_{s}^{−}.

We have to show that

Φ_{t_{l}+t_{λ_{0}}}^{λ_{0}} (0, x) ∈ U_{s}^{−}  (25)

From the assumption of our theorem concerning η and since δ < η it follows that there exists t′ > 0 such that

Φ_{t_{l}+t_{λ}+t}^{λ}(0, x) ∉ U, for t ∈ (0, t′]  (26)
Φ_{t_{l}+t_{λ}+t}^{λ}(0, x) ∈ B(U_{s}^{−}, δ/2), for t ∈ (0, t′)  (27)
ρ(Φ_{t_{l}+t_{λ}+t′}^{λ}(0, x), U) = δ/2  (28)

From conditions (28) and (19) we see

Φ_{t_{l}+t_{λ}+t′}^{λ_{0}}(0, x) ∉ U  (29)

hence

t_{λ} ≤ t_{λ_{0}} < t_{λ} + t′  (30)

From conditions (30), (27) and (19) it follows that

Φ_{t_{l}+t_{λ_{0}}}^{λ_{0}} (0, x) ∈ B(U_{s}^{−}, δ) ⊂ B(U_{s}^{−}, β/2)  (31)

This and (15) imply (25).

This finishes the proof of (20). □

We continue the proof of Theorem 4. Now we show that

H_{λ}(0, T, x) ≠ x, for x ∈ W_{α}(H_{λ}) \ D_{α}(H_{λ})  (32)

Let us take x ∈ W_{α}(H_{λ}) \ D_{α}(H_{λ}). Then for some j ∈ {0, 1, …, n} we have H_{λ}(0, t_{j}, x) ∈ W_{t_{j}} \ D_{j}. From the definitions of δ and D it follows that one of the following conditions holds

H_{λ}(0, t_{j}, x) ∈ \overline{B(W_{t_{j}}^{+}, δ)} ∩ W_{0}
H_{λ}(0, t_{j}, x) ∈ \overline{B(W_{t_{j}}^{−}, δ)} ∩ W_{0}

In the first case, by Lemma 5, H_{λ}(0, t_{j}, x) ∉ Im (W_{t_{j−1}}, H_{λ}, t_{j−1}, t_{j}), hence x cannot be a periodic point having an entire trajectory in W.

In the second case, by Lemma 5 (t_{j}, H_{λ}(0, t_{j}, x) leaves W in time less than t_{j+1}−t_{j}, so x cannot be a point having an entire trajectory in W. This finishes the proof of (32).

From Lemma 6 it follows that for λ, λ_{0} ∈ Λ we obtain

∂D_{α}(H_{λ}), ∂C_{α}(H_{λ}), ∂C_{α}(H_{λ_{0}}), ∂W_{α}(H_{λ}) ⊂ W_{α}(H_{λ}) \ D_{α}(H_{λ}).

So from (32) we see that the fixed point index for the maps H_{λ}(0, T, ·) relative to the sets D_{α}(H_{λ}), C_{α}(H_{λ}), C_{α}(H_{λ_{0}}), (int W)_{α}(H_{λ}) [D, Ch. VII.5] is well defined.

From the excision property of the fixed point index [D, Ch.VII Th. 5.4], (32) and Lemma 6 we conclude that

ind (H_{λ}(0, T, ·), D_{α}(H_{λ})) = ind (H_{λ}(0, T, ·), C_{α}(H_{λ_{0}})) =
ind (H_{λ}(0, T, ·), (int W)_{α}(H_{λ})), ∀λ ∈ Λ.  (33)

Substituting λ := λ_{0} we derive

ind (H_{λ_{0}}(0, T, ·), D_{α}(H_{λ_{0}})) = ind (H_{λ_{0}}(0, T, ·), C_{α}(H_{λ_{0}}))  (34)

From Lemma 6 and (32) it follows that

∀λ ∈ Λ ∀x ∈ ∂C_{α}(H_{λ_{0}}) H_{λ}(0, T, x) ≠ x

So from the homotopy invariance of the fixed point index [D, Ch.VII Th. 5.8] we obtain

ind (H_{λ}(0, T, ·), C_{α}(H_{λ_{0}})) = ind (H_{λ_{0}}(0, T, ·), C_{α}(H_{λ_{0}})) ∀λ ∈ Λ. (35)

From (33),(34),(35) we conclude

ind (H_{λ}(0, T, ·), D_{α}(H_{λ})) = ind (H_{λ_{0}}(0, T, ·), D_{α}(H_{λ_{0}})) ∀λ ∈ Λ.  (36)

From (36) we see that ind (H_{λ}(0, T, ·), D_{α}(H_{λ})) is locally constant. Hence

ind (H_{0}(0, T, ·), D_{α}(H_{0})) = ind (H_{1}(0, T, ·), D_{α}(H_{1})).

