Sharkovskii theorem for multidimensional perturbations of one-dimensional maps II

Piotr Zgliczyński ^{1} ^{2} Center for Dynamical Systems and Nonlinear Studies, School of Mathematics, Georgia Institute of Technology Atlanta, Georgia 30332 e-mail: piotrz@math.gatech.edu August 30, 1999

Keywords: fixed point index,chaos,Sharkovskii’s theorem

0 Introduction

We present a multidimensional generalization of the Sharkovskii’s Theorem concerning the appearance of periodic points for the self-maps on the real line.

Let f : X → X be a map. We say that x ∈ X is a periodic point of period n if f^{n}(x) = x and f^{k}(x) ≠ x for k = 1, 2, …, n − 1.

We begin with recalling the Sharkovskii’s Theorem.

Theorem 1 Let the ordering of positive integers be as follows:

3⊲5⊲7⊲…⊲2·3⊲2·5⊲⊲2·7 …⊲2^{2} ·3⊲2^{2} ·5 …⊲2^{3} ·3⊲2^{3} ·5 …⊲2^{3} ⊲2^{2} ⊲2⊲1

Let f : I → ℝ be a continuous map of an interval into the real line. If n ⊲ k and f has a periodic point of period n then f also has a periodic point of period k.

The ordering described in Theorem 1 is called the Sharkovskii’s ordering. We reserve the symbol ’⊲’ for this order.

Let f : ℝ → ℝ be a continuous map. Let V be a real Banach space and let the continuous decomposition V = ℝ ⊕ W be given. According to this decomposition we will represent elements v ∈ V as pairs v = (x, w), where x ∈ ℝ and w ∈ W. Let F : [0, 1]×V → V be a continuous and compact map. We will use the notation F_{λ} for the partial map with the fixed λ, so F_{λ}(v) := F (λ, v) for v ∈ V. Suppose that F_{0}(x, w) = (f(x), 0). We say that the maps F_{λ} are multidimensional perturbations of f. Let us recall that a continuous map is called compact, iff it maps bounded sets into relatively compact sets.

The main theorem proved in this paper is stated below

Theorem 2 Let f, F be as above. Suppose f has a point of period n.

For every r > 0 there exists λ_{0} > 0 such that, for λ ≤ λ_{0}, if n ⊲ m, m ≠ n then F_{λ} has a periodic point of period m in the set ℝ ⊕ B_{W} (r), where B_{W} (r) is an open ball in W of radius r.

1supported by Polish KBN grant 2P03A 021 15, 2P03A 029 12 and NSF–NATO grant DGE– 98–04459

2on leave from: Institute of Mathematics, Jagiellonian University, Cracow, Poland; e-mail:

zgliczyn@im.uj.edu.pl

Above theorem was proved in [Z1] with the additional assumption that F is uniformly continuous.

In this paper we present only modifications to the arguments from [Z1] required for the proof of Theorem 2, so without referring to [Z1] this note is rather unreadable.

Sections 1–5 correspond to analogous sections in [Z1]. In all these sections but 3 even the header of the section is preserved.

The idea of the modifications can be explained as follows: In study of periodic points it is enough to consider only points which are in the image of the map, which in our case is a compact set.

Another modification is a simpler proof of the existence of a nested sequence of topological horseshoes for one-dimensional maps (see section 3).

Section 6 contains some remarks concerning the relevance of the presented results in the context of ordinary differential equations.

1 Topological theorems

This section is a modification of analogous section in [Z1]. We will use the notations used there. The changes begin with new definition of covering relations.

Let us fix an r > 0. We define N, L(N), R(N), S_{L}(N), S_{R}(N) as in [Z1].

Definition 1 Let f : (−∞, ∞) × \overline{B_{W} (r)} → V be a continuous and compact map and A ⊂ V such that

\overline{f(N)} ∩ N ⊂ A, A ∩ L(N_{i}) ≠ ∅, A ∩ R(N_{i}) ≠ ∅.

We say that N_{i} f-covers horizontally N_{j} with respect to A iff

f(A ∩ N) ⊂ (−∞, ∞) × B_{W} (r)  (1)

and either

f(L(N_{i}) ∩ A) ⊂ S_{L}(j) and f(R(N_{i}) ∩ A) ⊂ S_{R}(j)  (2)

or

f(L(N_{i}) ∩ A) ⊂ S_{R}(j) and f(R(N_{i}) ∩ A) ⊂ S_{L}(j).  (3)

We will use the following graphical notation for this relation N_{i} ⟹^{f,A} N_{j}. In the case N ⊂ A we will often omit the set A (this coincides with the notation from [Z1] ).

