On periodic points for systems of weakly coupled 1-dim maps^{1}

Piotr Zgliczyński^{2} Center for Dynamical Systems and Nonlinear Studies, School of Mathematics, Georgia Institute of Technology Atlanta, Georgia 30332 e-mail: piotrz@math.gatech.edu October 10, 2001

Abstract

We consider the existence of periodic orbits and symbolic dynamics in high dimensional maps which are made up of weakly coupled 1-dimensional maps (tent and logistic maps). We give an explicit and dimension independent bound for the size of coupling, which guarantees the existence of symbolic dynamics for such map.

Keywords: chaos, periodic points, non-uniformly hyperbolic systems, topological hyperbolicity

1 Introduction

The aim of this paper is to develop the method of TS-maps (topological shifts) introduced by the author in [Z2] for proving existence of infinite number of periodic points in multidimensional dynamical systems.

The underlying idea behind the method presented in this paper is the topological hyperbolicity, which can be explained as follows in the context of maps (for ODE’s see [WZ]). Consider a map F. Classically the hyperbolicity means that there is a splitting of the tangent space into stable and unstable components invariant under applications of DF, which causes an uniform expansion in the unstable and a contraction in the stable direction for infinitesimal parallelograms. From these assumptions, one deduces a lot of interesting properties for F, like an existence of symbolic dynamics, a sensitive dependence on initial conditions etc. see for example [GH]. The problem with the hyperbolicity is that it is often very hard to verify it in the context of the concrete map.

In the method presented here we do not look what happens with infinitesimal parallelograms under the iterates of DF, but we investigate the images of small parallelograms under F. If these images behave apparently hyperbolically then we can apply our method to obtain a symbolic dynamics for F (see Section 2 for precise meaning of this statement). Since we work in C^{0} setting we are unable to say anything about the sensitive dependence on initial conditions, hence our results are weaker than the results based on hyperbolicity. However, assumptions of the presented method are also much easier to verify (see papers [GZ, MM, Z2, Z3, Z4]).

1Research supported by the Polish KBN grants 0892/P03/97/12, 2P03A 029 12 and 2P03A 021 15

2On leave from Insitute of Mathematics, Jagiellonian University, Cracow , Poland, e-mail:

zgliczyn@im.uj.edu.pl

We describe our results in a concrete example. Let us consider the map F : ℝ^{n} → ℝ^{n} given by

F (x_{1}, …, x_{n}) := (f(x_{1}), f(x_{2}), …, f(x_{n})) + G(x_{1}, …, x_{n})  (1)

where f(x) = 1 − ax^{2}, where a is close to 2 and G is a small perturbation. The map F is not hyperbolic. From f it inherits the coexistence of both, a hyperbolic dynamics in portions of phase space and a highly nonhyperbolic behavior in certain critical regions.

When G = 0 and a = 2 (or slightly less then 2) then the dynamics of F is given as the cartesian product of n independent copies of 1-dimensional chaotic maps, which are relatively well understood. There are two kind of results concerning dynamics of f(x) = 1 − ax^{2}. The ’ergodic’ results: concerning an existence of an absolutely continuous invariant measure for f plus a positive of Lyapunov exponent for positive Lesbegue measure in parameters space ([Ja],[CE1],[BC1]) and the ’topological’ results: kneading sequences, a symbolic dynamics, an infinite number of periodic points, a transivity (the existence of a dense orbit) (see for example [CE2]).

For a typical nonzero perturbation G, F is not a product of 1-dim maps and F has more than one expanding direction. The research concerning the ergodic properties of such non-uniformly hyperbolic maps is just starting, see paper [V] and references cited there.

Using our method we are able to show that the following topological properties hold for F if G is sufficiently small in C^{0} norm: F has a symbolic dynamics, the number of periodic points is infinite and there exists a region in the phase space that every not too small ball in this region contains a periodic point for F, see Theorem 11 for a precise result. The estimate for |G| is also given.

Section 2 contains topological theorems which form the base for the method. The main tool used to obtain periodic points is the fixed point index. The properties of the fixed point index and the proof of the main topological theorem are contained in the Appendix.

Section 3 contains a result concerning C^{0} small perturbations of cartesian products of tent maps.

Section 4 contains the above described result for the C^{0} small perturbations of cartesian products of logistic maps.

Section 5 contains an application of the method to the Rössler 3-dim ’folded towel’ map introduced in [R]. This map has two topologically expanding directions and one contracting direction. For this map we are able to prove the existence of symbolic dynamics with an infinite number of periodic points.

2 Topological theorems

First we introduce some notations. Let p ∈ ℝ^{n} by x_{i}(p) we will denote the i-th coordinate of the point p. We will use the max norm on ℝ^{n}, so

|(x_{1}, …, x_{n})| := max_{i} |x_{i}|  (2)

Let Z ⊂ ℝ^{n}, x ∈ ℝ^{n} then we will use the following notations dist (x, Z) = inf{|x − y||y ∈ Z}. B(x, ϵ) = {y||x − y| < ϵ}, B(Z, ϵ) = {x|dist (x, Z) < ϵ}, diam Z = sup_{x,y∈Z} |x − y|.

By C we will denote the set of parallelograms in ℝ^{n} with edges parallel to coordinate axes. Hence

C := {X ⊂ ℝ^{n} | X = ∏_{i=1}^{n}[x_{ai}, x_{bi}]}  (3)

Definition 1 Let f : ℝ → ℝ be continuous, X = [x_{a}, x_{b}] and Y = [y_{a}, y_{b}]. We say that X f-covers Y (with margin δ) iff there exist δ > 0 such that [y_{a} −δ, y_{b} +δ] is contained either in [f(x_{a}), f(x_{a})] or in [f(x_{b}), f(x_{a})].

Definition 2 Let X = ∏_{i=1}^{n}[x_{ai}, x_{bi}] For i ∈ {1, …, n} we define the i-th upper and lower edge of X respectively by

U_{i}(X) = {p ∈ X | x_{i}(p) = x_{bi}}  (4)
D_{i}(X) = {p ∈ X | x_{i}(p) = x_{ai}}  (5)

Definition 3 Let F : ℝ^{n} → ℝ^{n} be continuous, X = ∏_{i=1}^{n}[x_{ai}, x_{bi}] and Y = ∏_{i=1}^{n}[y_{ai}, y_{bi}]. Let 1 ≤ i ≤ n, we say that X F -covers Y in i direction (with margin δ) if there exists δ > 0 such that one of the two following conditions hold

[y_{ai} − δ, y_{bi} + δ] ⊂ [max x_{i}(F (D_{i}(X))), min x_{i}(F (U_{i}(X)))]  (6)
[y_{ai} − δ, y_{bi} + δ] ⊂ [max x_{i}(F (U_{i}(X))), min x_{i}(F (D_{i}(X)))]  (7)

If (6) is satisfied, then we will say that the orientation is preserved, if (7) is true then we will say that the orientation is reversed.