And finally, it follows from (33) that

ind (H_{0}(0, T, ·), (int W)_{α}(H_{0})) = ind (H_{1}(0, nT, ·), (int W)_{α}(H_{1})).

This finishes the proof. □

4 Model semiprocess

The aim of this section is to state the properties of the model semiprocess which was constructed in [WZ].

Let U ⊂ W be a periodic isolating segments for (1) constructed in [WZ]. Since we will use parameter R in this section let us remind that W_{0} = U_{0} = [−R, R]^{2}.

Let U^{+1}, U^{−1} be two connected components of U^{−}, the right one (x > 0) and the left one (x < 0), respectively.

The following theorem was proved in [WZ]

Theorem 7 There exists a semiprocess φ^{M} such that there are disjoint closed segments J_{−1} = [−b, −a], J_{0} = [−c, c], J_{1} = [a, b], J_{l} ⊂ (−R, R) for l = −1, 0, 1 and continuous function f : J_{−1} ∪ J_{0} ∪ J_{+1} → [−R, R] such that

Z := {p ∈ W_{0} | φ^{M}(0, t, p) ∈ W, t ∈ [0, T]} = {J_{−1} ∪ J_{0} ∪ J_{1}} × [−R, R] (37)
Z_{0} := {p ∈ W_{0} | φ^{M}(0, t, p) ∈ U, t ∈ [0, T]} = J_{0} × [−R, R]  (38)

Z_{l} := {p ∈ Z | p leaves U through U^{l} in time ≤ T } = J_{l} × [−R, R], for l = −1, 1 (39)

φ^{M}(0, T, (x, y)) = (f(x), 0), where (x, y) ∈ Z  (40)
f(−x) = −f(x), f(c) = R, f(a) = R, f(b) = −R.  (41)

The pairs (U, U^{−}) and (W, W^{−}) are periodic isolating segments for φ^{M}.

Consider the family of T–periodic semiprocesses for λ ∈ [0, 1]

H_{λ} = Ω({(φ^{M}, 0), (φ, λT)}, T).

Then for every λ ∈ [0, 1] H_{λ} is a semiprocess, (U, U^{−}) and (W, W^{−}) are periodic isolating segments and condition (vi) holds uniformly (see Theorem 4 for precise meaning of this statement).

To prove Theorem 1 we will need a bit more information about φ^{M}.

Theorem 8 Let φ^{M}, f, J_{−1}, J_{0}, J_{1} be as in the Theorem 7. Then the following conditions hold

1. f(−x) = −f(x)

2. f is strictly increasing on J_{0} and is strictly decreasing on J_{−1} and J_{1}

3. there exists γ > 0 such that φ^{M} restricted to the set

Y = {(t, x, y) | : (x, y) ∈ [−γ − r, r + γ]^{2}} ∪ U  (42)

is induced by the following system of differential equations

\frac{dx}{dt} = x, \frac{dy}{dt} = v(t, y)

where v(t, y) = −y for t ∈ D_{1} + kT and v(t, y) = −Lsgn (y)\sqrt{|y|} for t ∈

D_{2} + kT. L > 0 can be chosen arbitrary large and D_{1} ∪ D_{2} = [0, T]

Proof

During the whole construction the oddness of φ^{M} with respect to (x, y) was preserved. From this the oddness of f follows immediately.

To prove the monotonicity properties of f let us remind the construction of φ^{M} was done in the coordinates (t, x_{M}, y_{M}) (called the model coordinates in [WZ]), which coincide with the standard coordinates (t, x, y) for t = kT , where T is a period of (1) and k ∈ ℤ. In these coordinates f was given by (see stage 4 - collapse of the construction of φ^{M} in [WZ])

f(x) = \{ \begin{matrix}e^{T} x, & for x ∈ [−c, c] \\ e^{T −t_{4}}f_{1}(x), & otherwise \end{matrix}  (43)

where the function f_{1} was proved to be monotonically decreasing.