Let us fix an n > 0. Let α = (α_{0}, α_{1}, …, α_{n−1}) ∈ {0, 1, …, K − 1}^{n}.

Let Z = ⋃_{i=0}^{K−1} Z_{k} ⊂ V. For an indexed family of the continuous maps f_{i} : (−∞, ∞) × \overline{B_{W} (r)} → V, i = 1, 2, …, n define

Z_{α} := {v ∈ Z_{α_{0}}|f_{i} ∘ f_{i−1} ∘ … ∘ f_{1}(v) ∈ Z_{α_{i}}, for i = 1, 2, …, n − 1}

and similarly, for an indexed family of homotopies F_{i} : [0, 1]×(−∞, ∞)×\overline{B_{W} (r)} → V, i = 1, 2, …, n, F_{i,λ} we set

Z_{α}^{λ} := {v ∈ Z_{α_{0}}|F_{i,λ} ∘ F_{i−1,λ} ∘ … ∘ F_{1λ}(v) ∈ Z_{α_{i}}, for i = 1, 2, …, n − 1}

The following heorem is a generalization of theorems about topological horseshoes by Mischaikow & Mrozek [MM, Th. 2.3] and Zgliczynski [Z].

Theorem 3 Let α = (α_{0}, α_{1}, …, α_{n−1}) ∈ {0, 1, …, K − 1}^{n}. Let the maps F_{i} : [0, 1] × (−∞, ∞) × \overline{B_{W} (r)} → V, for i = 1, 2, …, n be continuous and compact. Let

Im F = ⋃_{i} \overline{F_{i}([0, 1] × N)}

Suppose that we have the following relations for every λ ∈ [0, 1]

N_{α_{i−1}} ⟹^{F_{i,λ},Im F} N_{α_{i}}, for i = 1, …, n − 1

N_{α_{n−1}} ⟹^{F_{n,λ},Im F} N_{α_{0}}

then the fixed point index I(F_{n,λ} ∘ F_{n−1,λ} ∘ … ∘ F_{1,λ}, intN_{α}^{λ}) is well defined and constant (i.e. does not depend on λ).

Proof

For δ > 0 define

L(N_{i}, δ) := [a_{2i}, a_{2i} + δ] × \overline{B_{W} (r)}  (4)
R(N_{i}, δ) := [a_{2i+1} − δ, a_{2i+1}] × \overline{B_{W} (r)}  (5)
V (N_{i}, δ) := L(N_{i}, δ) ∪ R(N_{i}, δ)  (6)
H(N_{i}, δ) := [a_{2i}, a_{2i+1}] × (\overline{B_{W} (r)} \ B_{W} (r − δ))  (7)
H(N, δ) := H(N_{0}, δ) ∪ H(N_{1}, δ) ∪ … ∪ H(N_{K−1}, δ)  (8)

For δ < a_{2i+1} − a_{2i} the sets L(N_{i}, δ), R(N_{i}, δ), H(N_{i}, δ) are contained in N_{i} and are equal to the δ-thickened left vertical, right vertical and the union of horizontal edges in N_{i}, respectively.

Let Z ⊂ N. We introduce the notation

Fix_{λ}(Z) := {x ∈ Z | F_{n,λ} ∘ F_{n−1,λ} ∘ … ∘ F_{1,λ}(x) = x}  (9)

It follows from the compactness of F_{i} that Im F is a compact set. It is easy to see that for λ ∈ [0, 1]

Fix_{λ}(N_{α}^{λ}) ⊂ Im F  (10)

Obviously Fix_{λ}(N_{α}^{λ}) is a compact set.

From the compactness of ImF and assumptions concerning covering relations we easily conclude that there exists δ > 0, such that for i = 1, …, n and λ ∈ [0, 1] the following conditions hold

F_{i,λ}(V (N_{α_{i−1}}, 2δ) ∩ Im F) ∩ N_{α_{i mod n}} = ∅  (11)
H(N_{α_{i}}, 2δ) ∩ F_{i,λ}(N_{α_{i−1}} ∩ Im F) = ∅  (12)

We define

C_{i} := (a_{2i} + δ, a_{2i+1} − δ) × B_{W} (r − δ)  (13)
D_{i} := (a_{2i} + 2δ, a_{2i+1} − 2δ) × B_{W} (r − 2δ)  (14)

We have

B(D_{i}, δ) ⊂ C_{i},  B(C_{i}, δ) ⊂ IntN_{i}  (15)

Obviously the sets D_{α}^{λ} and C_{α}^{λ} are both open and the set N_{α}^{λ} is closed for λ ∈ [0, 1].

