Georgian Mathematical Journal Volume 9 (2002), Number 2, 339–366

PERIODIC POINTS AND CHAOTIC-LIKE DYNAMICS OF PLANAR MAPS ASSOCIATED TO NONLINEAR HILL’S EQUATIONS WITH INDEFINITE WEIGHT

DUCCIO PAPINI AND FABIO ZANOLIN

Abstract. We prove some results about the existence of fixed points, periodic points and chaotic-like dynamics for a class of planar maps which satisfy a suitable property of “arc expansion” type. We also outline some applications to the nonlinear Hill’s equations with indefinite weight.

2000 Mathematics Subject Classification: 34C15, 34C25, 34C28, 37D45, 70KXX, 70K40. Key words and phrases: Planar maps, fixed points, periodic points, chaotic dynamics, nonlinear Hill’s equations.

1. Introduction

The study of the existence of oscillatory and periodic solutions of the nonlinear scalar second-order differential equation

ẍ + q(t)g(x) = 0  (1.1)

has motivated a great deal of research activities during the last fifty years.

Here and henceforth, we suppose that g : ℝ → ℝ is a locally Lipschitz continuous function and denote by

G(s) := ∫_{0}^{s} g(ξ) dξ

a primitive of g. The “weight” function q : ℝ → ℝ is assumed to be continuous. Further (mild) conditions on q(t) will be added later, if necessary. In the sequel, we will also confine our discussion to the situation in which q(t) changes its sign.

In [69], Waltman initiated the study of the oscillatory solutions of the nonlinear Hill’s equation

ẍ + q(t)x^{2n+1} = 0, n ≥ 1,  (1.2)

when the coefficient q(t) is not necessarily of definite sign. Contributions to this problem were also given by Kiguradze [36], Wong [76], Bobisud [10], Onose [48], Butler [12], [13], Kwong and Wong [39] and others. The interest in investigating the oscillatory (or nonoscillatory) solutions of equation (1.2) or equation (1.1)

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

for a sign indefinite weight is witnessed by a broad literature (see, e.g., [77], [78] and the references therein).

In the case of a periodic q(·), Butler in [14] first obtained the existence of infinitely many periodic solutions to equation (1.1) for a function g(x) having a superlinear growth at infinity, thus including the case of equation (1.2). Recent developments along Butler’s work have been obtained in [49], [50], [51].

In the past ten years, starting with Lassoued [40], the study of boundary value problems for superlinear equations with an indefinite weight has received much attention in the literature. Besides the case of partial differential equations that we don’t discuss here (see, e.g., [1], [7], [8]), general results have been obtained for the periodic solutions of Hamiltonian systems and, in particular, second order vector differential equations of the form ü + ∇_{u}V (t, u) = 0, using critical point theory (see [5], [6], [22], [25], [26], [43]). Other applications of variational methods to the periodic problem for the equation

ẍ + q(t)|x|^{p−1}x = 0  (1.3)

are contained in [41] and [57].

In [68], Terracini and Verzini showed that, for a sign changing weight, equation (1.2) (as well as its “forced” extension) may possess solutions which oscillate in a quite “wild” sense. In fact, in [68] the authors gave evidence of the possibility of a chaotic behavior for the solutions of (1.2), by proving that there is a double-sided sequence of positive integers m_{i}^{∗} (i ∈ ℤ), corresponding to a suitably labeled sequence of intervals J_{i}^{+} in which q > 0, such that, for every m_{i} ≥ m_{i}^{∗} , there are solutions having m_{i} zeros in the i-th interval of positivity J_{i}^{+} . These solutions vanish once in the intervals where q < 0 (see [68] for the precise statement).