To prove Assertion 3 let us remaind that the model coordinates are related to standard ones by

(t, x_{M}, y_{M}) ↦ (t, x, y) = (t, A_{t}(x, y))  (44)

From the construction of φ^{M} it follows that for γ ≤ ϵ, where ϵ is a small positive number defined in the third stage (the rotation), φ^{M} on Y is induced by the following system of the differential equations

x′_{M} = x_{M}, y′_{M} = v(t, y_{M})  (45)

where v is as in the assertion of the theorem.

It follows from Lemma 21 ( Assertion 5) that for sufficiently small δ A_{t} is an identity map on Y. Hence Equation (45) also holds in the standard coordinates. □

5 Construction of special isolating segments

5.1 Construction of segment V

Let T = 2π/κ. Let R, r, κ, Δ be as in Lemma 19 in [WZ]. As in section 5 in [WZ] we set

ω = \frac{R − r}{Δ}  (46)

We set for t ≥ 0

s_{V} (t) = \{ \begin{matrix}R − ωt, & for t ∈ [0, Δ] \\ r, & for t ∈ [Δ, T] \\ re^{−a(t−T)} & for t ≥ T \end{matrix}  (47)

where a > 0 is a real number. For t < 0 we set

s_{V} (t) = s_{V} (−t)  (48)

We define an isolating segment V by

V_{t} = [−s_{V} (t), s_{V} (t)]^{2}  (49)
V_{t}^{−} = {−s_{V} (t), s_{V} (t)} × [−s_{V} (t), s_{V} (t)]  (50)
V_{t}^{+} = [−s_{V} (t), s_{V} (t)] × {−s_{V} (t), s_{V} (t)}  (51)

Lemma 9 Let a = 0.25. Then V ⊂ U and V is an isolating segment for (1).

Moreover the function |x| − s_{v} is strictly increasing for trajectories of Φ in some neighborhood of V^{−}.

Proof

It is enough to show that V^{−} and V^{+} are local sections for the semiprocess induced by (1) through, which trajectories enter and leave V, respectively.

For this purpose it suffices to prove that the function |x(φ(t, x, y))|−s_{V} (t) is strictly increasing for (t, x, y) in some neighborhood of V^{−} and the function |y(φ(t, x, y))|− s_{V} (t) is strictly decreasing for (t, x, y) in some neighborhood of V^{+}. This is implied by the following conditions

\frac{d(|x| − s_{V})}{dt} > 0, for (t, x, y) ∈ V^{−}  (52)
\frac{d(|y| − s_{V})}{dt} < 0, for (t, x, y) ∈ V^{+}  (53)

here by \frac{df}{dt} we understand the derivative of a function f : ℝ^{3} → ℝ^{3} along the trajectory of the process φ induced by (1) i.e.

\frac{df}{dt} (t, x, y) = \frac{df(t, φ(τ; t, x, y))}{dτ}_{|τ=0} = \frac{∂f}{∂t} (t, x, y) + \frac{∂f}{∂x} (t, x, y)x′ + \frac{∂f}{∂y} (t, x, y)y′

We will prove (52–53) for t ≥ 0 only. The proof for t < 0 is analogous.

From Lemmas 14,16 and 19 in [WZ] it follows immediately that (52),(53) hold for t ∈ [0, T].

Assume that t ≥ T. Due to symmetry (x, y) ↦ (−x, −y) we see that it suffices to prove that

\frac{d(x − s_{V})}{dt} > 0, for t ≥ T, x = s_{V} (t), |y| ≤ x  (54)
\frac{d(y − s_{V})}{dt} < 0, for t ≥ T, y = s_{V} (t), |x| ≤ y  (55)

Observe that

s′_{V} (t) = −are^{−a(t−T)} = −as_{V} (t)  (56)

Let us consider (54). From Lemma 14 in [WZ] it follows that \frac{dx}{dt} > 0 for 0 < x ≤ r and |y| ≤ x. We have −\frac{ds_{V}}{dt} > 0, so (54) holds. Observe that