We will show now that for λ ∈ [0, 1]

F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}(x) ≠ x, provided x ∈ N_{α}^{λ} \ D_{α}^{λ}.  (16)

Suppose that (16) does not hold. Then there exists an x ∈ N_{α}^{λ} \ D_{α}^{λ} and 0 ≤ i_{0} ≤ n − 1 such that

F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}(x) = x  (17)
F_{i_{0},λ} ∘ F_{i_{0}−1,λ} ∘ … F_{1,λ}(x) ∈ (N_{α_{i_{0}}} \ D_{α_{i_{0}}} )  (18)

It follows from (12) and (14) that

F_{i_{0},λ} ∘ F_{i_{0}−1,λ} ∘ … F_{1,λ}(x) ∈ V (N_{α_{i_{0}}} , 2δ)  (19)

It follows from this and from (11) that

F_{i_{0}+1,λ} ∘ F_{i_{0},λ} ∘ … F_{1,λ}(x) ∉ N_{α_{(i_{0}+1) mod n}}  (20)

So if i_{0} < n − 1 then x ∉ N_{α}^{λ} and if i_{0} = n − 1 then it follows from (20) that x can not be a fixed point for the map F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}. In both cases we get a contradiction, so (16) holds.

From (16) it follows immediately that

Fix_{λ}(D_{α}^{λ}) = Fix_{λ}(C_{α}^{λ}) = Fix_{λ}(intN_{α}^{λ}) = Fix_{λ}(N_{α}^{λ}).  (21)

Let λ_{0} ∈ [0, 1]. It follows from the uniform continuity of continuous maps on compact sets that there exists an interval Λ open in [0, 1], λ_{0} ∈ \overline{Λ}, such that for every λ_{1}, λ_{2} ∈ \overline{Λ}, i = 1, …, n and x ∈ ImF holds

|F_{i,λ_{1}} ∘ F_{i−1,λ_{1}} ∘ … ∘ F_{1,λ_{1}}(x) − F_{i,λ_{2}} ∘ F_{i−1,λ_{2}} … ∘ F_{1,λ_{2}}(x)| ≤ δ.  (22)

We will show now the following inclusions for λ ∈ \overline{Λ}

Im F ∩ D_{α}^{λ} ⊂ Im F ∩ C_{α}^{λ_{0}} ⊂ Im F ∩ int(N_{α}^{λ}).  (23)

Let x ∈ Im F ∩ D_{α}^{λ}. Then

F_{i,λ} ∘ F_{i−1,λ} ∘ … F_{1,λ}(x) ∈ D_{α_{i}}, for i = 1, …, n − 1.

However, it follows from (15), (22) that

F_{i,λ_{0}} ∘ F_{i−1,λ_{0}} ∘ … F_{1,λ_{0}}(x) ∈ C_{α_{i}}, for i = 0, …, n − 1.

Thus x ∈ Im F ∩ C_{α}^{λ_{0}}. The proof of the second inclusion is analogous.

Because Fix_{λ}(C_{α}^{λ_{0}}) ⊂ ImF, we obtain from (21) and (23) for λ ∈ \overline{Λ}

Fix_{λ}(C_{α}^{λ_{0}}) = Fix_{λ}(N_{α}^{λ})  (24)

Since Fix_{λ}(N_{α}^{λ_{0}}) is a compact set then it follows from (21) and (24) that the fixed point indices of the map F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ} relatively to the sets intN_{α}^{λ}, C_{α}^{λ_{0}}, D_{α}^{λ} are well defined and from the excision property of the fixed point index [Z1] we conclude

I(F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}, D_{α}^{λ}) =
I(F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}, C_{α}^{λ_{0}}) =  (25)
I(F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}, intN_{α}^{λ}), ∀λ ∈ \overline{Λ}.