Definition 4 Let F : ℝ^{n} → ℝ^{n} be continuous, δ > 0, X = ∏_{i=1}^{n}[x_{ai}, x_{bi}] and Y = ∏_{i=1}^{n}[y_{ai}, y_{bi}]. Let 1 ≤ i_{1} < i_{2} < … < i_{k} ≤ n be a sequence of integers. We say that X F -covers Y in (i_{1}, i_{2}, …, i_{k})-direction (with margin δ) if the following conditions hold

• for every l = 1, …, k X F -covers Y in the i_{l} direction with margin δ

• for every j not in the sequence i_{1}, i_{2}, …, i_{k} hold

x_{j}(F (X)) ⊂ [y_{aj} + δ, y_{bj} − δ]  (8)

To illustrate the notions introduced above let us consider the following example, let n = 3 and f_{1}, f_{2}, f_{3} : ℝ → ℝ and the segments X_{i}, Y_{i} for i = 1, 2 be such that X_{i} f_{i}-covers Y_{i} with margin δ < 1 and be such that f_{3}(x) = 0 for x ∈ ℝ, we set X_{3} = Y_{3} = [−1, 1]. We set X = X_{1} × X_{2} × X_{3}, Y = Y_{1} × Y_{2} × Y_{3}. Consider the map F : ℝ^{3} → ℝ^{3} given by F (x_{1}, x_{2}, x_{3}) = (f_{1}(x_{1}), f_{2}(x_{2}), f_{3}(x_{3})). It is easy to see that the set X F -covers Y in (1, 2)-direction with margin δ. Consider now a perturbation F̃ : ℝ^{n} → ℝ^{n} of F such that |F̃ − F |_{|X} < δ. Then it is easy to see that X F̃-covers Y in (1, 2)-direction.

The basic theorem of the method is

Theorem 1 Suppose we have a family of sets N_{j} ∈ C, for j = 0, …, l − 1, continuous maps F_{j} : N_{j} → ℝ^{n} for j = 0, 1, …, l − 1 and a sequence 1 ≤ i_{1} < i_{2} < … < i_{k} ≤ n, such that for j = 0, …, l − 1 N_{j} F_{j}-covers N_{(j+1) mod l} in (i_{1}, …, i_{k})-direction, then there exists x ∈ int N_{0} such that

F_{j} ∘ F_{j−1} ∘ … F_{1} ∘ F_{0}(x) ∈ int N_{j+1}, for j = 0, 1, …, l − 2  (9)
F_{l−1} ∘ F_{l−2} ∘ … F_{1} ∘ F_{0}(x) = x  (10)

The proof of this theorem is based on the fixed point index and is contained in the Appendix.

The above theorem can be easily extended to maps on manifolds. To this end one needs for every set N_{j} a local chart ϕ_{j} in which this set is a parallelogram and the relation N_{j} F -covers N_{j+1} in (i_{1}, …, i_{k})-direction should be expressed using coordinates obtained from the charts ϕ_{j} and ϕ_{j+1}.

Also the requirement that F_{j} is defined on entire ℝ^{n} is unnecessary, it is enough to assume that the domain of F_{j} contains N_{j}.

Now we will describe two kinds of application of Theorem 1.

2.1 Symbolic dynamics. Topological shifts

We present here a generalization of the notion of TS-map (topological shifts) introduced (as window chains) in [E75], [E89] and in [Z2], to the situation of more then one expanding direction.

Let F : ℝ^{n} → ℝ^{n} be a continuous map. Let us fix a sequence 1 ≤ i_{1} < i_{2} < … < i_{k} ≤ n. Let {e_{i}} be a canonical basis in ℝ^{n}. Then we will call the linear subspace spanned by vectors {e_{i_{1}}, …, e_{i_{k}}} a topologically expanding direction. The reason for this name will be clear from Theorems 3, 4.

Definition 5 Let X, Y ∈ C. We will say that there exists an F -transition of length m from X to Y iff there exists a sequence of sets {N_{j}}_{j=0,1,...,l} ⊂ C and the sequence of integers {m_{j}}_{j=0,...,l−1}, such that

N_{0} ⊂ X, N_{l} = Y
N_{j} F^{m_{j}}-covers N_{j+1} in (i_{1}, …, i_{k})-direction, for j = 0, …, l − 1
m_{0} + m_{1} + … m_{l−1} = m

We will use the following graphical notation X ⟹^{F^{m}} Y.

The following proposition is an easy consequence of the above definition and Theorem 1.

Proposition 2 Suppose we have a family of sets N_{j} ∈ C, for j = 0, …, l − 1 and a sequence of positive integers m_{j} for j = 0, …, l, such that

N_{0} ⟹^{F^{m_{0}}} N_{1} ⟹^{F^{m_{1}}} ⋯ ⟹^{F^{m_{l−1}}} N_{l} ⟹^{F^{m_{l}}} N_{0}

Then there exists x ∈ int N_{0} such that

F^{m_{j}+m_{j−1}+...+m_{0}}(x) ∈ int N_{j+1}, for j = 1, …, l − 1  (11)
F^{m_{l}+m_{l−1}+...+m_{0}}(x) = x  (12)

Suppose now that we have a family of sets N_{j} ∈ C for j = 0, 1, …, l − 1 and an integer m such that

N_{j} ∩ N_{k} = ∅, for j, k = 0, …, l − 1, j ≠ k  (13)
N_{j} ⟹^{F^{m}} N_{k}, for j, k = 0, …, l − 1  (14)

We set N = ⋃_{j=0}^{l−1} N_{j}. An invariant part of the set N is defined by

Inv(N, F) := ⋂_{i∈ℤ} F_{|N}^{−i}(N)  (15)

Let Σ_{l} := {0, 1, …, l−1}ℤ, Σ_{l}^{+} := {0, 1, …, l−1}ℕ. Σ_{l}, Σ_{l}^{+} are topological spaces with the Tichonov topology. On Σ_{l}, Σ_{l}^{+} we have the shift map σ given by

(σ(c))_{i} = c_{i+1}

For i ∈ ℕ we define a map π_{i} : Inv(N, F^{m}) → {0, 1, …, l − 1} given by π_{i}(x) = j iff F^{mi}(x) ∈ N_{j}. Now we define a map π : Inv(N, F^{m}) → Σ_{l}^{+} by π(x) := (π_{i}(x))_{i∈ℕ}. The map π assigns to the point x the indices of N_{i}-s its F^{m}-trajectory goes through. It is easy to see that

π ∘ F^{m} = σ ∘ π.  (16)

If F is also a homeomorphism, then the definition of π_{i} can be extended to all integers and the domain of π is Σ_{l}.

Obviously semiconjugacy (16) alone is not a sign of complicated dynamics. The dynamics is complicated if the set Inv(N, F^{m}) is infinite. The following theorem gives a generalization of the results from [Z2]

Theorem 3 Let the family of sets {N_{j}}_{j=0,...,l−1} ⊂ C satisfies the conditions (13)(14). Then Σ_{l}^{+} = π(Inv(N, F^{m})). The preimage of any periodic sequence from Σ_{l}^{+} contains periodic points of F^{m}. If we additionally suppose that F is a homeomorphism, then Σ_{l} = π(Inv(N, F^{m})).

Proof Since periodic sequences are dense in Σ_{l}^{+}, then it is enough to show that every periodic sequence c ∈ Σ_{l}^{+} is realized as the image of some periodic point for F^{m}.

Let c ∈ Σ_{l}^{+} be periodic. Let (c_{0}, c_{1}, …, c_{k}) be a finite subsequence of c representing its period. From our assumptions it follows that

N_{c_{0}} ⟹^{F^{m}} N_{c_{1}} ⟹^{F^{m}} ⋯ ⟹^{F^{m}} N_{c_{k}} ⟹^{F^{m}} N_{c_{0}}  (17)

From proposition 2 it follows the existence of the point of x ∈ int N_{c_{0}} such that π(x) = c.

□

2.2 Distribution of periodic points in the phase space

Let F : ℝ^{n} → ℝ^{n} be continuous. Let (i_{1}, …, i_{k}) = (1, …, n). This mean that all directions are topologically expanding.