\frac{d(y − s_{V})}{dt} = (a − 1)y + (x^{2} + y^{2})(x sin(κt) − y cos(κt))  (57)

Hence to prove (55) we have to show that

L := (a − 1)s_{V} (t) + (x^{2} + s_{V}^{2} (t))(x sin(κt) − s_{V} (t) cos(κt)) < 0  (58)

for t ≥ T and |x| ≤ s_{V} (t). We will do it in two steps. First for T ≤ t ≤ T + \frac{π}{2κ} and then for t > T + \frac{π}{2κ}. Let T ≤ t ≤ T + \frac{π}{2κ}, then cos(κt) ≥ 0 and s_{V} (t) ≤ r. Hence we have L ≤ (a − 1)s_{V} (t) + 2s_{V}^{3} (t) sin(κt) − s_{V}^{3} (t) cos(κt) ≤ (a − 1)s_{V} (t) + 2s_{V}^{3} (t) =

s_{V} (t)(a − 1 + 2s_{V}^{2} (t)) ≤ s_{V} (t)(a − 1 + 2r^{2})

Since r < 0.6 then

a − 1 + 2r^{2} < a − 1 + 0.72 = a − 0.28  (59)

Hence if a < 0.28 then L < 0. This proves (58) for T ≤ t ≤ T + \frac{π}{2κ}. Assume t ≥ T + \frac{π}{2κ}. We have

s_{V} (t) ≤ re^{− \frac{aπ}{2κ}} < 0.6e^{−aπ} = 0.6e^{−π/4}  (60)

An easy computation shows that

L ≤ (a−1)s_{V} (t)+4s_{V}^{3} (t) = s_{V} (t)(a−1+4s_{V}^{2} (t)) ≤ s_{V} (t)(a−1+1.44e^{−π/2}) (61)

Observe that

a − 1 + 1.44e^{−π/2} < −0.75 + 1.44e^{−1} < −0.75 + 1.44/2 = −0.03  (62)

Hence L < 0. This finishes the proof of (58). □

Lemma 10 V is an isolating segment for φ^{M}. Moreover the function |x| − s_{v} is strictly increasing for trajectories of Φ^{M} in some neighborhood of V^{−}.

Proof The proof is almost the same as in the case of φ. The computations are even much easier, due to a simpler form of the equations describing φ^{M} as given in Theorem 8. □

5.2 Isolating segments W (n, k) and U(n, k)

Let n > 0, k ≥ 0. We define two (n + 2k)T –periodic isolating segments U(k, n) and W (k, n) as follows. To build one period of W (k, n) (U(k, n)) we take V_{[−kT,0]} followed by W_{[0,nT]}(U_{[0,nT]}) and then by V_{[0,kT]}. The formal definition is

Definition 8 Let T_{1} = (2k +n)T. The point (t+lT_{1}, x, y) ∈ W (k, n) (W (k, n)^{±}), where 0 ≤ t ≤ T_{1} and l ∈ ℤ, iff only one of the following conditions holds

if t ≤ kT, then  (t − kT, x, y) ∈ V (V^{±})
if kT ≤ t ≤ (k + n)T, then  (t, x, y) ∈ W (W^{±})
if (k + n)T ≤ t ≤ T_{1}, then  (t − (k + n)T, x, y) ∈ V (V^{±})

The definition of U(k, n) is analogous.

Lemma 11 Given n > 0, k ≥ 0.

Then for any λ ∈ [0, 1] the pairs (U(k, n), U(k, n)^{−}) and (W (k, n), W (k, n)^{−}) are (n + 2k)T – periodic isolating segments for H_{λ}.

Condition (vi) holds uniformly for H_{λ}, λ ∈ [0, 1] and the isolating segments (U(k, n), U(k, n)^{−}) and (W (k, n), W (k, n)^{−})

Proof

If k = 0, then (W (k, n), W (k, n)^{−}) = (W, W^{−}) and (U(k, n), U(k, n)^{−}) = (U, U^{−}) and the assertions follow from Theorem 7.

Assume now k > 0. Conditions (i)–(v) are obviously satisfied. It remains to show that (vi) holds uniformly.