Substituting λ := λ_{0} we derive

I(F_{n,λ_{0}} ∘ F_{n−1,λ_{0}} ∘ … F_{1,λ_{0}}, D_{α}^{λ_{0}}) = I(F_{n,λ_{0}} ∘ F_{n−1,λ_{0}} ∘ … F_{1,λ_{0}}, C_{α}^{λ_{0}}). (26)

From (21)(24) it follows that

Fix\overline{Λ}(C_{α}^{λ_{0}}) := ⋃_{λ∈\overline{Λ}} Fix_{λ}(C_{α}^{λ_{0}}) ⊂ C_{α}^{λ_{0}}  (27)
Fix\overline{Λ}(C_{α}^{λ_{0}}) = ⋃_{λ∈\overline{Λ}} Fix_{λ}(N_{α}^{λ})  (28)

From the above condition and the compactness of the maps F_{i,λ} we see that Fix\overline{Λ}(C_{α}^{λ_{0}}) is compact.

Now from the homotopy property of the fixed point index we obtain

I(F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}, C_{α}^{λ_{0}}) = I(F_{n,λ_{0}} ∘ F_{n−1,λ_{0}} ∘ … F_{1,λ_{0}}, C_{α}^{λ_{0}}) ∀λ ∈ \overline{Λ}. (29)

From (25),(26),(29) we conclude

I(F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}, D_{α}^{λ}) = I(F_{n,λ_{0}} ∘ F_{n−1,λ_{0}} ∘ … F_{1,λ_{0}}, D_{α}^{λ_{0}}) ∀λ ∈ \overline{Λ}. (30)

From the connectedness of [0, 1], (30) and (25) we get

I(F_{n,λ} ∘ F_{n−1,λ} ∘ … F_{1,λ}, IntN_{α}^{λ}) = I(F_{n,0} ∘ F_{n−1,0} ∘ … F_{1,0}, IntN_{α}^{0}) ∀λ ∈ [0, 1]

This finishes the proof. □

The following theorem calculates the fixed point index of the composition F_{n,λ} ∘ F_{n−1,λ} ∘ … ∘ F_{1,λ} on the set int N_{α}^{λ} for multidimensional perturbations of onedimensional maps.

Theorem 4 Let α = (α_{0}, α_{1}, …, α_{n−1}) ∈ {0, 1, …, K −1}^{n}. Let F_{i} : [0, 1]×N → V, i = 1, …, n be continuous and compact. Suppose that there exist the onedimensional maps f_{i} : ℝ → ℝ such that

F_{i,0}(x, y) = (f_{i}(x), 0), for i = 1, …, n

Let

Im F = ⋃_{i} \overline{F_{i}([0, 1] × N)}

Suppose that the we have the following relations for every λ ∈ [0, 1]

N_{α_{i−1}} ⟹^{F_{i,λ},Im F} N_{α_{i}}, for i = 1, …, n − 1

N_{α_{n−1}} ⟹^{F_{n,λ},Im F} N_{α_{0}}

then

I(F_{n,λ} ∘ F_{n−1,λ} ∘ … ∘ F_{1,λ}, intN_{α}^{λ}) = ±1

Proof

It was shown in [Z1, th. 4] that

I(F_{n,0} ∘ F_{n−1,0} ∘ … ∘ F_{1,0}, intN_{α}^{0}) = ±1

The assertion of the theorem follows now from Theorem 3. □

Now we are going to define notions of a topological horseshoe and a topological prehorseshoe

Let a_{0} < a_{1}, a_{2} < a_{3}, (a_{0}, a_{1}) ∩ (a_{2}, a_{3}) = ∅ and r > 0. N_{0} = [a_{0}, a_{1}] × \overline{B_{W} (r)} and N_{1} = [a_{2}, a_{3}] × \overline{B_{W} (r)}. N = N_{0} ∪ N_{1} Definition 2 A continuous and compact map f : N → V will be called a topological horseshoe iff there exists a compact homotopy F : [0, 1] × N → V, such that F_{1} = f and F_{0} is one-dimensional, such that for A := \overline{F ([0, 1] × N)} the following covering relations hold

N_{i} ⟹^{F_{λ},A} N_{j}, for i, j = 0, 1

Definition 3 A continuous and compact map f : N → V will be called a topological prehorseshoe iff there exists a compact homotopy F : [0, 1]×N → V, such that F_{1} = f and F_{0} is one-dimensional, such that for A := \overline{F ([0, 1] × N)} the following covering relations hold

N_{0} ⟹^{F_{λ},A} N_{0}, N_{0} ⟹^{F_{λ},A} N_{1}, N_{1} ⟹^{F_{λ},A} N_{0}

With these definitions the theorems from [Z1] concerning the existence of symbolic dynamics for topological (pre)horseshoes are also true.