In [52], we considered some boundary value problems (including the Sturm-Liouville boundary conditions) for equation (1.1) with q(t) a sign changing function defined in a interval and satisfying suitable assumptions in order to have, like in [14], the continuability of the solutions across the sub-intervals where q(t) ≥ 0. Therein, under the hypothesis that

(g_{0})  g(s)s > 0 for s ≠ 0

and assuming also that g satisfies a condition of superlinear growth at infinity, we proved the existence of solutions having an arbitrarily large number of zeros in the intervals where q(t) > 0 and having precisely one zero or no zeros at all (according to any sequence of 1’s and 0’s fixed in advance) in the intervals where q(t) < 0. Further results in this direction and for the superlinear indefinite case have been obtained in [16], [50], [51]. In particular, a detailed investigation concerning the existence of chaotic-like oscillatory solutions of (1.1) is contained in the recent article [16] by Capietto, Dambrosio and Papini.

With this respect, the aim of this paper is not that of reproducing the argument which has been already used in [16], yet we would like first to focus our attention to the properties of the planar transformations which are associated to the solutions of equation (1.1) and then to prove the existence of a chaotic-like dynamics for a broad class of planar maps which emulate the Poincaré map of (1.1).

The key properties of the solutions of (1.1) which were used in the proof of the main results in [52] are described in the following lemmas taken from [52, Lemma 3, Lemma 4], that we recall now for the reader’s convenience.

For the purposes of the present work it is not necessary to repeat here the precise assumptions of superlinear growth at infinity which were considered in [52]. For simplicity in the exposition, we restrict ourselves to the special case of

(g_{mon})  g is monotone nondecreasing in a neighborhood of ± ∞

and

(g_{∞})  lim_{s→±∞} \frac{g(s)}{s} = +∞, | ∫^{±∞}|G(s)|^{− \frac{1}{2}} ds| < ∞.

Clearly, all the conditions (g_{0}), (g_{mon}) and (g_{∞}) are satisfied for

g(x) = |x|^{p−1}x, p > 1,

which is the case of equation (1.3). However, we point out that in our results the oddness or the homogeneity of g(x) are not required. More general assumptions for g(x) in terms of time-mappings, like in [23], are given in [52].

Following the notation in [52] we denote by A_{i}, for i = 1, 2, 3, 4, the open quadrants of the plane, counted in the counterclockwise sense starting from A_{1} = {(x, y) : x > 0, y > 0}. We also denote by z(·; t_{0}, z_{0}) the solution of

ẋ = y, ẏ = −q(t)q(x)  (1.4)

satisfying the initial condition z(t_{0}) = (x(t_{0}), y(t_{0})) = z_{0} . We observe that, due to the fact that q < 0 in some intervals, we have that z(·; t_{0}, z_{0}) may be not globally defined for some z_{0} ∈ ℝ^{2} (see [11], [15] ). On the other hand, we assume (like in [14], [52] ) the continuability of the solutions across the sub-intervals where q ≥ 0.

Lemma 1 ([52, Lemma 3]). Let a < b < c, be such that

q ≥ 0 and q ≢ 0 on [a, b], q ≤ 0 and q ≢ 0 on [b, c].

Then there is a constant R^{∗} (depending only on g and q|_{[b,c]}) such that the following holds:

For each R > 0, there is n^{∗} = n_{R}^{∗} > 0 such that, for each n > n^{∗}, and for each continuous curve γ : [α, β[ → Ā_{1} (respectively, γ : [α, β[ → Ā_{3}), with

|γ(α)| ≤ R and |γ(s)| → ∞, as s → β^{−},

there is an interval [α_{n}, β_{n}] ⊂]α, β[ such that for each s ∈]α_{n}, β_{n}] we have: • z(t; a, γ(s)) is defined for all t ∈ [a, c],

• x(·; a, γ(s)) has exactly n zeros in ]a, b[ , no zeros in [b, c] and exactly one change of sign of the derivative in ]b, c[ .

Moreover, setting

γ_{n}(s) := z(c; a, γ(s)), ∀ s ∈]α_{n}, β_{n}],

we have that

|γ_{n}(β_{n})| ≤ R^{∗} and |γ_{n}(s)| → ∞, as s → α_{n}^{+},

and γ_{n} lies in Ā_{1} or in Ā_{3} according to the fact that n is even or odd (respectively, γ_{n} lies in Ā_{3} or in Ā_{1} according to the fact that n is even or odd).

