Strong Pairs of Periodic Segments

We introduce the notion of a strong pair of periodic segments over [0, T] and we show its applications in detecting chaotic dynamics. We prove a various number of fixed point index formulas concerning periodic points of the Poincaré map.


Introduction
The machinery of periodic segments was introduced by Srzednicki in order to investigate the existence of periodic points of the Poincaré map associated to periodic in time ODE [10]. It is based on the Ważewski retract method and closely related to the Conley index theory [1,11]. A geometric method for detecting chaotic dynamics based on the existence of special configurations of periodic segments was introduced in [12] and developed in [6-9, 13, 16, 18, 20]. We very briefly describe the notion of periodic segment. If v : R × M −→ T M is a smooth T -time periodic vector field on manifold M then the system of equationṡ generates a local flow on the extended phase space R × M. Let W ⊂ [0, T ] × M be such that W 0 = W T , where W t ⊂ M is a t-section of W . By W − we denote the exit set of W i.e., W − is the set of boundary points of W at which the vector field (1, v) is pointing out with respect to W . We say that W is a periodic segment over [0, T ] iff the following conditions hold • W and W − are compact ENRs, • there exists a compact subset W −− of W − (called the essential exit set) such that A homeomorphism h in a natural way induce a corresponding monodromy homeomorphism m W : W 0 −→ W 0 . It follows that m W (W −− 0 ) = W −− 0 . We put m W := m W | W −− 0 . Let P be the Poincaré map associated to the vector field v. Srzednicki proved that if L(m W ) = L(m W ) (L is a Lefschetz number) then P has a fixed point x ∈ W 0 \ W −− 0 . Moreover, the orbit of x with respect to φ is contained in W . Motivated by the results in [12,13,16,18] we introduce the notion of a strong pair of periodic segments. We say that (W , Z ) is a strong pair of periodic segments if the following conditions hold: We very briefly describe the way in which a strong pair of periodic segments (W , Z ) is related to symbolic dynamics. Let I ⊂ W 0 be the set of all points x ∈ W 0 whose full orbit with respect to φ are contained in the translated copies of the bigger segment W . It follows that I is compact and invariant for the Poincaré map P. Let 2 be the space of the sequences of two symbols. There is a natural, geometric way to define a continuous map q : I −→ 2 such that where σ : 2 −→ 2 is the shift map. This is a natural question in the context of chaotic dynamics. We recall that if q is surjective then P is chaotic in the sense of Devaney [3]. One way to achieve this is to show that q(I ) contains a dense set of periodic sequences. We use the fixed point index theory and the Lefschetz fixed point theorem [4] to attack Problem 1. It turns out that the homeomorphism m W : plays a key role here. It follows by Theorem 16 that the answer to the Problem 1 is positive provided where c W := card {s ∈ {0, . . . , k − 1} : c s = 1} ≥ 1.