2 Nested sequences of topological horseshoes

Definition 4 Let f : (−∞, ∞) × \overline{B_{W} (r)} → V be continuous and compact. Let l ∈ ℕ, k ∈ ℕ ∪ {∞} 0 < l < k. Let I_{i}^{s} for i = 0, 1 and s = l, …, k be closed intervals such that I_{i}^{s} ⊃ I_{i}^{s+1} and intI_{0}^{l} ∩ intI_{1}^{l} = ∅. Let N_{i}^{s} := I_{i}^{s} × \overline{B_{W} (r)}.

We say that f has an (l, k)-nested sequence of topological horseshoes ( of vertical size r), iff there exists a continuous and compact homotopy map F : [0, 1] × (−∞, ∞) × \overline{B_{W} (r)} → V such that F_{1} = f and F_{0} is one-dimensional and for A := \overline{F ([0, 1], N_{0}^{l} ∪ N_{1}^{l})} holds

N_{i}^{s} ⟹^{F_{λ}^{s},A} N_{j}^{l}, for i, j = 0, 1 and s = l, …, k

Definition 5 Let f : (−∞, ∞) × \overline{B_{W} (r)} → V be continuous and compact. Let l ∈ ℕ, k ∈ ℕ ∪ {∞} 1 < l < k. Let I_{i}^{s} for i = 0, 1 and s = l − 1, …, k be closed intervals such that I_{i}^{s} ⊃ I_{i}^{s+1} and intI_{0}^{l} ∩ intI_{1}^{l} = ∅. Let N_{i}^{s} := I_{i}^{s} × \overline{B_{W} (r)}.

We say that f has an (l, k)-nested sequence of topological horseshoes with a prehorseshoe (of vertical size r) , iff there exists a continuous and compact homotopy map F : [0, 1]×(−∞, ∞)×\overline{B_{W} (r)} → V such that F_{1} = f and F_{0} is one-dimensional and for A := \overline{F ([0, 1], N_{0}^{l−1} ∪ N_{1}^{l−1})} holds

N_{i}^{s} ⟹^{F_{λ}^{s},A} N_{j}^{l−1}, for i, j = 0, 1 and s = l, …, k

N_{0}^{l−1} ⟹^{F_{λ}^{l−1},A} N_{j}^{l−1}, for i, j = 0, 1

N_{1}^{l−1} ⟹^{F_{λ}^{l−1},A} N_{0}^{l−1}

With these definitions the results concerning existence of all but a finite number of periods for nested sequences of topological horseshoes from [Z1] are true.

3 Nested sequences of topological horseshoes in dimension one

In this section we present a modification to the argument from [Z1] concerning the existence of the nested sequences of the topological horseshoes for one-dimensional maps which have a point of odd period greater than 1.

In [Z1] the case of period 3 was treated as follows: from the Sharkovskii theorem we know that there exist a point of period 5 and then we considered case by case all possible permutations, which can be induced by the orbit of period 5 . This was the content of the proof of Theorem 23 in [Z1].

Here we consider period 3 separately. Due to this for period 5 we are left with only one case – described by the Stefan diagram [Ste] and treated in Theorem 25 in [Z1].

Lemma 5 Let I ⊂ ℝ be a closed segment. Let f : I → ℝ be continuous.

Suppose that f has a periodic point of period 3, then f has an (4, ∞)-nested sequence of topological horseshoes.

Proof

Let x_{0} < x_{1} < x_{2} be an orbit of period 3 for f. Without any loss of generality we may assume that I = [x_{0}, x_{2}].

Without any loss of generality we can also assume that

f(x_{0}) = x_{1}, f(x_{1}) = x_{2}, f(x_{2}) = x_{0}  (31)

because the case of f(x_{0}) = x_{2} can be obtained from this via symmetry x ↦ −x.

Since I ⊂ f([x_{1}, x_{2}]) we can define

y_{1} := inf{y|y > x_{1}, f(y) = x_{1}}  (32)

It follows immediately from the Darboux theorem that

x_{1} < y_{1} < x_{2}, f(y_{1}) = x_{1}
[x_{1}, x_{2}] ⊂ f([x_{1}, y_{1}]), I ⊂ f^{2}([x_{1}, y_{1}])

Observe that y_{1} ∈ (x_{1}, x_{2}) ⊂ f((x_{0}, x_{1})), hence we can define

y_{2} = sup{y|y < x_{1}, f(y_{2}) = y_{1}}  (33)

We have

x_{0} < y_{2} < x_{1} < y_{1} < x_{2}  (34)

We set

I_{0} := [x_{1}, y_{1}], I_{1} := [y_{2}, x_{1}]  (35)