Lemma 2 ([52, Lemma 4]). Let a < b < c, be such that

q ≥ 0 and q ≢ 0 on [a, b], q ≤ 0 and q ≢ 0 on ; [b, c].

Then there is a constant R^{∗} (depending only on g and q|_{[b,c]}) such that the following holds:

For each R > 0, there is n^{∗} = n_{R}^{∗} > 0 such that, for each n > n^{∗}, and for each continuous curve γ : [α, β[ → Ā_{1} (respectively, γ : [α, β[ → Ā_{3}), with

|γ(α)| ≤ R and |γ(s)| → ∞, as s → β^{−},

there is an interval [α_{n}, β_{n}] ⊂]α, β[ such that for each s ∈ [α_{n}, β_{n}[ we have: • z(t; a, γ(s)) is defined for all t ∈ [a, c],

• x(·; a, γ(s)) has exactly n zeros in ]a, b[ , exactly one zero in ]b, c[ and no

zeros of the derivative in [b, c]. Moreover, setting

γ_{n}(s) := z(c; a, γ(s)), ∀ s ∈ [α_{n}, β_{n}[ ,

we have that

|γ_{n}(α_{n})| ≤ R^{∗} and |γ_{n}(s)| → ∞, as s → β_{n}^{−}

and γ_{n} lies in Ā_{3} or in Ā_{1} according to the fact that n is even or odd (respectively, γ_{n} lies in Ā_{1} or in Ā_{3} according to the fact that n is even or odd).

Let us define, for 0 < r_{1} < r_{2} , the closed annulus

A[r_{1}, r_{2}] := {z ∈ ℝ^{2} : r_{1} ≤ |z| ≤ r_{2}}

and, for r > 0, the circumference C_{r} := {z ∈ ℝ^{2} : |z| = r}. Now, from Lemma 1 and Lemma 2 we can obtain the following.

Proposition 1. Assume (g_{0}), (g_{mon}) and (g_{∞}). Let a < b < c, be such that

q ≥ 0 and q ≢ 0 on [a, b], q ≤ 0 and q ≢ 0 on [b, c]

and suppose that the solutions of (1.1) are continuable across [a, b]. Then there is a constant R^{∗} (depending only on g and q|_{[b,c]}) such that for each R ≥ R^{∗}, there is n^{∗} = n_{R}^{∗} > 0 such that for each j > n^{∗} there is R_{j} > R in order to satisfy the following arc expansion property:

Let us fix i = 1, 3 and δ = 0, 1. Given any path σ contained in Ā_{i} ∩ A[R, R_{j}] and such that

σ ∩ C_{R} ≠ ∅, σ ∩ C_{R_{j}} ≠ ∅,

there is a sub-path σ′ such that, for each z_{0} ∈ σ′ : z(·; a, z_{0}) is defined for all t ∈ [a, c], x(b)x(c)ẋ(b)ẋ(c) ≠ 0, x(·; a, z_{0}) has exactly j zeros in ]a, b[ , exactly δ zeros in ]b, c[ and exactly 1 − δ changes of sign of the derivative in ]b, c[ .

Moreover, for the map ϕ : z_{0} ↦ z(c; a, z_{0}), we have that:

ϕ(σ′) ∩ C_{R} ≠ ∅, ϕ(σ′) ∩ C_{R_{j}} ≠ ∅

and

ϕ(σ′) ⊂ Ā_{ℓ} ∩ A[R, R_{j}],

with ℓ = i or ℓ = i + 2 (mod. 4) according to the fact that j + δ is even or odd.

Proof (sketched). The proof of Proposition 1 is a straightforward consequence of Lemma 1 and Lemma 2 using the following two facts.