Problem 2
Give the conditions ensuring that the answer to Problem 1 is positive for each periodic sequence c ∈ 2 .
Unfortunately, L((μ W − I H (W −− 0 ) ) c W ) = 0 in many natural cases, so we need a different approach. We will use the connected components of Z −− . We show (Theorem 40) that it is sufficient to assume that The main tool in the proof is the fixed point index formula (5) in Theorem 31.
Assume that E(1), . . . , E(n) are the connected components of Z −− . If the orbit of x ∈ I leaves the smaller segment Z it has to do it through one of components of Z −− . This gives us a possibility to define geometrically a continuous map q :
We prove (Proposition 44) that the answer to the Problem 3 is positive for each periodic sequence c ∈ 3 provided Z −− has exactly two connected components E(1), E (2) such that the following conditions hold and 2χ(E(1) 0 ) / ∈ {0, χ(W 0 )}. We compare our main results to the techniques developed in [10-13, 18, 19]. We first focus on the geometric method for detecting fixed points of the Poincaré map P presented in [10,11]. Assume that (W , Z ) is a strong pair of periodic segments and L(μ W ) = 0. It follows by Theorem 7.7 in [11] (compare Theorem 7.1 in [10]) that P has a fixed point provided χ(W −− 0 ) = 0. Theorem 24 allows us to show a multiplicity result. More precisely, if W −− 0 has k connected components that are not invariant for m W and having non-zero Euler characteristic then P has at least k fixed points.
The continuation method was used in [18] to show that the Poincaré map P associated to the local process 0 generated by the planar periodic systeṁ z = (1 + e iκt |z| 2 )z is 3 -chaotic for κ > 0 sufficiently small. The proof of the main Theorem 8 in [18] rely on the following ingredients: (i) The existence of a strong pair of periodic segments (W , Z ) such that W −− 0 consists of two contractible, connected components E 0 , F 0 and m W (E 0 ) = F 0 , (ii) A construction of a model semi-process 1 having the same strong pair of periodic segments (W , Z ) (Theorem 20 in [18]), (iii) The existence of some appropriate homotopy between processes 0 and 1 (see Remark 33 for more details), (iv) Theorem 10 in [18] concerning the continuation of the involved fixed point indices, (v) A computation of the fixed point indices of the model Poincaré map for 1 .
The proof of Theorem 10 given in [18] is quite difficult and technically complicated. Fortunately, it turns out that it is a simple consequence of our Corollary 32. Moreover, Proposition 44 shows that all what is needed in order to get a conclusion of Theorem 8 in [18] is the point (i). In particular, we can avoid the challenging points (ii-iii) (see Sect. 6 in [18]). This is particularly important from the point of view off the construction of model processes, which may be particularly difficult for the general strong pairs of periodic segments.
The fixed point index formula in Theorem 31 is purely topological and is not a consequence of analytical results in [18,19]. It allows us to replace the construction of a model processes with combinatorial and number-theoretic problems involving the fixed point indices.

Periodic Segments
In this section we give a brief exposition of the Srzednicki method. For the proofs we refer the reader to [12,13]. Let X be a metric space and let D ⊂ [0, ∞) × X be an open set. A continuous map φ : D −→ X is called a local semi-flow on X if for every x ∈ X the set {t ∈ R : (t, x) ∈ D} is equal to an interval I x = [0, ω x ) with 0 < ω x ≤ ∞, and the following conditions hold: and if (t, x) ∈ D, (s, φ(t, x)) ∈ D then (t + s, x) ∈ D and Let W ⊂ X . Define the exit set of W as We call W a Ważewski set for φ if it is closed and its exit set W − is closed as well.
A compact Ważewski set is called a block. In the case φ is a local flow we say that a block W is isolating if the boundary W is equal to the union of W − and the entry set W + defined as the exit set of W with respect to the local flow with reversed time. The isolating blocks are basic notion for the Conley index theory of the isolated invariant sets [1,2,11]. If W is a Ważewski set and then the escape-time map is continuous. Consequently, W − is a strong deformation retract of W * . Deformation is given by the flow. This is a famous Ważewski retract theorem: if W − is not a strong deformation retract of a Ważewski set W then φ + (x) ⊂ W for some x ∈ W . By a local semi-process on X we mean a continuous map : for each σ and t. In that case the map P = (0,T ) , called the Poincaré map, satisfies (0,nT ) = n (0,T ) .
it is a block with respect to φ such that the following conditions hold: Obviously, W and W −− are topological periodic sets over [0, T ]. Let m W : W 0 −→ W 0 be the corresponding monodromy homeomorphism given by In particular, W and W −− are the topological periodic sets over [0, T ]. A homeomorphismm W induces the isomorphism in the singular homologies (with a coefficients in Q):μ It can be proved that a different choice of the homeomorphism h provides the monodromy map homotopic tom W , so μ W is an invariant of the segment W [17]. We put If W and W −− are ENRs (i.e., euclidean neighborhood retracts), then H (W 0 , W −− 0 ) is of finite type and the Lefschetz number is correctly defined. In particular, where χ is the Euler-Poincaré characteristic. Let k ∈ Z. By τ k we denote the translation i.e., W ∞ is obtained by gluing together the translated copies of W . For a T -periodic local proces and a periodic segment W over [0, T ] we define the set of all points in W 0 whose full trajectories are contained in W ∞ i.e., Assume that X is an ENR and f : U −→ X is a continuous map, where U ⊂ X is open. If the set of fixed points of f is compact then the fixed point index ind( f ) is well defined [4]. Sometimes, it is more convenient to use the notation ind an open set isolating a compact set of fixed points Fix ( f ). We will use the following properties of the fixed point index: is open in W 0 , the set of fixed points of the restriction P| U W : In particular, if L(μ W ) = 0 then P has a fixed point in U W .