Definition 6 Let X ∈ C, δ > 0. We will say that F has TE property (topological expansion) for X and δ, if there exists a finite covering {M_{j}} ⊂ C of X, (i.e. ⋃ M_{j} = X), such that for every j diam (M_{j}) ≤ δ and there exists an integer l(M_{j}, δ) such that

M_{j} ⟹^{F^{l(M_{j} ,δ)}} X  (18)

Theorem 4 Assume that F has TE property for X and δ. Then there exists x ∈ X such that

• x is a periodic point for F • for every closed ball Z = cl B(z, 2δ), Z ⊂ X there exists m such that

F^{m}(x) ∈ Z

Proof Let {M_{j}}_{j=0,...,k} be a finite covering of X, which exists due to TE property for X and δ. We have

M_{0} ⟹^{F^{l(M_{0},δ)}} M_{1} ⟹^{F^{l(M_{1},δ)}} M_{2} … ⟹^{F^{l(M_{k−1},δ)}} M_{k} ⟹^{F^{l(M_{k},δ)}} M_{0}  (19)

From proposition 2 it follows the existence of x with desired properties. □

Obviously in the above theorem to get something nontrivial we need that δ should be small when compared with the size of the set X. We will see that this is indeed the case in the examples considered in the next two sections.

Theorem 5 Suppose that F has TE property for X ∈ C and every δ > 0. Then

• the set of periodic points is dense in X

• there exists x ∈ X such that a positive orbit of x is dense in X.

Proof

The assertion concerning the density of periodic points follows directly from the previous theorem. It remains to prove the existence of dense orbit.

For every l ∈ ℕ let M^{l} = {M_{i}^{l}}_{i=0,...,i_{l}}, M_{i}^{l} ⊂ X be covering of X by closed balls of radius equal to or less than 1/2^{l}, whose existence follows from TE property for δ = 1/2^{l}. We define a new sequence {M_{i}}, i ∈ ℕ which is an union of sequences M^{l}. From TE property it follows that for every i ∈ ℕ there exists l_{i} such that

M_{i} ⟹^{F^{l_{i}}} X  (20)

Hence we obtain

M_{i} ⟹^{F^{l_{i}}} M_{i+1}  (21)
M_{i} ⟹^{F^{l_{i}}} M_{0}  (22)

Let us set L_{i} = Σ_{k=0}^{i−1}l_{k}. Consider now a finite subsequence M_{0}, M_{1}, …, M_{i}. We have

M_{0} ⟹^{F^{l_{0}}} M_{1} ⟹^{F^{l_{1}}} … ⟹^{F^{l_{i−1}}} M_{i} ⟹^{F^{l_{i}}} M_{0}  (23)

From proposition 2 it follows that there exists a periodic point x_{i} ∈ M_{0} such that

F^{L_{k}}(x_{i}) ∈ M_{k}, for k = 1, 2, …, i  (24)

Let x be a limit of a convergent subsequence of {x_{i}}. Passing to the limit in (24) we obtain

F^{L_{k}}(x) ∈ M_{k}, for k ∈ ℕ  (25)

To finish the proof let us remark that for every ball B(p, r) ⊂ N there exists k such that M_{k} ⊂ B(p, r). □

3 Infinite number of periodic points for perturbations of products of tent maps

In this section we consider perturbations of products of tent maps. We show that Theorems 3, 4 can be applied to these maps. Another reason for investigation of these maps is the existence of the conjugacy between the tent map t_{2} and the logistic map 1−2x^{2}, so the results concerning an existence of transitions (as defined in the previous section) between some intervals for the tent map are true also for the logistic map. We will exploit this conjugacy in Section 4.

In the sequel we will use the following convention: maps f, t_{μ}, g denoted with small letters are one dimensional; maps F, G denoted by capitals are multidimensional.

Let us define a tent map t_{μ} : ℝ → ℝ by

t_{μ}(x) := \{ \begin{matrix}1 + μx, & for x ≤ 0 \\ 1 − μx, & for x ≥ 0 \end{matrix}  (26)

Let us denote by

I_{μ,ϵ} := [t_{μ}(1 − ϵ) + ϵ, 1 − ϵ] = [1 − μ + ϵ(1 + μ), 1 − ϵ].  (27)

We will assume further that μ and ϵ are such that

1 − μ + ϵ(1 + μ) < 0 < 1 − ϵ  (28)
|1 − μ + ϵ(1 + μ)| < 1 − ϵ  (29)

Condition (28) means that 0 ∈ int I_{μ,ϵ}. Condition (29) means that the interval I_{μ,ϵ} ∩ ℝ_{−} is shorter than I_{μ,ϵ} ∩ ℝ_{+}.

Conditions (28), (29) hold, for example, for μ close to 2 and ϵ small.

Let f be a continuous perturbation of t_{μ} given by

f(x) = t_{μ}(x) + g(x), where |g(x)| < ϵ  (30)

Let F : ℝ^{n} → ℝ^{n} be a perturbation of the cartesian product of n tents given by

F (x_{1}, …, x_{n}) = (t_{μ}(x_{1}), t_{μ}(x_{2}), …, t_{μ}(x_{n})) + G(x_{1}, …, x_{n})  (31)

where |G([−1, 1]^{n})| < ϵ.

Let us remark that form (28) it follows

I_{μ,ϵ} ⊂ f(I_{μ,ϵ})  (32)

moreover it is easy to see that the interval I_{μ,ϵ} is a maximal set in [−1, 1] which has this property. Namely the following proposition holds

Proposition 6 If J ⊂ [−1, 1] such that for every continuous map g : [−1, 1] → ℝ, such that |g| < ϵ holds

J ⊂ (t_{μ} + g)(J)  (33)

then J ⊂ I_{μ,ϵ}

Proof

Consider the function f = t_{μ} + g such that

f(x) = min{t_{μ}(x) + ϵ, 1 − ϵ}  (34)

It is easy to see that

f([−1, 1]) = [t_{μ}(−1) + ϵ, 1 − ϵ]  (35)

Since we have

t_{μ}(1 − ϵ) = t_{μ}(−1 + ϵ), and − 1 + ϵ < t_{μ}(−1) + ϵ  (36)

we obtain

f^{2}([−1, 1] = [t_{μ}(1 − ϵ) + ϵ, 1 − ϵ] = I_{μ,ϵ}  (37)

By our assumption J ⊂ f^{2}([−1, 1]), so the assertion follows. □

In the multidimensional case we obtain the following

Proposition 7 If J ⊂ [−1, 1]^{n} such that for every continuous map G : [−1, 1]^{n} → ℝ^{n}, such that |G| < ϵ holds

J ⊂ F (J)  (38)

then J ⊂ I_{μ,ϵ}^{n}

In this section we define the expanding directions as (i_{1}, i_{2}, …, i_{k}) = (1, …, n).

The aim of this section is to prove the following theorem

Theorem 8 Let F, G satisfy (31) and μ, ϵ be such that (28),(29) are satisfied. Then there exists δ = δ(μ, ϵ) such that

1. there exists x ∈ ∏_{i=1}^{n} I_{μ,ϵ} such that

• x is a periodic point for F

• for every closed ball Z = cl B(z, 2δ) , Z ⊂ ∏_{i=1}^{n} I_{μ,ϵ} there exists m such that F^{m}(x) ∈ Z

2. for every finite family of sets {N^{s}}_{s=0,...,k−1}, N^{s} ⊂ ∏_{i=1}^{n} I_{μ,ϵ}, of the form

N^{s} = J^{s,1} × J^{s,2} × … × J^{s,n}  (39)

where J^{s,i} are closed intervals of length greater than δ and 0 ∉ intJ^{s,i} and

N^{s} ∩ N^{s′} = ∅, for s ≠ s′.  (40)

There exists m and a continuous map π : Inv(⋃_{s} N^{s}, F^{m}) → Σ_{k}^{+}, such that

σ ∘ π = π ∘ F^{m}  (41)

π is onto. The preimage of any periodic sequence in Σ_{k}^{+} contains a periodic point for F^{m}.