First, by our assumptions, we have the continuability of the solutions across the intervals of positivity of q(t). From this and the fact that g is lipschitzean around zero, it follows that solutions with an initial point z(a) ≠ 0 in a compact set have a number of zeros in [a, b] which is (uniformly) upper bounded. Indeed, for the number of zeros n(x; a, b) of x(·) in [a, b[ , we have that ⌊2rot(z; a, b)⌋ ≤ n(x; a, b) ≤ 2rot(z; a, b) + 1 where

rot(z; a, b) = \frac{1}{2π} ∫_{a}^{b} \frac{y(t)^{2} + q(t)g(x(t))x(t)}{y(t)^{2} + x(t)^{2}} dt.

Second, thanks to the assumption of superlinear growth at infinity for g(x), we know (see, e.g., [14], [29], [33], [44], [65] ) that in the interval of positivity of q(t) solutions having at most j zeros are uniformly bounded (by a constant depending on j) in the C^{1}-norm on [a, b].

By virtue of these facts, we can “cut” the paths γ and γ_{n} of Lemma 1 and Lemma 2 in order to reduce the corresponding results to compact sets, provided that the number of the zeros of the solutions is sufficiently large but bounded.

As to the precise dynamics of the trajectories we have that the following four situations are possible for a solution z(·) with z(a) belonging to a suitable sub-path σ′ of σ in Ā_{i} ∩ A[R, R_{j}] :

if j is even and δ = 0, we have that x(·) has exactly j-zeros in ]a, b[ , with z(b) ∈ A_{i−1} (mod. 4), x(t) ≠ 0 for all t ∈ [b, c] and ẋ changes sign exactly once in ]b, c[ , with z(c) ∈ A_{i}; if j is even and δ = 1, we have that z(·) has exactly j-zeros in ]a, b[ , with z(b) ∈ A_{i−1} (mod. 4), ẋ(t) ≠ 0 for all t ∈ [b, c] and x vanishes exactly once in ]b, c[ , with z(c) ∈ A_{i+2} (mod. 4); if j is odd and δ = 0, we have that z(·) has exactly j-zeros in ]a, b[ , with z(b) ∈ A_{i+1} , x(t) ≠ 0 for all t ∈ [b, c] and ẋ changes sign exactly once in ]b, c[ , with z(c) ∈ A_{i+2} (mod. 4); if j is odd and δ = 1, we have that z(·) has exactly j-zeros in ]a, b[ , with z(b) ∈ A_{i+1} , ẋ(t) ≠ 0 for all t ∈ [b, c] and x vanishes exactly once in ]b, c[ , with z(c) ∈ A_{i} .

With respect to Lemma 1 and Lemma 2 we have added here also the information that z(c) is not on the coordinate axes. This would follow by reworking along the argument in [52], or by a corresponding remark in [51].

Note that from ẋ = y and by an elementary analysis of the direction of the vector field in system (1.4), it follows that all the zeros of x(·) are simple and that (roughly) the transition from a quadrant to another is of the type A_{2} → A_{1} , A_{4} → A_{3} and, moreover, A_{1} → A_{4} , A_{3} → A_{2} for t ∈]a, b[ , A_{4} → A_{1} , A_{2} → A_{3} for t ∈]b, c[ . This way, we have a complete control of the position of z(c) according to the number of the zeros of the solution. In particular, from the above four cases, we have also that ϕ(z_{0}) ∈ A_{i} or ϕ(z_{0}) ∈ A_{i+2} (mod. 4) according to the fact that j +δ is even or odd. Hence the conclusion follows.

Now, our aim will be that of showing that the arc expansion property described in Proposition 1 implies the existence of fixed points and of periodic points of arbitrary order for the Poincaré map associated to (1.1) as well as of a chaotic-like dynamics of coin tossing type [37]. ^{1} This goal will be achieved by means of some theorems about the dynamics of a planar map which entails some properties of the Poincaré map associated to the nonlinear Hill’s equations with indefinite weight. We stress that, even if in the present article we have made our main reference to the case of a restoring field g(x) having superlinear growth at infinity (like in [16] and in [52] ), in principle (and due to the more general setting of planar maps instead of that of a special class of differential equations) our results can be applied to a wide family of nonlinear scalar ordinary differential equations, including some cases of asymptotically linear systems (provided that a suitable gap between zero and infinity is assumed). More details and further results in this direction will be given in [53].