Strong Pair of Periodic Segments
Let be a local semi-process on a metric space X .
It follows that We define a homotopy It follows that so the proof is complete.

Definition 7
We say that (W , Z ) is a strong pair of periodic segments over [0, T ] if the following conditions hold: We assume a technical condition (iv) in order to simplify formulation of some further results.

Moreover, if the sequence L(μ n W ) is non-constant, then the Poincaré map has a periodic point in W
where F i are the Nielsen fixed point classes. We say that a fixed point class

Definition 9
The relative Nielsen number ofm W on the closure of the complement is the number of the Nielsen fixed point classes of m W that do not assume its index in W −− 0 [21].
The following was proved in [17].

Fixed Point Index Formula and Chaotic Dynamics
Assume that n = {0, . . . , n − 1} N and σ : n −→ n is the shift map. Let P ⊂ n = {1, . . . , n} N be the set of all periodic sequences of n symbols.

Definition 13
Let be a T -periodic local semi-process. We say that the corresponding Poincaré map P has a n -weak chaotic dynamics on some compact, invariant set I if the following two conditions hold: (ch1) there exists a continuous, surjective map q : I −→ n such that q • P| I = σ • q, (ch2) there exists D ⊂ P such that D is dense in n and if c ∈ D is a k-periodic sequence, then g −1 (c) contains a k-periodic point of P.
We say that P has a n -chaotic dynamics if it has a n -weak chaotic dynamic with D = P.
Let (W , Z ) be a strong pair of periodic segments over [0, T ]. We define a Ważewski set W ∞ ⊂ [0, ∞) × X by: i.e., W ∞ is obtained by gluing together the translated copies of W . The set I is the set of all points in W 0 whose full trajectories are contained in W ∞ i.e., It follows that I is compact and invariant for P. For x ∈ I we define q(x) ∈ 2 by the following rule: The map q is continuous and satisfies (ch1) [13].

Corollary 14
Let g : I −→ 2 be a continuous map defined above. If (ch2) holds then P has a 2 -weak chaotic dynamics.
Proof It remains to show that g is surjective. It follows that D ⊂ g(I ), so g(I ) = 2 by density of D in 2 .
Assume that (W , Z ) is a strong pair of periodic segments over

It follows that (W [c], Z k ) is a strong pair of periodic segments over [0, kT ] and
We say that a point x ∈ W [c] follows a sequence c if the following condition holds: for some t ∈ (i T , (i + 1)T ). We put  [13].
We are ready to formulate the main result of this section.

Theorem 16 Assume that (W , Z ) is a strong pair of periodic segments and c
For n ≥ 1 we have because m W id W 0 by Lemma 6. We putμ 0 := I H (W 0 ,W −− 0 ) and μ 0

Corollary 17 Assume that (W , Z ) is a strong pair of periodic segments over [0, T ] for the local semi process . Let
If D ⊂ P is dense in 2 then the Poincaré map P has a 2 -weak chaotic dynamics.
Proof Let c ∈ D. It follows by Theorem 16 that g −1 (c) contains a fixed point of P k , so (ch2) holds.