We have

[x_{1}, x_{2}] ⊂ f(I_{0}), I ⊂ f^{2}(I_{0})

For I_{1} we obtain

[y_{1}, x_{2}] ⊂ f(I_{1}), [x_{0}, x_{1}] ⊂ f^{2}(I_{1}), [x_{1}, x_{2}] ⊂ f^{3}(I_{1}), I ⊂ f^{4}(I_{1})

We can now easily construct inductively the sets I_{j}^{s} for s = 4, 5, … and j = 0, 1 with the following properties

I_{j}^{4} ⊂ I_{j}, j = 0, 1
I_{j}^{s+1} ⊂ I_{j}^{s}, j = 0, 1
f^{s}(I_{j}^{s}) = I, j = 0, 1

This gives us an (4, ∞)-nested sequence of topological horseshoes.

□

We summarize Theorems 22 and 25 from [Z1] and the above lemma into the following statement.

Theorem 6 Let I ⊂ ℝ be a closed segment. Let f : I → ℝ be continuous.

Suppose f has a periodic point of odd period n, n > 1. Then f has a (p + 1, ∞)- nested sequence of topological horseshoes with a prehorseshoe.

4 Existence of infinitely many periodic points for perturbations

Let W, V = ℝ ⊕ W be Banach spaces. Suppose that we have a continuous and compact map F : [0, 1] × V → V such that F_{0}(x, w) = (f(x), 0). Contrary to the analogous section in [Z1] we do not require for F to be uniformly continuous.

Theorem 7 Suppose f has a periodic point of odd period n, n > 1. k > 0. There exists p ≤ max{n, 5}, such that for every r > 0 there exists λ_{0} > 0 such that for λ < λ_{0} F_{λ} has a (p + 1, p + k)-nested sequence of topological horseshoes with a prehorseshoe of vertical size r.

Proof

Let p be the smallest odd period for f greater than 3.

It follows from Theorem 6 that f as a one-dimensional map has a (p+1, ∞)-nested sequence of topological horseshoes with a prehorseshoe.

Let the segments I_{i}^{s}, where s = p, p + 1, … and i = 0, 1 be as in Definition 4.

Let us fix an r > 0. We define N_{i}^{s} := I_{i}^{s} × \overline{B_{W} (r)}, for i = 0, 1 and s = p, p + 1, . . .. We set N^{s} := N_{0}^{s} ∪ N_{1}^{s}. Let

A = \overline{F ([0, 1], ⋃ N^{p})}

Observe that from the compactness of F it follows that the set A is compact. From a uniform continuity of F on the compact set [0, 1] × A for sufficiently small λ we have

\overline{F_{λ}^{s}(N^{p} ∩ A)} ⊂ (−∞, ∞) × B_{W} (r), s = p, p + 1, …, p + k  (36)

The remaining conditions of the definition of the covering relations are obtained in the same way for sufficiently small λ.

Hence F_{λ} is a (p + 1, p + k)-nested sequence of topological horseshoes with prehorseshoe for λ sufficiently small λ. □

Proceeding further as in [Z1] we obtain the following

Theorem 8 Suppose f has a periodic point of odd period n, n > 1.

There exists a integer M(n) such that for every r > 0 there exists λ_{0} > 0 such that for λ ≤ λ_{0} and m > M(n) F_{λ} has a periodic point of period m in the set ℝ ⊕ B_{W} (r).

5 Continuation of individual periodic orbits

In this section we assume that W and V = ℝ ⊕ W are Banach spaces. F : [0, 1] × V → V is a continuous and compact map, such that F_{0}(x, w) = (f(x), 0).

Contrary to the analogous section in [Z1] we do not assume a uniform continuity of F.

Suppose that f has a nontrivial periodic point x, so we can define segments I, I_{i} and D(f, x) as in [Z1].

The aim of this section is to show that many periodic orbits, which exist for f by the Sharkovski theorem, continue for small λ.

The only modification in the comparison to [Z1] is in the proof of the following theorem

Theorem 9 Let p ∈ ℕ, p > 0. Suppose that there exist i_{0}, i_{1}, i_{2}, …, i_{p−1}, i_{p} where i_{0} = i_{p} such that

D_{i_{p}i_{p−1}}D_{i_{p−1}i_{p−2}} … D_{i_{1}i_{0}} = −1.