At the end of this introductory section, we should also mention two recent articles by Kennedy and Yorke [35] and Kennedy, Koçak and Yorke [34] in which the authors develop a general topological framework in order to deduce the presence of chaotic dynamics for a broad variety of different situations. In particular, in [34], the authors define the concept of a “family of expanders” and it seems that it should be possible to show that some of the results following from the arc expansion property of Proposition 1 (and that we are going to develop in the next sections) are likely to be obtainable as a consequence of the results [34] or [35]. On the other hand, in our setting, we are able to prove the existence of fixed points and of periodic points, via an elementary fixed point theorem for planar maps (see Theorem 1 and Theorem 2[case (w_{2})]). It is therefore our hope that the approach proposed in the present paper (even if limited to the study of a two-dimensional map) may be of some independent interest as well. In the last section, we try to describe the presence of an underlying general topological structure for our setting, in order to make more transparent the connections between our results for the nonlinear Hill equation and some arguments which belongs to the recent research on “topological horseshoes”.

1 Recall that, according to [37], a discrete dynamical system (X, ψ) is chaotic in the sense of coin tossing, if there are two disjoint compact sets X_{0} and X_{1} of X such that, given any two-sided sequence (…, s_{−2}, s_{−1}, s_{0}, s_{1}, s_{2}, …), with s_{i} ∈ {0, 1}, there is an orbit (x_{n})_{n} of ψ with x_{n} ∈ X_{s_{n}} for each n ∈ ℤ.

Throughout the paper, we recall that by an arc (respectively, a path) we mean the homeomorphic image (respectively, the image through a continuous map) of a compact nondegenerate interval of the real line. A continuum is a compact connected set. In the sequel, all the arcs, paths and continua that will be considered are subsets of the plane ℝ^{2}. It will be also useful to have in mind that a plane continuum is not necessarily a path and that each path joining two different points contains an arc joining the same two points [24], [79]. Moreover, if Γ ⊂ ℝ^{2} is a continuum, then, for any two points P, Q ∈ Γ, with P ≠ Q and any ε > 0, there is a path (or, if we like, an arc) σ = σ_{(P,Q)}^{ε} joining P with Q and contained in the ε-neighborhood B(Γ, ε) of Γ. This property clearly allows to suitably approximate a continuum with arcs.

As a last definition, we call a two-dimensional cell R ⊂ ℝ^{2} the homeomorphic image of the square Q = [−1, 1]^{2}.

Finally, and in order to avoid misunderstanding, we notice that we denote by ℕ = {1, 2, … } the set of positive integers and when we speak of a map ψ we implicitly assume that it is a continuous one (unless when the contrary is explicitly mentioned).

2. Fixed Points and Dynamics of Planar Maps

We start this section with some results on the existence of fixed points for a (continuous) map ψ : ℝ^{2} ⊃ dom(ψ) := D_{ψ} → ℝ^{2} .

Let v = (v_{1}, v_{2}) and w = (w_{1}, w_{2}) be two unit vectors of the plane with v ≠ w. We denote by ℓ_{v} = {tv : t > 0} and ℓ_{w} = {tw : t > 0} the (open) half-lines from the origin through v and w. The 0-pointed angle v̂0w is the union of ℓ_{v}, ℓ_{w} with all the half-lines ℓ_{u} such that the triple (v, u, w) is positively oriented. Given an angle v̂0w we denote by ν_{v} = (v_{2}, −v_{1}) and ν_{w} = (−w_{2}, w_{1}) the outer normals to v̂0w at the points of ℓ_{v} and ℓ_{w} (clearly, these points lie on the boundary of v̂0w).

Note that in the above definition, 0 ∉ v̂0w. Clearly, the closure of v̂0w is v̂0w ∪ \{0\}, that will be denoted by \overline{v0w}.