The crucial step in the proof of the above theorem is the following lemma, which shows that F for sufficiently small ϵ and δ has TE property for ∏_{i=1}^{n} I_{μ,ϵ}. Namely

Lemma 9 Let F, G satisfy (31) and μ, ϵ be such that (28),(29) are satisfied. Then there exists δ = δ(μ, ϵ) such that for every set M ⊂ ∏_{i=1}^{n} I_{μ,ϵ}, of the form

M = J^{1} × J^{2} × … × J^{n}  (42)

where J^{i} are closed intervals of length greater than δ and 0 ∉ intJ^{i} there exists m such that

M ⟹^{F^{m}} ∏_{i=1}^{n} I_{μ,ϵ}  (43)

Before we prove this lemma we need the following

Lemma 10 There exists δ = δ(μ, ϵ), such that for every closed interval J ⊂ I_{μ,ϵ}, such that 0 ∉ int J and length of J is greater than δ there exists s and a sequence of closed intervals {J_{i}}_{i=0,...,l} such that

J_{0} := J, J_{l} = [0, 1 − ϵ]  (44)
0 ∉ int J_{i},  (45)
J_{i} t_{μ}-covers J_{i+1}with margin ϵ, for i=0,…,l-1  (46)

Proof

We will denote by d([a, b]) = b − a the length of [a, b].

Let us consider the following procedure for constructing J_{i+1} = [a_{i+1}, b_{i+1}] from a given J_{i} = [a_{i}, b_{i}].

Step 1. Since 0 ∉ J_{i}, so t_{μ} is strictly monotone on J_{i} and t_{μ}(J_{i}) = [a, b], where either a := t_{μ}(a_{i}) and b := t_{μ}(b_{i}) or a := t_{μ}(b_{i}) and b := t_{μ}(a_{i}).

Step 2. We set a := a+ϵ, b := b−ϵ. If not a < b then we interrupt the procedure returns fail. If a < 0 < b then we go to step 3, otherwise we go to step 4.

Step 3. The cut If b ≥ |a|, then we set a := 0. In the opposite case we set b := 0. This means that we choose the longer one of two segments (a, 0) and (0, b).

Step 4. We set a_{i+1} := a, b_{i+1} := b and we return success.

It is clear from the construction of the above procedure that if success is returned then

0 ∉ int J_{i},  (47)
J_{i} t_{μ}-covers J_{i+1}with margin ϵ  (48)

It is easy to see that this procedure returns fail iff

|t_{μ}(a_{i}) − t_{μ}(b_{i})| ≤ 2ϵ  (49)

and since t_{μ} is a linear function on J_{i} it is easy that this happens iff

μd(J_{i}) ≤ 2ϵ  (50)

Let us remark that it follows from (28-29) that a result of an application of this procedure to a segment [0, 1 − ϵ] is [0, 1 − ϵ], so this interval is the fixed point for this procedure.

Suppose that we start with J_{0} and we apply recursively the above procedure to produce a sequence J_{i} until we obtain fail message or J_{i} = [0, 1 − ϵ].

Now we will show that if d(J_{0}) is sufficiently big then we always get J_{i} = [0, 1 − ϵ] for some i.

It is easy to see that if the above procedure returns success then we have

d(J_{i+1}) ≥ μd(J_{i})/2 − ϵ, if cut was executed  (51)
d(J_{i+1}) = μd(J_{i}) − 2ϵ, otherwise.  (52)

From (50) it follows that the necessary condition for constructing J_{i+1} is

d(J_{i}) > 2ϵ/μ  (53)

We show now that two consecutive cuts starting from J_{i−1} are possible only if J_{i+1} = [0, 1 − ϵ]. For the proof suppose that J_{i} is obtained after the cut. Then either J_{i} = [0, x] or J_{i} = [−x, 0] for some x > 0. But the image of [−x, 0] is the same as the image of the segment [0, x], so we can assume that J_{i} = [0, x], where 0 < x < 1 − ϵ . Consider now the next cut, which takes place when we construct J_{i+1}. After execution of step 2 we have a = t_{μ}(x) + ϵ and b = 1 − ϵ. The condition for the cut is

t_{μ}(x) + ϵ < 0 < 1 − ϵ  (54)

Let us observe that from 0 < x < 1 − ϵ and since of t_{μ} is strictly decreasing on ℝ_{+} we get

t_{μ}(1 − ϵ) + ϵ < t_{μ}(x) + ϵ < 0  (55)

From this and (29) it follows that

|t_{μ}(x) + ϵ| < 1 − ϵ  (56)

So executing step 3 and 4 we obtain J_{i+1} = [0, 1 − ϵ].

Let us fix q > 1, but close to 1. We set

δ > max\{\frac{2ϵ}{μ − q} , 2ϵ(2/μ^{2} + 1/μ), ϵ\frac{μ + 2}{μ^{2}/2 − q} \}  (57)

Suppose that d(J_{0}) ≥ δ, then we show that one of the following condition holds

J_{1} = [0, 1 − ϵ]  (58)
J_{1}, J_{2} exist, J_{2} = [0, 1 − ϵ]  (59)
J_{1}, J_{2} exist, d(J_{2})/d(J_{0})) > q  (60)

Suppose that (58),(59) do not hold, we have to prove (60). Since the case of two consecutive cuts was shown to lead to (59) we need to consider the following three cases.

1. there is no cut in the construction of J_{1} and J_{2}

2. there is a cut in the construction of J_{1}

3. there is a cut in the construction of J_{2}

ad 1. we have d(J_{1}) = μd(J_{0}) − 2ϵ. Hence d(J_{1})/d(J_{0}) ≥ μ − 2ϵ/δ ≥ q > 1 if δ ≥ \frac{2ϵ}{μ−q}. Since length of J_{1} is greater than J_{0} construction of J_{2} is possible

and we obtain d(J_{2})/d(J_{0}) > q^{2} > q.

ad 2. d(J_{1}) ≥ μd(J_{0})/2−ϵ. From (53) it follows that to construct J_{2} the following

inequality must hold

d(J_{1}) > 2ϵ/μ  (61)

This forces the following condition for δ to satisfy

δ > 2ϵ(2/μ^{2} + 1/μ).  (62)

For J_{2} we get

d(J_{2}) ≥ μ(μd(J_{0})/2 − ϵ) − 2ϵ = μ^{2}d(J_{0})/2 − ϵ(μ + 2)  (63)

d(J_{2})/d(J_{0}) ≥ μ^{2}/2 − ϵ(μ + 2)/δ  (64)

and finally

d(J_{2})/d(J_{0}) ≥ q, if δ ≥ ϵ\frac{μ + 2}{μ^{2}/2 − q}  (65)

ad 3.

d(J_{1})  = μd(J_{0}) − 2ϵ  (66)
d(J_{2})  ≥ μd(J_{1})/2 − ϵ  (67)
d(J_{2}) ≥ μ^{2}d(J_{0})/2 − ϵ(μ + 1)  (68)

We obtain

d(J_{2})/d(J_{0}) ≥ μ^{2}/2 − \frac{ϵ}{δ} (μ + 1)  (69)

and to have d(J_{2})/d(J_{0}) ≥ q we obtain for δ the condition

δ ≥ ϵ\frac{μ + 1}{μ^{2}/2 − q}  (70)

Now it is easy to see that the sequence J_{i} will reach [0, 1 − ϵ] if d(J_{0}) ≥ δ.