We will show it for (W (k, n), W (k, n)^{−}) only. The reasoning for (U(k, n), U(k, n)^{−}) is analogous. Since by Theorem 7 (vi) holds uniformly for (W, W^{−}) and since V ⊂ U ⊂ W it follows immediately that (vi) holds uniformly with some η_{1} > 0 for p = (t, x, y) ∈ W (k, n)^{−} such that t ∈ [kT, (k + n)T] + lT_{1}.

Since for t ∈ ([kT − Δ, kT] ∪ [(k + n)T + Δ]) + lT_{1} W^{−}(n, k)_{t} = U_{t}. Therefore, condition (vi) holds uniformly with the same η_{1} for t ∈ [kT −Δ, (k+n)T +Δ]+lT_{1}.

It remains to consider (t, x, y) ∈ W (k, n)^{−}, which belong to the shifted set V_{t′}^{−} for some t′ and V_{t′}^{−} ⊂ int U_{t′}.

It follows from Theorem 8 and the proof of Lemma 9 that |x|−s_{V} is locally strictly increasing for φ^{M} and φ on B(V_{[−T_{1},T_{1}]}^{−}, η_{2}). Since every trajectory of H_{λ} consists from pieces of trajectories of φ and φ^{M}, we see that |x| − s_{V} is strictly increasing for H_{λ}, λ ∈ [0, 1] on B(V_{[−T_{1},T_{1}]}^{−}, η_{2}). Hence the assertion holds for η = min(η_{1}, η_{2}).

□

6 Construction of symbolic dynamics

In this section we will prove Theorem 1. The symbolic dynamics is defined as in [WZ]. We recall it here for the convenience of the reader, because the notation in this paper and [WZ] differ slightly. Let

S := I_{W} = {x ∈ W_{0} (t, φ(0, t, x) ∈ W, for t ∈ ℝ}  (63)

The map g : S → Σ_{3} = {−1, 0, 1}ℤ (denoted in [WZ] by g_{3}) is given by

g(x)_{l} := \{ \begin{matrix}0, & if (t, φ(0, t, x)) ∈ U for t ∈ (lT, (l + 1)T \\ −1, & if (lT, φ(0, lT, x)) leaves U in time less than T through U^{−1} \\ 1, & if (lT, φ(0, lT, x)) leaves U in time less than T through U^{+1} \end{matrix}

(64)

Let κ, W, U, H_{λ}, φ^{M} be as in [WZ] and V be the isolating segment constructed in section 5.1. We assume that we modify Equation (1) in outside some large ball (of radius A), to ensure that the process obtained by the modified equation is a global one. We will still call this process φ, because the symbolic dynamics we defined above is unaffected by this modifications if A > 2R.

6.1 The existence of homoclinic solutions

Lemma 12 For any n > 0, k ≥ 0 and β = (β_{k}, β_{k+1}, …, β_{k+n−1}) ∈ {−1, 0, 1}^{n}, there exists a point x ∈ [−s_{V} (nT), s_{V} (nT)]^{2} such that

φ(0, (2n + k)T, x) = x  (65)

and for i = k, …, k + n − 1 the following conditions hold

φ(0, t_{l}, x) ∈ int W (n, k)_{t_{l}}, for l = 0, …, 2k + n (66)
(t, φ(0, t, x)) ∈ int W (n, k), for t ∈ [0, (2n + k)T] (67)
if β_{i} = 0, then Φ_{iT +t}(0, x) = (iT + t, φ(0, iT + t, x)) ∈ int U
for t ∈ (0, T) (68)

if β_{i} ≠ 0, then Φ_{iT} (0, x) = (iT, φ(0, iT, x)) leaves U through U^{β_{i}} in time

less than T (69)

Proof

The idea of the proof is to apply the continuation theorem to the (n+2k)T -periodic isolating segments (W (k, n), W (k, n)^{−}) and (U(k, n), U(k, n)^{−}) and H_{λ}, and then to compute explicitly the appropriate fixed point index for φ^{M}.

Let us fix n, k, β. We set T_{1} = (2k + n)T.