Lemma 3. Let v̂0w be a pointed angle at the origin. Assume that there is a connected set Γ ⊂ D_{ψ} ∩ v̂0w, with Γ ∩ ℓ_{v} ≠ ∅ and Γ ∩ ℓ_{w} ≠ ∅ and such that

(ψ(z)|ν_{v}) ≤ 0, ∀ z ∈ Γ ∩ ℓ_{v} and (ψ(z)|ν_{w}) ≤ 0, ∀ z ∈ Γ ∩ ℓ_{w} (2.5)

(or

(ψ(z)|ν_{v}) ≥ 0, ∀ z ∈ Γ ∩ ℓ_{v} and (ψ(z)|ν_{w}) ≥ 0, ∀ z ∈ Γ ∩ ℓ_{w} , (2.6)

respectively.) Then, there is at least one point z̃ ∈ Γ such that ψ(z̃) = λz̃ for some λ ∈ ℝ.

The interested reader is invited to draw a picture in order to visualize the geometrical meaning of conditions (2.5), (2.6). Proof. Assume (2.5). Let ψ = (ψ_{1}, ψ_{2}) and consider the map h : z = (z_{1}, z_{2}) ↦ z_{1}ψ_{2}(z) − z_{2}ψ_{1}(z), for z ∈ Γ. If z ∈ Γ ∩ ℓ_{v} , we have that z = tv for some t > 0. Hence, h(z) = t(v_{1}ψ_{2}(tv) − v_{2}ψ_{1}(tv)) = −t(ψ(tv)|ν_{v}) ≥ 0. With analogous computation, we can see that h(z) ≤ 0 for z ∈ Γ ∩ ℓ_{w} . By the continuity of ψ and the connectedness of Γ, we conclude that there is z̃ ∈ Γ such that h(z̃) = 0 (Bolzano theorem). Since z̃ ≠ 0 (because 0 ∉ Γ), this implies that there is a λ ∈ ℝ such that ψ(z̃) = λz̃ and the proof is complete.

Corollary 1. Let v̂0w be a pointed angle at the origin such that its closure \overline{v0w} is convex. Assume that there is a connected set Γ ⊂ D_{ψ} ∩ v̂0w, with Γ ∩ ℓ_{v} ≠ ∅ and Γ ∩ ℓ_{w} ≠ ∅ and such that

ψ(Γ) ⊂ v̂0w.  (2.7)

Suppose further that

\begin{matrix}ψ(z) ≠ μv, & ∀ z ∈ Γ ∩ ℓ_{v} , ∀μ < 0, \\ ψ(z) ≠ μw, & ∀ z ∈ Γ ∩ ℓ_{w} , ∀μ < 0, \end{matrix}  (2.8)

is satisfied. Then, there is at least one point z̃ ∈ Γ such that ψ(z̃) = λz̃ for some λ > 0.

Proof. Since \overline{v0w} is closed and convex, the assumption that ψ(Γ) ⊂ v̂0w, implies that (ψ(z)|ν_{v}) ≤ 0, for all z ∈ Γ ∩ ℓ_{v} and (ψ(z)|ν_{w}) ≤ 0, for all z ∈ Γ ∩ ℓ_{w} . Hence, we can apply Lemma 3 and find z̃ ∈ Γ and λ ∈ ℝ such that ψ(z̃) = λz̃. As a consequence of ψ(Γ) ⊂ v̂0w, we find that λz̃ ∈ v̂0w with z̃ ∈ v̂0w and hence λ ≠ 0 for 0 ∉ v̂0w. If, by contradiction, λ < 0, we have that both z̃ and −z̃ belong to v̂0w and this is possible only if z̃ ∈ ℓ_{v} ∪ ℓ_{w} , but then, we contradict (2.8).

Remark 1. In the frame of Corollary 1, we notice that condition (2.8) is always fulfilled when w ≠ −v (that is, v̂0w ∪ {0} is a cone [21, Ch.6]) or when ψ(Γ) ⊂ int(v̂0w).

It will be also useful to introduce the following definition.

Let v̂0w be a pointed angle at the origin. For r_{1} and r_{2} with 0 < r_{1} < r_{2} , we define the (closed) set

[v̂0w]_{r_{1}}^{r_{2}} := {z ∈ v̂0w : r_{1} ≤ |z| ≤ r_{2}} = v̂0w ∩ A[r_{1}, r_{2}].

