Generalization of Sharkovskii’s theorem to multidimensional perturbations of

1-dim maps

Piotr Zgliczyński

IMUJ preprint 1997/14

Abstract

In paper we present a generalization of Sharkovski theorem for multidimensional perturbations of one-dimesional maps. We show that if unperturbed one-dimensional map has a periodic point, then all sufficiently close multidimesional perturbations of this map have periodic points of all periods which are allowed by Sharkovski theorem.

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

0 Introduction

This paper is a second part to the paper [?]. In this paper we present a generalization of the famous Sharkovskii’s Theorem [?](see also [?],[?],[?], [?]) 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 Sharkovskii’s Theorem.

Theorem 1 Let be the ordering of positive integers:

3⊲5⊲7⊲…⊲2·3⊲2·5⊲2·⊲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 ⊲ m and f has a periodic point of period n then f also has a periodic point of period m.

The ordering described in Theorem ?? 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 a continuous decomposition V = ℝ ⊕ W be given. According to this decomposition we will often 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. In the sequel we will often use the notation F_{λ} for the partial map with 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. If dim V = ∞ then we additionally assume that F is uniformly continuous and compact.

If f has a point of period n then for every r > 0 there exists λ_{0} > 0, such that for λ < λ_{0} and every m ∈ ℕ such that n ⊲ m and n ≠ m F_{λ} has periodic point of period m in ℝ ⊕ B_{W} (r), where B_{W} (r) is an open ball in W of radius r.

In [?] this theorem was proved for n odd.

In our theorem for infinite dimensional case we have assumed uniform continuity and compactness. We believe that compactness alone is sufficient, but we are unable to prove it, yet.

1 1-dim maps with even periods

The aim of this section is to investigate a form of 1-dim maps, for which all periodic points which are not fixed point have an even period. The results of this section follow essentially from the ramification of arguments presented in [?] or [?] for odd periods, but they are probably not stated there explicitly as they are not required for the proof of Sharkovskii’s theorem in one-dimensional case.

In this section we assume that f : ℝ → ℝ is a continuous map. Let x_{0} < x_{1} < … < x_{n−1}, n > 0 be a periodic trajectory. Like in section 3 in [?] we set I = [x_{0}, x_{n−1}] and we define I_{i} = [x_{i−1}, x_{i}]. We will use notions and notation introduced in section 3 from [?].

Theorem 3 Assume that n is even and there is in I no periodic point of period odd greater than one. Then

I_{n/2} →^{f−} I_{n/2}  (1)
f^{2j+1}([x_{0}, x_{n/2−1}]) ⊂ (x_{n/2−1}, ∞), for j ∈ ℕ  (2)
f^{2j}([x_{0}, x_{n/2−1}]) ⊂ (−∞, x_{n/2}), for j ∈ ℕ  (3)
f^{2j+1}{x_{0}, x_{1}, …, x_{n/2−1}} = {x_{n/2}, x_{n/2+1}, …, x_{n−1}}, for j ∈ ℕ  (4)
f^{2j}{x_{0}, x_{1}, …, x_{n/2−1}} = {x_{0}, x_{1}, …, x_{n/2−1}}, for j ∈ ℕ  (5)

Proof

Let i_{0} := min{i = 0, 1, …, n − 1|f(x_{i}) < x_{i}}. Obviously i_{0} > 0. From the definition of i_{0} follows immediately that

I_{i_{0}} →^{f−} I_{i_{0}}  (6)

We define inductively the sets J_{l} by

J_{0} = {i_{0}}  (7)
i ∈ J_{l+1} iff ∃j ∈ J_{l} I_{j} →^{f} I_{i}  (8)

We define |J_{l}| = ⋃_{i∈J_{l}} I_{i}.

We show that for some l ≤ n J_{l} = {0, 1, …, n − 1}.

To see this we replace f with a map f̃, which agrees with f on points x_{i} and is affine on every I_{i}. It is easy to see that

I_{i} →^{f} I_{j} iff I_{i} →^{f̃} I_{j}

But for f̃ holds

I_{i} →^{f̃} I_{j} iff I_{j} ⊂ f̃(I_{i})

From this and (??) follows immediately that J_{l} ⊂ J_{l+1}, |J_{l}| = f̃^{l}(I_{i_{0}}). Entire trajectory of x_{i_{0}} must be contained in J_{n}, hence from connectedness of |J_{l}| follows that |J_{n}| = I and J_{n} = {0, 1, …, n − 1}.