Example 19
For a given l ∈ N we consider the basic sequence reg l defined as Assume that L(m n W ) = reg l (n) for n ≥ 1. It follows by Remark 3.2 in [17] that L((m W − I H (W −− 0 ) ) m ) = 0 iff l is odd and m is odd multiplicity of l.
In order to shed more light on behavior of the sequence L((μ W − I H (W −− 0 ) ) n ), we will focus on the sequence tr(A − I k ) n , where A is k × k-integer matrix. If λ ∈ σ (A) is the eigenvalue of A, then by m A (λ) we denote its algebraic multiplicity. Let R ⊂ C be the set of all complex roots of unity i.e., λ ∈ R iff λ n = 1 for some n ∈ N. If A is an integer matrix then the following conditions are equivalent ( [5]): (a) the sequence of traces tr A n is bounded, (b) the sequence tr A n is periodic,

Using the Connected Components of Z −−
Assume that (W , Z ) is a strong pair of periodic segments over [0, T ] for a local semiprocess. Let E be the connected component of Z −− . We say that Observe that m 0 = m W and m T = id W 0 . (1) g E is continuous,

Lemma 21 Assume that (W , Z ) is a strong pair of periodic segments over [0, T ] and E is a connected component of Z
It is sufficient to show that g E is continuous in This leads to a contradiction because Z −− \ E is closed. It follows that (0,σ Z (0,x n )) (x n ) ∈ W T \W −− T , so x n ∈ U Z . In particular, σ Z (0, x n ) = T and g E (x n ) = P(x n ). On the other hand, we have g E (x 0 ) = P(x 0 ). Consequently, g E (x n ) → g E (x 0 ) and g E is continuous. We show that g E is homotopic to id W 0 . Let H : [0, 1] × W 0 −→ W 0 be given by Consequently, H is continuous. We have m 0 = m W and m 1 = g E , hence g E m W and consequently, g E id W 0 by Lemma 6.

Example 22 Assume that Z −− has a three connected components E(1), E(2), E(3),
Both sides restricted to E(3) 0 are the identity maps. We have . On the other hand, .

Corollary 23
Let E be a connected component of Z −− . The following conditions hold:

leaves Z through E in time less then T ,
We do not provide a proof of the above result here since it is a corollary of a more general Theorem 31. Let E, F be some two connected components of Z −− . We have the following four possibilities:

Lemma 26
Assume that E, F are connected components of Z −− and l ≥ 1. Then

Lemma 27
Assume that E, F are connected components of Z −− and l ≥ 1. Then Proof It follows by the same arguments like the ones used in the proof of Lemma 26.
Proof The condition (1) is a consequence of Corollary 23. Assume that (c1) holds.
In particular, if c W = 1 then

Moreover
hence the result follows.
and is independent on the local semi-processes i (i = 1, 2). Consequently, the result follows by Theorem 31 and the induction on c W .

Remark 33
Corollary 32 is a generalization of Theorem 10 in [18]. The equality (6) was proved there under the following assumptions: • i (i = 0, 1) are local semi-processes on R d joined by a continuous family of local semi-processes t (t ∈ [0, 1]), • (W , Z ) is a strong pair of periodic sequences for each t , • there exists η > 0 such that for every λ ∈ [0, 1] and for every .
Proof We use the induction with respect to n = c W . If n = 1 then the result follows by Theorem 31. Assume that formula holds for all sequencesb with b W ≤ n. We prove it for a sequencec with c W = n + 1. It follows that everyb <c is a hereditary sequence with L(gb) = L(gc). By Theorem 31 and the inductive step we get

Corollary 35 Assume that E is a connected component of Z
In particular, the following conditions are equivalent:  ).
Let S ∈ {E, F}. We denote by 1 ≤ |S| ≤ n the number of appearances of S inc. It follows by Corollary 35 and where κ > 0 is a real parameter. The Eq. (7) is T = 2π/κ periodic. Let be a generated T -periodic local process. It follows by Example (6.8) in [13] that for sufficiently small κ the process admits a strong periodic segment W (n) over [0, T ]. We describe briefly how the segments W (n) look like. The segment W (n) is a twisted prism with a 2(n + 1)-gon base W (n) 0 centered at origin. Its time sections W (n) t are obtained by rotating W (n) 0 with the angular velocity κ n+1 over the the time interval