Letting q → 1 we can set

δ(μ, ϵ) = max\{\frac{2ϵ}{μ − 1}, 2ϵ(2/μ^{2} + 1/μ), ϵ\frac{μ + 2}{μ^{2}/2 − 1}\}  (71)

□

Proof of lemma 9 Let δ be that obtained in lemma 10. From lemma 10 it follows that for i = 1, …, n there exists a sequence of segments J_{j}^{i} such that

J_{0}^{i} = J^{i}  (72)
J_{j}^{i} t_{μ}-covers J_{j+1}^{i} with margin ϵ, for j = 0, …, l_{s,i} − 1  (73)
J_{l_{i}}^{i} = [0, 1 − ϵ]  (74)

It follows from (28) that

[0, 1 − ϵ] ⊂ I_{μ,ϵ}  (75)
[0, 1 − ϵ] t_{μ}-covers I_{μ,ϵ} with margin ϵ  (76)

We set l = max_{i} l_{i} + 1. From conditions (75–76) it follows that we can prolongate the sequences J^{i} to get

J_{0}^{i} = J^{i}  (77)
J_{j}^{i} t_{μ}-covers J_{j+1}^{i} with margin ϵ, for j = 0, …, l − 1  (78)
J_{l}^{i} = I_{μ,ϵ}  (79)

We define

M_{j} := J_{j}^{1} × J_{j}^{2} × … × J_{j}^{n}  (80)

Since by assumption |G|_{[−1,1]} < ϵ, hence from (77–79) and the definition of M_{j} it follows

M_{0} = M  (81)
M_{j} F -covers M_{j+1} for j = 0, …, l_{s} − 1  (82)
M_{l} = I_{μ,ϵ}^{n}  (83)

□ Proof of Theorem 8 Let δ be that obtained in lemma 9. To obtain the first assertion let us observe that by lemma 9 F has TE property for ∏_{i=1}^{n} I_{μ,ϵ} and δ, so the assertion follows from Theorem 4. Now we prove the second assertion. From lemma 9 it follows that there exists a sequence of integers l_{s}, such that

N^{s} ⟹^{F^{ls}} ∏_{i=1}^{n} I_{μ,ϵ}  (84)

It follows from (28) that

[0, 1 − ϵ]^{n} ⊂ ∏_{i=1}^{n} I_{μ,ϵ}  (85)
[0, 1 − ϵ]^{n} F -covers ∏_{i=1}^{n} I_{μ,ϵ} in (1, …, n-direction  (86)

We set m = max l_{s}. From (84),(85),(86) it follows that

N^{s} ⟹^{F^{m}} ∏_{i=1}^{n} I_{μ,ϵ}  (87)

We can apply now Theorem 3 to get the desired statement. □ Suppose now that μ = 2. We are interested in the range of ϵ and δ for which assertion 2 of Theorem 8 guarantee an existence of an infinite number of periodic points. From the proof of lemma 10 ( condition (71)) it follows that

δ(2, ϵ) = 4ϵ  (88)

Form (28) it follows that

ϵ < \frac{1}{3}  (89)

It is easy to check that when (89) is true then (29) is satisfied. Observe that I_{2,ϵ} = (−1 + 3ϵ, 1 − ϵ). The condition for assertion 2 with k = (2l)^{n} is −1 + 3ϵ + lδ(2, ϵ) < 0, since in both segments (−1 + 3ϵ, 0) and (0, 1 − ϵ) we can insert the same number l of segments of length slightly greater than δ(2, ϵ). We obtain

ϵ < \frac{1}{4l + 3}  (90)

4 Perturbations of products of map 1 − 2x^{2}

The aim of this section is to extend results of Section 3 to the products of maps x ↦ 1 − 2x^{2}. Let l : ℝ → ℝ be given by

l(x) := 1 − 2x^{2}  (91)

Let F : ℝ^{n} → ℝ^{n} be a continuous function such that

F (x_{1}, x_{2}, …, x_{n}) = (l(x_{1}), l(x_{2}), …, l(x_{n})) + G(x_{1}, x_{2}, …, x_{n})  (92)

where |G|_{[−1,1]^{n}} < ϵ. We intent to prove for such F a theorem analogous to Theorem 8.

First, we need to introduce notations. Let us denote by

I_{ϵ} := [l(1 − ϵ) + ϵ, 1 − ϵ] = [−1 + 4ϵ − 2ϵ^{2}, 1 − ϵ].  (93)

The conditions corresponding to (28-29) are

−1 + 4ϵ − 2ϵ^{2} < 0 < 1 − ϵ  (94)
| − 1 + 4ϵ − 2ϵ^{2}| < 1 − ϵ  (95)

These inequalities give

ϵ < 1 − \frac{1}{\sqrt{2}}  (96)

Our task is to prove the following

Theorem 11 Let F, G satisfy (92) and ϵ be such that (94)(95) are satisfied. Then there exists δ = δ(ϵ) such that

1. there exists x ∈ ∏_{i=1}^{n} I_{ϵ} such that

• x is a periodic point for F

• for every closed ball Z = cl B(z, 2δ) , Z ⊂ ∏_{i=1}^{n} I_{ϵ} there exists m such that F^{m}(x) ∈ Z

2. for every finite family of sets {N^{s}}_{s=0,...,k−1}, N^{s} ⊂ ∏_{i=1}^{n} I_{ϵ}, of the form

N^{s} = J^{s,1} × J^{s,2} × … × J^{s,n}  (97)

where J^{s,i} are closed intervals of length greater than δ and 0 ∉ intJ^{s,i} and

N^{s} ∩ N^{s′} = ∅, for s ≠ s′.  (98)

There exists m and a continuous map π : Inv(⋃_{s} N^{s}, F^{m}) → Σ_{k}^{+}, such that

σ ∘ π = π ∘ F^{m}  (99)

π is onto. The preimage of any periodic sequence in Σ_{k}^{+} contains a periodic point for F^{m}.

We prove this theorem using the fact that the tent map t_{2} is conjugated to the map l on the interval [−1, 1]. The coordinate transformation is given by

x = ϕ(θ) = sin(\frac{π}{2} θ)  (100)

In the sequel we will use the following convention: the symbols with tilde denote objects expressed using θ coordinates. So according to this convention we have

l̃(θ) = t_{2}(θ).  (101)

To express ϵ and δ using θ coordinates we need the following

Lemma 12 Let x_{1} = ϕ(θ_{1}), x_{2} = ϕ(θ_{2}). x_{1}, x_{2} ∈ [−1, 1]. Then

\frac{π}{4} |θ_{1} − θ_{2}|^{2} ≤ |x_{1} − x_{2}| ≤ \frac{π}{2} |θ_{1} − θ_{2}|  (102)

Proof The right inequality follows directly from the mean value theorem applied to ϕ.

Suppose x_{2} > x_{1}. Let Δx := x_{2} − x_{1}. It is easy to observe that

arcsin(x_{2}) − arcsin(x_{2} − Δx) ≤ arcsin(1) − arcsin(1 − Δx)  (103)

Hence

θ_{2} − θ_{1} ≤ \frac{2}{π} (arcsin(1) − arcsin(1 − Δx)) = θ(1) − θ(1 − Δx)  (104)

We set Δθ := θ(1) − θ(1 − Δx). We have

1 − Δx = sin(\frac{π}{2} (1 − Δθ)) = cos(\frac{π}{2} Δθ)  (105)

We have the following estimate

cos x ≤ 1 − x^{2}/π, for x ∈ [−π/2, π/2].  (106)

From (105) and (106) we obtain

(\frac{π}{2} Δθ)^{2}/π ≤ Δx  (107)

and finally

Δθ ≤ \frac{2}{\sqrt{π}} \sqrt{Δx}  (108)

□

Proof of Theorem 11

Let R : ℝ^{n} → [−1, 1]^{n} be a retraction. It is easy to see that it is enough to prove the theorem for the map F̄ := R ∘ F as F̄ = F on [−1, 1]^{n}. Let us remark that applying the retraction R to F does not increase ϵ (C^{0} bound on the perturbation). Hence we can assume that F ([−1, 1]^{n}) ⊂ [−1, 1]^{n}.