When k = 0 then we set m = n and we define

t_{i} = iT, for i = 0, …, m  (70)
α_{i} = β_{i}, for i = 0, …, m − 1  (71)

For k > 0 we set m = n + 2 and we define

t_{0} = 0, t_{m} = T_{1}, t_{i} = kT + (i − 1)T, for i = 1, …, n + 1  (72)
α_{0} = 0, α_{m−1} = 0, α_{i} = β_{k+i−1}, for i = 1, …, n  (73)

Observe that U(k, n) ⊂ W (k, n) and

U(k, n)_{t_{i}} = W (k, n)_{t_{i}}, U(k, n)_{t_{i}}^{−} = W (k, n)_{t_{i}}^{−}, i = 0, …, m − 1

Now we apply the continuation theorem (Thm. 4) to H_{λ}, the isolating segments (W (k, n), W (k, n)^{−}) and (U(k, n), U(k, n)^{−}), {t_{i}} and α.

Hence we obtain

ind (φ_{(0,T_{1})}, (int W(k, n))_{α}(φ))) = ind (φ_{(0,T_{1})}^{M}, (int W(k, n))_{α}(φ^{M})))  (74)

We will now compute the index for φ^{M}. We will say that α is odd (even), if the number of its nonzero elements is odd (even). From Theorems 7 and 8 it follows immediately that there exists interval J_{α} = [a_{α}, b_{α}], where −s_{V} (kT) < a_{α} < b_{α} < s_{V} (kT), such that

(int W(k, n))_{α}(φ^{M})) = int J_{α} × [−s_{V} (kT), s_{V} (kT)]  (75)

and there exists a continuous function f_{α} : J_{α} → [−s_{V} (kT), s_{V} (KT)] such that

φ^{M}(0, T_{1}, (x, y)) = (f_{α}(x), 0), for (x, y) ∈ J_{α} × [−s_{V} (kT), s_{V} (kT)] (76)
f_{α} is strictly monotone  (77)
if α is odd, then f_{α}(a_{α}) = s_{V} (kT), f_{α}(b_{α}) = −s_{V} (kT)  (78)
if α is even, then f_{α}(a_{α}) = −s_{V} (kT), f_{α}(b_{α}) = s_{V} (kT)  (79)

It is now an easy exercise in the fixed point index computation to show that

ind (φ_{(0,T_{1})}^{M}, (int W(k, n))_{α}(φ^{M}))) = ind (f_{α}, int J_{α}) = ±1  (80)

The value −1 is obtained for even α and +1 for odd α. From (74)(75) and (80) the assertion of the lemma follows immediately. □

Lemma 13 Let α ∈ Σ_{3}. Assume that there exist j_{0} and i_{0} such that α_{k} = 0 if k < j_{0} or k > i_{0}. Then there exists a point x ∈ W_{0} such that, for any k ∈ ℤ the following conditions hold

lim_{|t|→∞} φ(0, t, x) = (0, 0) (81)
Φ_{t}(0, x) = (t, φ(0, t, x)) ∈ int W, for t ∈ ℝ (82)
if α_{i} = 0, then Φ_{iT +t}(0, x) = (iT + t, φ(0, iT + t, x)) ∈ int U
for t ∈ (0, T) (83)

if α_{i} ≠ 0, then Φ_{iT} (0, x) = (iT, φ(0, iT, x)) leaves U through U^{α_{i}} in time

less than T (84)

Moreover the following estimates are satisfied

|φ(0, t, x)|_{max} ≤ s_{V} (t − i_{0}T), for t ≥ i_{0}T  (85)
|φ(0, t, x)|_{max} ≤ s_{V} (j_{0}T − t), for t ≤ j_{0}T  (86)

Proof Without any loss of generality we may assume that j_{0} = 0 and i_{0} > 0. Let us set n = i_{0}. For any k > 0 we define β = (β_{k}, β_{k+1}, …, β_{k+n−1}) by

β_{k+i} = α_{i}, i = 0, …, n − 1 Let y_{k} be a point obtained from Lemma 12 applied to k, n, β. Observe that x_{k} = φ(0, kT, y_{k}) satisfies the following conditions for i = −k, −k + 1, …, k + i_{0}