From this follows that for every j = 1, …, n − 1 exist i_{1}, i_{2}, …, i_{l} such that

I_{i_{0}} →^{f} I_{i_{1}} →^{f} … →^{f} I_{j}  (9)

We show now that there does not exists any j ≠ i_{0} such that I_{j} →^{f} I_{i_{0}}. To prove this assume the contrary. Then from (??) and follows (??) that we can construct a closed loop

I_{i_{0}} →^{f} I_{i_{1}} →^{f} … →^{f} I_{j} →^{f} I_{i_{0}} →^{f} I_{i_{0}} →^{f} … →^{f} I_{i_{0}}  (10)

which have prime number p of arrows. From this loop we can deduce the existence of point x ∈ I_{i_{0}} such that f^{p}(x) = x and f^{s}(x) ∈ I_{j}. As x is different from x_{i} for every i then it follows that period of x is equal to p. But this in condradiction with assumption that no odd period exists.

Nonexistence of I_{j} →^{f} I_{i_{0}}, for i ≠ i_{0} means that for i ≤ i_{0} − 1 and i ≥ i_{0} + 1 both ends of the segment I_{i} are mapped on the same side of the middle point of the segment I_{i_{0}}. So the set {x_{0}, x_{1}, …, x_{i_{0}−1}} must be mapped onto the set {x_{i_{0}}, x_{i_{0}+1}, …, x_{n−1}} and vice versa. From this it follows that i_{0} = n/2, so conditions (??–??) are satiesfied.

Suppose that (??) is not true for some j. Hence there exists y ≤ x_{n/2−1}, such that y ∈ f^{2j+1}([x_{0}, x_{n/2−1}]). From (??) follows that x_{n/2} ∈ f^{2j+1}([x_{0}, x_{n/2−1}]). By connectedness of f^{2j+1}([x_{0}, x_{n/2−1}]) we obtain

I_{n/2} ⊂ f^{2j+1}([x_{0}, x_{n/2−1}])  (11)

Hence there exists K ⊂ [x_{0}, x_{n/2−1}] such that

K ⟶^{f^{2j+1}} I_{n/2}  (12)

From |J_{n}| = I it follows that there exists segment L ⊂ I_{n/2} such that

L →^{f^{n}} K  (13)

Now we can construct loop of arrows as follows

L →^{f^{n}} K ⟶^{f^{2j+1}} I_{n/2} →^{f} … I_{n/2} →^{f} L  (14)

such that the sum of the powers of f appearning above arrows is equal to some prime number p. In this way from lemma 1.2 from [?] we obtain a periodic point of odd period p. This is in contradiction with our asumptions. So (??) is true. The proof of (??) is analogous.

□

Suppose now that n = 2^{q}n′, where q > 0, n′ > 0. Suppose that f do not have on I any point of period of the form 2^{q−1}m with m odd and m > 1. Then from the above theorem it follows that x_{0}, x_{1}, …, x_{n/2−2} are points from 2^{q−1}n′ periodic trajectory for the map f^{2} and if q − 1 > 0 then f^{2} has no periodic point in [x_{0}, x_{n/2−1}] of period of the form 2^{q−2}m, where m is odd and m > 1. If q − 1 > 0 then we can continue this construction and we see that x_{0}, …, x_{n/2^{2}−1} are points from 2^{q−2}m periodic trajectory for the map f^{2^{2}} and if q − 2 > 0 then the map f^{2^{2}} has no periodic points in [x_{0}, x_{2^{2}} −1] of the form 2^{q−3}m where m is odd and m > 1. We can continue this induction q times. In this way we obtain the following

Theorem 4 Suppose that n = 2^{q}n′, where q > 0, n′ > 0. Suppose that f do not have on I any point of period of the form 2^{q−1}m with m odd and m > 1. Then

I_{n/2^{i+1}} ⟶^{f^{2^{i}} −} I_{n/2^{i+1}}, for i = 0, …, q − 1

f^{j·2^{i}+s}([x_{0}, x_{\frac{n}{2^{i}} −1}]) ⊂ (x_{\frac{n}{2^{i}} −1}, ∞), for i = 1, …, q, j = 0, 1, 2 … and 0 < s < 2^{i}

f^{j·2^{i}}([x_{0}, x_{\frac{n}{2^{i}} −1}]) ⊂ (−∞, x_{\frac{n}{2^{i}}}), for i = 1, …, q , j = 0, 1, 2 …
f^{2^{i}}(\{x_{0}, x_{1}, …, x_{\frac{n}{2^{i}} −1}\}) = \{x_{0}, x_{1}, …, x_{\frac{n}{2^{i}} −1}\}, for i = 1, …, q

2 Proof of main theorem

Before we present a proof let us explain, why we cannot not follow directly the proof of 1-dim Sharkovskii’s theorem. The situation is as follows: we have the Sharkovskii’s theorem for odd periods and we consider now period of the form n2^{q}, where n > 1 is odd. In onedimensional case one consider now the map g := f^{2^{q}}, obviously g have a point of odd period and in this way one obtains points x_{m} which are fixed points for the maps f^{m2^{q}}, where n ⊲ m, but one have no need to bother himself with the problem: what is the period of x_{m}, because from Sharkovskii’s ordering follows that if the period is not m2^{q}, there must exist a periodic point of period m2^{q}.