The strategy of the proof is obvious. We have to go to θ variables, calculate ϵ̃ and δ̃ and then go back to x-variables.

The change of variables (x_{1}, …, x_{n}) = (ϕ(θ_{1}), …, ϕ(θ_{n})) transforms the map F into a map F̃

F̃(θ_{1}, …, θ_{n}) = (t_{2}(θ_{1}), t_{2}(θ_{2}), …, t_{2}(θ_{n})) + G̃(θ_{1}, …, θ_{n})  (109)

such that |G̃|_{|[−1,1]} < ϵ̃, where by lemma 12

ϵ̃ = 2\sqrt{ϵ/π}  (110)

From (89) it follows that ϵ̃ < 1/3, so we have

ϵ < π/36  (111)

From (88) we have

δ̃ = 4ϵ̃  (112)

From lemma 12 it follows that

δ = \frac{2}{π} δ̃ = \frac{16}{π}\sqrt{ϵ/π} = 16\sqrt{ϵ/π^{3}} > 2.9\sqrt{ϵ}  (113)

□ To obtain point 2 of the assertion we have to check the following condition kδ < 1 − ϵ, where k > 1. When we insert for δ the value 2.9\sqrt{ϵ} we obtain the inequality to solve

2.9k\sqrt{ϵ} < 1 − ϵ  (114)

We obtain

ϵ < (−2.9k + \sqrt{2.9^{2}k^{2} + 4})^{2} /4  (115)

For k = 2 we obtain ϵ < 2.8 · 10^{−2}

5 3-dim Rössler ’folded towel’ map

Let us define

f_{μ}(x) := μx(1 − x)  (116)

We consider the following folded towel map introduced by Rössler in [R] F (x, y, z) = (x̄, ȳ, z̄)(x, y, z)

x̄(x, y, z) = f_{3.8}(x) − 0.05(y + 0.35)(1 − 2z)  (117)
ȳ(x, y, z) = 0.1[(y + 0.35)(1 − 2z) − 1](1 − 1.9x)  (118)
z̄(x, y, z) = f_{3.78}(z) + 0.2y  (119)

For the parameter values considered here this map has two topologically expanding directions (1, 3). Our aim is to apply Theorem 3 to this map to obtain a symbolic dynamics.

Let us set ϵ = 0.022, ϵ_{1} = 0.01, ϵ_{2} = 0.02.

Let us define

x_{max} := max_{x∈[0,1]} f_{3.8}(x) − ϵ_{1} = 3.8/4 − ϵ_{1} = 0.94  (120)
x_{min} := f_{3.8}(x_{max}) = 0.21432  (121)
y_{max} := 0.1095  (122)
z_{max} := max_{z∈[0,1]} f_{3.78}(z) − ϵ = 3.78/4 − ϵ = 0.923  (123)
z_{min} := f_{3.8}(z_{max}) = 0.26864838  (124)

Let D := [x_{min}, x_{max}] × [−y_{max}, y_{max}] × [z_{min}, z_{max}].

We show that

|ȳ| < y_{max}, for (x, y, z) ∈ D  (125)

|ȳ| = 0.1|[(y + 0.35)(1 − 2z) − 1](1 − 1.9x)| ≤
0.1|(y_{max} + 0.35)(1 − 2z_{max}) − 1||1 − 1.9x_{max}| <
0.1(0.46 · 0.846 + 1)0.786 = 0.1 · 1.38916 · 0.786 < 0.1092

We show now

|x̄(x, y, z) − f_{3.8}(x)| < ϵ_{2}, for (x, y, z) ∈ D  (126)
|z̄(x, y, z) − f_{3.78}(z)| < ϵ, for (x, y, z) ∈ D  (127)

Equation (127) follows immediately from (119). To get (126) we compute

|x̄(x, y, z) − f_{3.8}(x)| = |0.05(y + 0.35)(1 − 2z)| ≤
0.05(y_{max} + 0.35)|1 − 2z_{max}| < 0.05 · 0.46 · 0.846 < 0.0195

We set

Z_{0} := [0.295, 0.5], Z_{1} := [0.809, 0.922]

It is easy to check that

Z_{0} f_{3.78}-covers Z_{1} with margin ϵ  (128)
Z_{1} f_{3.78}-covers Z_{0} with margin ϵ  (129)

For (x, y, z) ∈ [x_{min}, x_{max}] × [−y_{max}, y_{max}] × Z_{0} we have

|x̄(x, y, z) − f_{3.8}(x)| = |0.05(y + 0.35)(1 − 2z)| ≤
0.05(y_{max} + 0.35)|1 − 2 · 0.295| < 0.05 · 0.46 · 0.41 < 0.0095

Hence

|x̄(x, y, z) − f_{3.8}(x)| < ϵ_{1}, for (x, y, z) ∈ [x_{min}, x_{max}] × [−y_{max}, y_{max}] × Z_{0} (130)

We set

X_{00} = [0.2347, 0.5],  X_{01} = [0.5, 0.7653]  (131)
X_{1} = [0.6927, 0.94].  (132)

Observe that X_{01} is an image of X_{00} under reflection x → 1 − x.

It is easy to check that

X_{00}, X_{01} both f_{3.8}-cover X_{1} with margin ϵ_{1}  (133)
X_{1} f_{3.8}-covers X_{00} ∪ X_{01} with margin ϵ_{2}  (134)

We set

N_{00} = X_{00} × [−y_{max}, y_{max}] × Z_{0}  (135)
N_{01} = X_{01} × [−y_{max}, y_{max}] × Z_{0}  (136)
N_{1} = X_{1} × [−y_{max}, y_{max}] × Z_{1}  (137)

From (125), (126), (127),(128), (129),(130), (133) and (133) it follows that N_{00} and N_{01} F -cover N_{1} in (1, 3)-direction, and N_{1} F -covers in (1, 3)-direction both sets N_{00} and N_{01}.

We want to apply Theorem 3 to F, m = 2 and the sets N_{00}, N_{01}, but N_{00}∩N_{01} ≠ ∅. We overcome this problem by observing that there exist the sets Ñ_{00} ⊂ int N_{00} and Ñ_{01} ⊂ int N_{01}, such that Ñ_{00} and Ñ_{01} F -cover N_{1} in (1, 3)-direction. We have

Ñ_{00} ∩ Ñ_{01} = ∅  (138)
Ñ_{00} ⟹^{F^{2}} Ñ_{00}, Ñ_{01} Ñ_{01} ⟹^{F^{2}} Ñ_{00}, Ñ_{01}  (139)

and by applying Theorem 3 we obtain the following

Theorem 13 There exists a continuous map π : Inv(Ñ_{00} ∪ Ñ_{01}, F^{2}) → Σ_{2},

π ∘ F^{2} = σ ∘ π  (140)

such that Σ_{2} = π(Inv(Ñ_{00} ∪ Ñ_{01}, F^{2})). The preimage of any periodic sequence from Σ_{2} contains periodic points of F^{2}.

6 Appendix

6.1 Fixed point index

The tool used in the proof of Theorem 1 is the fixed point index. Below we list the basic properties of this notion. The proofs can be found in [D, Ch. VII, sec. 5].