φ(0, lT, x_{k}) ∈ W_{0}, for l = −k, …, i_{0} + k  (87)
(t, φ(0, t, x_{k})) ∈ int W, for t ∈ [−kT, (n + k)T]  (88)
if α_{i} = 0, then Φ_{iT +t}(0, x) = (iT + t, φ(0, iT + t, x_{k})) ∈ int U
for t ∈ (0, T)  (89)
if α_{i} ≠ 0, then Φ_{iT} (0, x) = (iT, φ(0, iT, x_{k}))  (90)

leaves U through U^{α_{i}} in time less

than T  (91)
|φ(0, t, x_{k})|_{max} ≤ s_{V} (t − i_{0}T), for t ∈ [i_{0}T, (k + i_{0})T]  (92)
|φ(0, t, x_{k})|_{max} ≤ s_{V} (t) = s_{V} (−t), for t ∈ [−kT, 0]]  (93)

Since W_{0} is compact, hence without any loss of generality we can assume that lim_{k→∞} x_{k} → x. Observe that for a fixed k the conditions (87–91) define by Lemma 3 a closed set. Hence (82–84) are satisfied also for x. The remaining assertions are true by (92–93). □

6.2 Orbits with prescribed one–sided asymptotic behavior

As was mentioned in the introduction Assertions 1–3 are already proved in [WZ]. First we will show that a strengthened form of Assertion 6 implies Assertions 4 and 5. This is a content of the following

Lemma 14 Let X be a topological space and let P : X → X be a continuous map.

Assume that S ⊂ X is a compact set, such that P (S) = S and there exists a continuous map g : S → Σ_{3} and a point q ∈ S, such that

6’. For any α ∈ Σ_{3} such that α_{i} = 0 for i > i_{0} and i < j_{0} g^{−1}(α) contains a point x such that lim_{n→±∞} P^{n}(x) = q and

ρ(P^{n}(x), q) < c(n − i_{0}), for n > i_{0}  (94)
ρ(P^{n}(x), q) < c(j_{0} − n), for n < j_{0}  (95)
where c : ℕ → ℝ_{+}, lim_{n−>∞} c(n) = 0

then

4’. For any α ∈ Σ_{3} such that α_{i} = 0 for i > i_{0} , g^{−1}(α) contains a point x such that lim_{n→∞} P^{n}(x) = q

5’. For any α ∈ Σ_{3} such that α_{i} = 0 for i < i_{0} , g^{−1}(α) contains a point x such that lim_{n→−∞} P^{n}(x) = q

Proof For i = −1, 0, 1 we set

S_{i} = {x ∈ S | g(x)_{0} = i}  (96)

where the subscript 0 denote the 0-th coordinate of the point g(x) in Σ_{3}.

It is easy to see that q ∈ S_{0} and

g(x) = α  iff  P^{n}(x) ∈ S_{α_{n}}, for n ∈ ℤ  (97)

To show 4’ we fix α ∈ Σ_{3}, such that α_{i} = 0 for i > i_{0}. For any k > 0 we define β^{k} ∈ Σ_{3} by

β_{i}^{k} = \{ \begin{matrix}α_{i}, & for i > −k \\ 0, & otherwise \end{matrix}  (98)

It is easy to see that β^{k} has only finitely many nonzero elements. Hence we may apply to it assumption 6’ to obtain a point x_{k} such that

P^{n}(x_{k}) ∈ S_{α_{n}}, for n > −k

Passing eventually to a subsequence we may assume that the sequence x_{k} converges to x. Hence in the limit we obtain

P^{n}(x) ∈ S_{α_{n}}, for n ∈ ℤ  (99)

It remains to show that P^{n}(x) → q. Observe that for k > i_{0} we have an estimate

ρ(P^{n}(x_{k}), q) ≤ C(n − i_{0})  (100)

This estimate is preserved in the limit, hence P^{n}(x) → q.

The proof of 5’ is analogous. □

6.3 Proof of the main theorem

Proof:

The compactness, the invariance of S and the continuity of P and Assertion 1 are obvious.

Assertion 3 follows immediately from Lemma 12 applied to k = 0 and n being equal to the principal period of α.

Assertion 2 follows from Assertion 3 by observing that g(S) is compact and periodic sequences are dense in Σ_{3}.

Observe that Assertion 6 follows from Lemma 13. Assertions 4 and 5 follow from Assertion 6 and Lemma 14. □

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