We say that [v̂0w]_{r_{1}}^{r_{2}} is a conical shell if it is determined by a pointed angle v̂0w at the origin such that \overline{v0w} is convex and also w ≠ −v. This definition corresponds to that given in [21, p. 239].

Corollary 2. Let W = [v̂0w]_{r_{1}}^{r_{2}} be a conical shell and suppose that there is a connected set Γ ⊂ D_{ψ} ∩ W with Γ ∩ ℓ_{v} ≠ ∅ and Γ ∩ ℓ_{w} ≠ ∅ and such that ψ(Γ) ⊂ W and |ψ(z)| = |z| for all z ∈ Γ. Then, the map ψ has at least one fixed point z̃ ∈ Γ.

Proof. By Corollary 1 and Remark 1 we have that there is z̃ ∈ Γ such that ψ(z̃) = λz̃ for some λ > 0. But, evidently, λ = 1 as |ψ(z)| = |z| on Γ.

For the next steps, we recall a slight variant of a result from plane topology previously used also in [14], [20] (more or less explicitly). Details can be found in [54].

Lemma 4. Let R ⊂ ℝ^{2} be a two-dimensional cell and let h : Q = [−1, 1]^{2} → R, be a surjective homeomorphism. Let us define H^{±} := h([−1, 1] × {±1}) and V^{±} := h({±1} × [−1, 1]). Let S ⊂ R be a closed set such that

σ ∩ S ≠ ∅,

for each path σ contained in R and joining H^{−} with H^{+}. Then, S contains a closed connected set C joining V^{−} with V^{+}.

Using the property that any path joining two distinct points contains an arc joining the same two points (as recalled at the end of the Introduction), it is easy to see that it is completely equivalent if in the assumption of Lemma 4 we suppose that σ is an arc.

Example 1. Let us consider the set S = S_{0} ∩ Q, where

S_{0} = \{(x, y) : (x ∈ ℚ, y ∈ ℚ, −\frac{1}{2} ≤ y ≤ 0)∨(x ∈ ℝ\ℚ, y ∈ ℝ\ℚ, 0 < y < \frac{1}{2})\}.

The set S is not closed, not connected and does not contain any connected subset joining V^{−} with V^{+}. Though, any path from H^{−} to H^{+} has nonempty intersection with S.

For the subsequent results, we also need a property of the domain (or a part of the domain) on which we want to have the map ψ defined.

Let us consider a set D ⊂ D_{ψ}. We say that the (continuous) map is proper on D if ψ^{−1}(K) ∩ D is compact for each compact set K ⊂ ℝ^{2}. Equivalently, we have that whenever there is a sequence z_{n} ∈ D and a point w ∈ ℝ^{2} such that ψ(z_{n}) → w, then, there is a subsequence z_{i_{n}} of z_{n} and a point z^{∗} ∈ D such that z_{i_{n}} → z^{∗}.

In this situation, we will also say that the pair (D, ψ) is proper.

To be used in the next steps, we note that a composition of proper maps is proper and that if (D, ψ) is proper, then ψ is proper on D ∩ C, for each closed set C ⊂ ℝ^{2} (see the Appendix for more details).

Given a map ψ and a set D ⊂ D_{ψ} , we also define (D, ψ) proper on compact sets if, for each K compact, ψ is proper on D ∩ K.

Example 2. Let f : ℝ × ℝ^{2} → ℝ^{2} be a Carathéodory function and assume the uniqueness of the solutions for the Cauchy problems

\{ \begin{matrix}ż = f(t, z) \\ z(t_{0}) = z_{0} \end{matrix}  (2.9)

with (t_{0}, z_{0}) ∈ ℝ × ℝ^{2}.