Let D ⊂ ℝ^{d} be an open set and f : D → ℝ^{d} be continuous and compact. Suppose that

Fix(f, D) = {x ∈ D | f(x) = x} is compact.  (141)

Then the fixed point index for f in D is defined. We denote it by I(f, D).

If \overline{D} ⊂ dom(f) then (141) follows from the condition

Fix(f, \overline{D}) ∩ ∂D = ∅  (142)

Let us summarize the properties of the fixed point index (see [D, Ch. VII, sec. 5]).

Index is an integer.

I(f, D) ∈ ℤ.  (143)

Homotopy property. Let H : [0, 1] × D → ℝ^{d} be continuous. Suppose that

Fix(H, D) := ⋃_{λ∈[0,1]} Fix(H_{λ}, D) is compact  (144)

Then

∀λ ∈ [0, 1] I(H_{λ}, D) = I(H_{0}, D)  (145)

If [0, 1] × \overline{D} ⊂ dom(H) then (144) follows from the following condition

Fix(H, \overline{D}) ∩ ∂D = ∅  (146)

Excision property. Suppose that C ⊂ D, C is open and

Fix(f, D) ⊂ C  (147)

Then

I(f, C) = I(f, D).  (148)

Fixed point property.

if I(f, D) ≠ 0, then ∃x ∈ D f(x) = x  (149)

in particular

I(f, ∅) = 0  (150)

Fixed point index for affine maps. Suppose that f(x) = A(x−x_{0})+x_{0}, where A is a linear map and x_{0} ∈ ℝ^{d}. If the equation A(x) = x has no nontrivial solutions (i.e. if Ax = x, then x = 0) and x_{0} ∈ int(D), then

I(f, D) = sgn(det(Id − A)), where Id denotes the identity map (151)

6.2 Continuation lemma

Let G_{j} : [0, 1] × ℝ^{n} → ℝ^{n}, j = 0, 1, …, l − 1 be a family of continuous homotopies. In the sequel we will use notation G_{j,λ}(v) := G_{j}(λ, v). For an indexed family of sets Z_{j}, j = 0, 1, …, l − 1 we introduce notation

Z^{λ} := {v ∈ Z_{0}|G_{j−1,λ} ∘ G_{j−2,λ} ∘ … ∘ G_{0,λ}(v) ∈ Z_{j}, for j = 1, 2, …, l − 1}

The following lemma plays a crucial role in the proof of Theorem 1 (see also [Z1, Th. 2.2] for the similar result in a slightly different situation).

Lemma 14 Suppose that for j = 0, 1, …, l −1 the set N_{j} G_{j,λ} covers N_{(j+1) mod l} in the (i_{1}, i_{2}, …, i_{k})-direction, then the fixed point index I(G_{l−1,λ} ∘ G_{l−2,λ} ∘ … ∘ G_{0,λ}, int N^{λ}) is well defined and constant (i.e. does not depend on λ).

Proof

Without any loss of the generality we can assume that (i_{1}, …, i_{k}) = (1, …, k).

We will call the linear subspace spanned by the vectors {e_{1}, …, e_{k}} a horizontal subspace and the one generated by {e_{k+1}, …, e_{n}} will be called a vertical subspace.

Since N_{j} ∈ C, hence

N_{j} = ∏_{i=1}^{n}[v_{ji} − r_{ji}, v_{ji} + r_{ji}].  (152)

For δ > 0 we define

V (N_{j}, δ) := ∏_{i=1}^{k}[v_{ji} − r_{ji}, v_{ji} − r_{ji} + δ] ∪ [v_{ji} + r_{ji} − δ, v_{ji} + r_{ji}] ×

∏_{i=k+1}^{n}[v_{ji} − r_{ji}, v_{ji} + r_{ji}]

H(N_{j}, δ) := ∏_{i=1}^{k}[v_{ji} − r_{ji}, v_{ji} + r_{ji}] ×

∏_{i=k+1}^{n}[v_{ji} − r_{ji}, v_{ji} − r_{ji} + δ] ∪ [v_{ji} + r_{ji} − δ, v_{ji} + r_{ji}]

The letters V, H used in the definition of the above sets are shortcuts for vertical and horizontal faces in the parallelogram N_{j}, with width δ toward interior of N_{j}.

For δ sufficiently small the sets V (N_{j}, δ), H(N_{j}, δ) are nonempty and

∂N_{j} ⊂ V (N_{j}, δ) ∪ H(N_{j}, δ)  (153)

Since N_{j} G_{j,λ}-covers N_{(j+1) mod l} in (1, …, k), then we have for j = 0, 1, …, l − 1 and λ ∈ [0, 1] the following conditions

G_{j,λ}(V (N_{j}, 0)) ∩ N_{(j+1) mod l} = ∅  (154)
H(N_{(j+1) mod l}, 0) ∩ G_{j,λ}(N_{j}) = ∅  (155)

From the compactness of N_{j} we easily conclude that there exists δ > 0, such that for j = 0, 1, …, l − 1 and λ ∈ [0, 1] the following conditions hold

G_{j,λ}(V (N_{j}, 2δ)) ∩ N_{(j+1) mod l} = ∅  (156)
H(N_{(j+1) mod l}, 2δ) ∩ G_{j,λ}(N_{j}) = ∅  (157)

We define

C_{j} := ∏_{i=1}^{n}(v_{ij} − r_{ij} + δ, v_{ji} + r_{ji} − δ)  (158)
D_{j} := ∏_{i=1}^{n}(v_{ji} − r_{ji} + 2δ, v_{ji} + r_{ji} − 2δ)  (159)

We have

B(D_{j}, δ) ⊂ C_{j},  B(C_{j}, δ) ⊂ int N_{j}  (160)

Obviously the sets D^{λ}, C^{λ} are both open and the set N^{λ} is closed for λ ∈ [0, 1].

We will show now that for λ ∈ [0, 1]

G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ}(x) ≠ x, provided x ∈ N^{λ} \ D^{λ}.  (161)

Suppose that (161) does not hold. Then there exists an x ∈ N^{λ} \ D^{λ} such that

G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ}(x) = x  (162)

and at least one the following conditions holds

x ∈ N_{0} \ D_{0}  (163)
G_{j_{0}−1,λ} ∘ G_{j_{0}−2,λ} ∘ … G_{0,λ}(x) ∈ (N_{j_{0}} \ D_{j_{0}}), for some 1 ≤ j_{0} ≤ l − 1(164)

Assume (164), the proof for (163) is analogous. It follows from (157) and (159) that

G_{j_{0}−1,λ} ∘ G_{j_{0}−2,λ} ∘ … G_{0,λ}(x) ∈ V (N_{j_{0}}, 2δ)  (165)

It follows from this and from (156)

G_{j_{0},λ} ∘ G_{j_{0}−1,λ} ∘ … G_{0,λ}(x) ∉ N_{(j_{0}+1) mod l}  (166)

So if j_{0} < l − 1 then x ∉ N^{λ} and if j_{0} = l − 1 then it follows from (166) that x can not be a fixed point for a map G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ}. In both cases we get contradiction, so (161) holds.

Let Z ⊂ ℝ^{n}. We introduce the notation

Fix_{λ}(Z) := {x ∈ Z | G_{l−1,λ} ∘ G_{l−2,λ} ∘ … ∘ G_{0,λ}(x) = x}  (167)

From (161) it follows immediately that

Fix_{λ}(D^{λ}) = Fix_{λ}(C^{λ}) = Fix_{λ}(intN^{λ}) = Fix_{λ}(N^{λ}).  (168)

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}, j = 0, 1, …, l − 1 and x ∈ ⋃ N_{j} holds

|G_{j,λ_{1}} ∘ G_{j−1,λ_{1}} ∘ … ∘ G_{0,λ_{1}}(x) − G_{j,λ_{2}} ∘ G_{j−1,λ_{2}} … ∘ G_{0,λ_{2}}(x)| ≤ δ. (169)

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

D^{λ} ⊂ C^{λ_{0}} ⊂ int(N^{λ}).  (170)

Let x ∈ D^{λ}. Then

G_{j,λ} ∘ G_{j−1,λ} ∘ … G_{0,λ}(x) ∈ D_{j+1}, for j = 0, 1, …, l − 2.