In our case we need to show that the fixed points for F_{λ}^{m2^{q}} we find in the proof have exactly period equal to m2^{q}.

Proof

For n odd and n > 1 this theorem was proved in [?].

Let x_{0} < x_{1} < … < x_{n−1}, n > 0 be a periodic trajectory. As usual we set I = [x_{0}, x_{n−1}] and we define I_{i} = [x_{i−1}, x_{i}]. We can assume that n = 2^{q}n′, where n′ is odd, and in I there no periodic point of period 2^{q}m, where m is odd and m > 1.

Let us fix r > 0.

Assume first that n′ > 1. We define J := [x_{0}, x_{n′}]. From theorem ?? it follows that the set {x_{0}, x_{1}, …, x_{n′}} is the periodic trajectory of the map f^{2^{q}} . From the theorem ?? for odd periods applied to f^{2^{q}} and the homotopy F_{λ}^{2^{q}} follows that there exists λ_{1} > 0 such that for every m ≠ n′ and n′ ⊲ m there exists an m-periodic point x_{m} ∈ J × B_{W} (r) for a map F_{λ}^{2^{q}}, such that

F_{λ}^{j·2^{q}}(x_{m}) ∈ int J × B_{W} (r), for j = 1, 2, …, m  (15)
F_{λ}^{j·2^{q}}(x_{m}) ≠ x_{m}, for j = 1, …, m − 1  (16)
F_{λ}^{m·2^{q}}(x_{m}) = x_{m}  (17)

It remains to show that the period of x_{m} with regard to the map F_{λ} is exactly m2^{q} for λ sufficiently small.

From the uniform continuity of F and theorem ?? it follows that there exists λ_{2} > 0, such that

F_{λ}^{s}(J × \overline{B_{W} (r)}) ∩ J × \overline{B_{W} (r)} = ∅, for s = 1, 2, …, 2^{q} − 1  (18)

We set λ_{0} = min{λ_{1}, λ_{2}}. Now is easy to see that for λ < λ_{0} the period of x_{m} is exactly m2^{q}. Suppose that there exists k another period for x_{m}, such that k < m2^{q}. k must be a number of the form k = m′2^{q′}. From condition (??) it follows that q′ = q. From (??) it follows that m′ = m. This finishes the proof of existence of all periods of the form 2^{l}m, where l = q and m > n′ or l > q, when n′ > 1.

We release now the assumption n′ > 1. It remains to show that numbers of the form 2^{i} for i = 0, 1, …, q − 1 can be realised as periods of some points in J for sufficiently small λ. It follows from theorem ?? that for i = 0, 1, …, q − 1 holds

I_{n/2^{i+1}} ⟶^{f^{2^{i}} −} I_{n/2^{i+1}}  (19)
f^{s}(I_{n/2^{i+1}}) ⊂ f^{s}([x_{0}, x_{n/2^{i}−1}]) ⊂ (x_{n/2^{i}−1}, ∞), for s = 1, …, 2^{i} − 1 (20)

Obviously

I_{n/2^{i+1}} ∩ (x_{n/2^{i}−1}, ∞) = ∅, for s = 1, …, 2^{i} − 1.  (21)

From uniform continuity of F and conditions (??) (??) it follows that there exists λ_{3} > 0 such that for λ < λ_{3}

F_{λ}^{s}(I_{n/2^{i+1}} × \overline{B_{W} (r)}) ∩ I_{n/2^{i+1}} × \overline{B_{W} (r)} = ∅, for s = 1, …, 2^{i} − 1  (22)

From the proof of the theorem 29 from [?] and condition (??) it follows that there exists λ_{4} > 0, such that for λ < λ_{4} there exists y_{i} fixed point for the map F_{λ}^{2^{i}}. If λ < min{λ_{3}, λ_{4}} then from (??) it follows immediately that y_{i} has period 2^{i}.

This finishes the proof of theorem ??. □

References

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[St] P. Stefan A theorem of Sarkovskii on the existence of periodic orbits of contnuous endomorphisms of real line , Comm. Math. Phys. 54, 237–248, 1977

[Z] P. Zgliczyński Odd period implies chaos for multidimensional perturbations of 1-D maps., submitted Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland. E-mail: zgliczyn@im.uj.edu.pl