Then for every r > 0 there exists λ_{0} > 0 such that for λ < λ_{0} there exists a periodic point z_{λ} such that

F_{λ}^{l}(z_{λ}) ∈ IntI_{i_{l}} × B_{W} (r), for l = 0, 1, 2, …, p − 1  (37)
F_{λ}^{p}(z_{λ}) = z_{λ}  (38)

Before we prove the above theorem we need one technical lemma, which was proved in [Z1]

Lemma 10 Suppose that assumptions of Theorem 9 are satisfied. Then there exists a family of segments J_{l} = [d_{l}, u_{l}] for l = 0, 1, …, p − 1 such that

J_{l} ⊂ intI_{i_{l}}  (39)
f(I_{l−1}) = I_{l}  (40)
f^{p}(d_{0}) > d_{0},  f^{p}(u_{0}) < u_{0}  (41)

Proof of Theorem 9

Let us fix r > 0. Let

Im F = \overline{F ([0, 1] × I × \overline{B_{W} (r)})}  (42)

It follows from a uniform continuity of F on [0, 1] × Im F that for sufficiently λ small we have

\overline{F_{λ}^{s}((I × \overline{B_{W} (r)}) ∩ Im F)} ⊂ (−∞, ∞) × B_{W} (r), for s = 1, 2, …, p  (43)

Let J_{l} = [d_{l}, u_{l}], for l = 0, 1, …, p − 1 be a family of segments constructed in lemma 10.

We show now that

I(F_{0}^{p}, IntJ_{0} × B_{W} (r)) = 1  (44)

To this end we use the homotopy property of the fixed point index (see [Z1][D][G]). We define the homotopy H : [0, 1] × V → V by

H(λ, (x, y)) = (1 − λ)(f^{p}(x), 0) + λ((d_{0} + u_{0})/2, 0)  (45)

Let us remark that it follows from (41) for λ ∈ [0, 1] and y ∈ W that

x(H_{λ}(d_{0}, y)) > d_{0},  x(H_{λ}(u_{0}, y)) < u_{0}  (46)

Fix(H, J_{0} × \overline{B_{W} (r)}) ∩ ∂(J_{0} × \overline{B_{W} (r)}) = ∅  (47)

Therefore, by the homotopy property of the fixed point index and by the formula for an index of affine maps [Z1] we obtain (44).

Let λ_{0} > 0 be such that for λ < λ_{0} holds

x(F_{λ}^{p}(d_{0}, y)) > d_{0},  x(F_{λ}^{p}(u_{0}, y)) < u_{0},  (48)
for |y| ≤ r, (d_{0}, y) ∈ Im F, (u_{0}, y) ∈ Im F

The existence of such λ_{0} follows from (41) and a uniform continuity of F^{p} on the set [0, 1] × \{(J_{0} × \overline{B_{W} (r))} ∩ Im F \}.

From (48) and (43) we get

Fix(F_{λ}^{p}, J_{0} × \overline{B_{W} (r)}) ∩ ∂(J_{0} × \overline{B_{W} (r)}) = ∅  (49)

for λ < λ_{0}. Therefore, we can apply again the homotopy property to F_{λ}. We obtain

I(F_{λ}^{p}, intJ_{i_{0}} × B_{W} (r)) = 1  (50)

Now to obtain (37) we find λ_{1} ≤ λ_{0} such that for λ < λ_{1} holds

F_{λ}^{l}(\{J_{l} × \overline{B_{W} (r)}\} ∩ Im F) ⊂ IntI_{l+1} × B_{W} (r), for l = 1, …, p − 1  (51)

Again the existence of λ_{1} follows from (39), (40) and the uniform continuity of F on compacts.

□

Proceeding further as in [Z1] we obtain the following theorem

Theorem 11 Suppose f has a point of odd period n, n ≥ 3. Then for every M ∈ ℕ and every r > 0 there exists λ_{0} such that for λ ≤ λ_{0} holds if m ≤ M and n⊲m, m ≠ n then F_{λ} has a periodic point of period m in ℝ⊕B_{W} (r).

Proof of Theorem 2

Observe that Theorem 2 for n odd follows immediately from the above theorem and Theorem 8. The proof of Theorem 2 for even periods is exactly the same as in [Z1] with the similar modifications as these presented in this section.

□

6 Conclusions and outlook

Theorem 2 presented here is an example of the following prototype theorem:

Theorem 12 Let I ⊂ ℝ be a closed segment and f : I → ℝ be a continuous map.

Suppose that f has some interesting dynamical property A.