But it follows from (160), (169) that

G_{j,λ_{0}} ∘ G_{j−1,λ_{0}} ∘ … G_{0,λ_{0}}(x) ∈ C_{j+1}, for j = 0, 1, …, l − 2.

Thus x ∈ C^{λ_{0}}. The proof of the second inclusion is analogous.

It follows from (168)(170) for λ ∈ \overline{Λ} that

Fix_{λ}(C^{λ_{0}}) = Fix_{λ}(N^{λ})  (171)

Since Fix_{λ}(N_{α}^{λ_{0}}) is a compact set and from (168)(171) it follows that the fixed point indices of the map G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ} relatively to the sets intN^{λ}, C^{λ_{0}}, D^{λ} are well defined and from the excision property of the fixed point index [D, Ch.VII Th. 5.4] we conclude

I(G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ}, D^{λ}) =
I(G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ}, C^{λ_{0}}) =  (172)
I(G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ}, intN^{λ}), ∀λ ∈ \overline{Λ}.

Substituting λ := λ_{0} we derive

I(G_{l−1,λ_{0}} ∘ G_{l−2,λ_{0}} ∘ … G_{0,λ_{0}}, D^{λ_{0}}) = I(G_{l−1,λ_{0}} ∘ G_{l−2,λ_{0}} ∘ … G_{0,λ_{0}}, C^{λ_{0}}). (173)

From (168)(171) it follows that

Fix\overline{Λ}(C^{λ_{0}}) := ⋃_{λ∈\overline{Λ}} Fix_{λ}(C^{λ_{0}}) ⊂ C^{λ_{0}}  (174)

Moreover, from (168)(171) it follows that

Fix\overline{Λ}(C^{λ_{0}}) = ⋃_{λ∈\overline{Λ}} Fix_{λ}(N^{λ})  (175)

From the above condition and the compactness of maps G_{j,λ} we see that Fix\overline{Λ}(C^{λ_{0}}) is compact.

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

I(G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ}, C^{λ_{0}}) = I(G_{l−1,λ_{0}} ∘ G_{l−2,λ_{0}} ∘ … G_{0,λ_{0}}, C^{λ_{0}}) ∀λ ∈ \overline{Λ}.

(176)

From (172),(173),(176) we conclude

I(G_{l−1,λ} ∘ G_{l−2,λ} ∘ … G_{0,λ}, D^{λ}) = I(G_{l−1,λ_{0}} ∘ G_{l−2,λ_{0}} ∘ … G_{0,λ_{0}}, D^{λ_{0}}) ∀λ ∈ \overline{Λ}.

(177)

From the connectedness of [0, 1], (177) and (172) we get

I(G_{l−1,λ}∘G_{l−2,λ}∘… G_{0,λ}, intN^{λ}) = I(G_{l−1,0}∘G_{l−2,0}∘… G_{0,0}, intN^{0}) ∀λ ∈ [0, 1]

This finishes the proof. □

6.3 Proof of topological theorem

Proof of Theorem 1

We can assume that that (i_{1}, …, i_{k}) = (1, …, k). Let

N_{j} = ∏_{i=1}^{n}[v_{ji} − r_{ji}, v_{ji} + r_{ji}].  (178)

We want to connect maps F_{j} by homotopy with affine maps M_{j} for which we can easily calculate the fixed point index of interest.

Let us choose d < u ∈ ℝ such that for every j N_{j} ⊂ [d, u]^{n}.

We define an affine map M_{j} : ℝ^{n} → ℝ^{n} as a cartesian product of one-dimensional affine maps M_{j}(x_{1}, …, x_{n}) = (m_{j1}(x_{1}), …, m_{jn}(x_{n}). First we define m_{ji} for i > k (i is not an expanding dimension). In this case we set m_{ji}(x_{i}) = v_{ji}.

If i ≤ k (i is an expanding direction), then if N_{j} F_{j}-covers N_{(j+1) mod l} in the i-direction preserving orientation then we define

m_{ji}(v_{ji} − r_{ji}) := d, m_{ji}(v_{ji} + r_{ji}) := u  (179)

in the other case (orientation reversed) we set

m_{ji}(v_{ji} − r_{ji}) := u, m_{ji}(v_{ji} + r_{ji}) := d  (180)

We now extend the maps m_{ji} to affine maps defined on ℝ.

We define a family of homotopies G_{j} : [0, 1] × ℝ^{n} → ℝ^{n} as a convex combination of M_{j} and G_{i}, so

G_{j}(λ, x) := (1 − λ)F_{j}(x) + λM_{j}(x)  (181)

It follows easily from the construction of M_{j} and G_{j} that for every λ ∈ [0, 1]

N_{j} G_{j,λ}-covers the set N_{(j+1) mod l} in (1, …, k)-direction  (182)

So we can apply lemma 14 to G_{j} to obtain

I(F_{l−1} ∘ F_{l−2} ∘ … ∘ F_{0}, intN^{0}) = I(M_{l−1} ∘ M_{l−2} ∘ … ∘ M_{0}, intN^{1})  (183)

To finish the proof it remains to calculate the fixed point index of the superposition of M_{j}’s.

Let us set for p = 1, …, l − 2

Y_{p} := {x ∈ N_{0} | M_{s} ∘ M_{s−1} ∘ … ∘ M_{0}(x) ∈ N_{s+1}, for s = 1, …, p} (184)

For p = 0 we set Y_{0} = N_{0}. Obviously Y_{l−2} = N^{1}.

It is easy to prove inductively for p = 0, …, l − 2,

Y_{p} = ∏_{i=1}^{k} J_{p,i} × ∏_{i=k+1}^{n}[v_{0,i} − r_{0,i}, v_{0,i} + r_{0,i}]  (185)
M_{p}∘ M_{p−1} ∘ … ∘ M_{0}(Y_{p}) =
∏_{i=1}^{k}[v_{p+1,i} − r_{p+1,i}, v_{p+1,i} + r_{p+1,i}] × ∏_{i=k+1}^{n}{v_{p+1,i}}  (186)

where J_{p,i} ⊂ [v_{0,i} − r_{0,i}, v_{0,i} + r_{0,i}] for i = 1, …, k.

From the definition of M_{j} it follows that there exist a family of affine maps a_{i} : ℝ → ℝ, i = 1, …, n such that

M_{l−1} ∘ M_{l−2} ∘ … ∘ M_{0}(x_{1}, …, x_{n}) = (a_{1}(x_{1}), a_{2}(x_{2}), …, a_{n}(x_{n}))  (187)

Moreover, it follows from (186) and (182) for λ = 1 that

M_{l−1} ∘ M_{l−2} ∘ … ∘ M_{0}(Y_{l−2}) =
∏_{i=1}^{k}[d, u] × ∏_{i=k+1}^{n}{v_{0,i}}  (188)

So we obtain

J_{l−2,i} ⊂ int a_{i}(J_{l−2,i}), if i ≤ k  (189)
a_{i}(x_{i}) = v_{0i} ∈ int J_{l−2,i}, otherwise  (190)

From this and (151) it follows that

I(M_{l−1} ∘ M_{l−2} ∘ … ∘ M_{0}, intN^{1}) = ±1  (191)

□

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