Then there exists δ = δ(f, A), n = n(f) such that every continuous and compact map P : I ⊕ W → ℝ ⊕ W, satisfying

|P_{(}^{i}x, y) − (f^{i}(x), 0)| ≤ δ, (x, y) ∈ Z, i = 1, …, n  (52)

where Z is a suitable compact set containing the set

\overline{P (I ⊕ \overline{B}_{W} (r))} ∩ I ⊕ \overline{B}_{W} (r))

has some interesting property A’.

To see that Theorem 2 indeed an example of realization of the above prototype theorem observe that λ_{0} from its conclusion is given by the following condition

|F_{λ}^{i}(x, y) − F_{0}^{i}(x, y)| ≤ δ(n, F_{0}_{|I}), (x, y) ∈ Z, i = 1, …, M(n), λ ≤ λ_{0} (53)

where n is the smallest (in the Sharkovskii ordering) period for F_{0}, I is a closed interval spanned by the periodic orbit of period n, 2p is obtained in Theorem 8 and

Z = \overline{F ([0, λ_{0}] × I ⊕ \overline{B}_{W} (r))} ∩ I ⊕ \overline{B}_{W} (r))

Given the map P, we define homotopy F as follows

F_{λ}(x, y) = λP (x, y) + (1 − λ)(f(x), 0).  (54)

In applications the map P can be given as a Poincaré map for an ordinary differential equation. If the map P posses an nearly one-dimensional attractor, then we can define a one-dimensional model map f and we can treat P as a multidimensional perturbation of one-dimensional map f through the homotopy (54).

In fact a direct application of Theorem 2 in this case is usually impossible, because the bounds (53) are very hard to verify even with computer assistance. However, the methods used in the proof of Theorem 2: topological horseshoes and a continuation of individual orbits can be applied under much weaker conditions. For an example of such approach to the studies of strange attractors for Rössler and Lorenz equations reader is referred to papers [Z2] and [GZ], respectively. In those papers the existence of symbolic dynamics was proved with computer assistance, we used computer to rigorously check assumptions of our theorems concerning topological horseshoes.

We think that the the methods developed in [Z1] and the present paper one can be applied to prove prototype theorem also for other dynamical properties of topological origin like: topological entropy, other forcing relations between periodic orbits, kneading sequences.

For development concerning topological horseshoes in the case of more than one expanding direction reader is referred to other papers by author [Z3, Z4, Z5].

References

[D] A. Dold, Lectures on Algebraic Topology. 1972. Springer-Verlag, Berlin, Heidelberg, New York

[GZ] Z. Galias and P. Zgliczynski Computer assisted proof of chaos in the Lorenz system. 165–188, Physica D, 115, 1998

[G] A. Granas The Leray-Schauder index and the fixed point theory for arbitrary ANR’s. 209–228, Bull. Soc. Math. France , 100, 1972

[MM] K. Mischaikow and M. Mrozek Isolating neighborhoods and chaos 205–236, Japan Journal of Industral and Applied Mathematics, 1995, vol. 12, No. 2

[Sh] A. N. Sharkovskii Coexistence of cycles of a continuous map of a line into itself. (Russian) 61–71, Ukr. Math. Z., 16, 1964, ( (English) Int. J. of Bifurcations and Chaos, 1263–1274, 5, 1995)

[S] S. Smale Differentiable Dynamical Systems 747–817, Bull. Amer. Math. Soc., 73, 1967 [Ste] P. Stefan A Theorem of Sarkovskii on the Existence of Periodic Orbits of

Continuous Endomorphism of the Real Line 237–248, Comm. Math. Phys. 54, 1977

[Z] P. Zgliczyński Fixed point index for iterations of maps, topological horseshoe and chaos., Topological Methods in Nonlinear Analysis, 1996, Vol. 8, No. 1, 169–177

[Z1] P. Zgliczyński Sharkovskii theorem for multidimensional perturbations of 1dim maps., Erg. Th. and Dyn. Systems, to appear

[Z2] P. Zgliczyński Multidimensional perturbations of one-dimensional maps and stability of Sharkovskii ordering.,Int. J. of Bifurcations and Chaos , Vol. 7, No.9, 1999, to appear

[Z3] P. Zgliczyński Symbolic dynamics for the Rössler folded towel map, in Conley Index Theory, Warsaw 1999, Banach Center Publication, editors K. Mischaikow, M. Mrozek and P. Zgliczyński

[Z4] P. Zgliczyński On periodic points for systems of weakly coupled 1-dim maps, submitted, http://www.im.uj.edu.pl/˜ zgliczyn

[Z5] P. Zgliczyński Topological hyperbolicity, in preparation