If we denote by z(·; t_{0}, z_{0}) the noncontinuable solution of (2.9), defined on its maximal interval of existence I_{(t_{0},z_{0})} , we have that for each t_{0}, t ∈ ℝ, the map ϕ_{t_{0}}^{t} : z_{0} ↦ z(t; t_{0}, z_{0}) is continuous and is defined in a open (possibly empty) set D(t_{0}, t) ⊂ ℝ^{2} (by the Peano’s theorem for the Carathéodory equations [28] D(t_{0}, t) ≠ ∅ if |t − t_{0}| is sufficiently small). We also have that ϕ_{t_{1}}^{t_{2}} ∘ ϕ_{t_{0}}^{t_{1}} = ϕ_{t_{0}}^{t_{2}} .

We look now for conditions to be satisfied by the differential equation in order that the pair (D(t_{0}, t), ϕ_{t_{0}}^{t}) is proper on compact sets. In fact, in general, the map ϕ_{t_{0}}^{t} may fail to be proper even when it is restricted to the compact sets. An example of this fact is shown, for instance, by the the planar system ẋ = 1, ẏ = −xy^{2} .

Let t_{0} < t_{1} and let K be a compact set. Suppose that (2.9) has solutions defined on [t_{0}, t_{1}] for each z_{0} ∈ K. Then it is easy to prove that ϕ_{t_{0}}^{t_{1}} is proper on K ⊂ D_{t_{0}}^{t_{1}} . A more general result is the following.

Claim 1. Assume that there is a continuous function η : ℝ^{+} → ℝ^{+} such that

|z(s; t_{0}, z_{0})| ≤ η( max{|z_{0}|, |z(t; t_{0}, z_{0})|}), ∀ t_{0} ≤ s ≤ t, ∀ z_{0} ∈ D(t_{0}, t). (2.10)

Then, for any compact set K ⊂ ℝ^{2}, the map ϕ_{t_{0}}^{t} is proper on D(t_{0}, t) ∩ K. Proof. Let t > t_{0} be fixed and suppose that D(t_{0}, t) ≠ ∅ (otherwise the claim is vacuously true). Let E = K ∩ D(t_{0}, t), with K a compact subset of ℝ^{2}.

Let z_{0}^{n} ∈ E be a bounded sequence of initial points and, without loss of generality, assume that z_{0}^{n} → z_{0} ∈ K and w_{n} := z(t; t_{0}, z_{0}^{n}) → w_{0} ∈ ℝ^{2}. Then, we have that |z_{0}^{n}|, |w_{n}| ≤ M, for all n and for some M > 0, so that, by (2.10),

|z(s; t_{0}, z_{0}^{n})| ≤ η(M) := M_{1}, ∀ s ∈ [t_{0}, t], n ∈ ℕ.

By the Carathéodory assumption, there is ρ ∈ L^{1}([t_{0}, t], ℝ^{+}) such that

|f(s, z(s; t_{0}, z_{0}^{n}))| ≤ ρ(s), for a.e. s ∈ [t_{0}, t], ∀ n ∈ ℕ.

Using the Ascoli–Arzelá theorem and the Lebesgue dominated convergence theorem in the integral relation

z(s; t_{0}, z_{0}^{n}) = z_{0}^{n} + ∫_{t_{0}}^{t} f(ξ, z(ξ; t_{0}, z_{0}^{n})) dξ, ∀ s ∈ [t_{0}, t],

(cf.[28, p.29]) we can prove that there is a continuous function ẑ : [t_{0}, t] → ℝ^{2} such that

ẑ(s) = z_{0} + ∫_{t_{0}}^{t} f(ξ, ẑ(ξ)) dξ, ∀ s ∈ [t_{0}, t].

From this, it follows that z_{0} ∈ D(t_{0}, t), ẑ(s) = z(s; t_{0}, z_{0}) and w_{0} = ẑ(t). Hence the claim is proved.

Example 3. Let g : ℝ → ℝ be a locally Lipschitzean function satisfying (g_{0}) and let q : ℝ → ℝ be a locally summable function.

• Let [a, b] be an interval such that q ≥ 0 on [a, b] and q is continuous and of bounded variation on [a, b]. Suppose that the set {t ∈ [a, b] : q(t) > 0} is the union of a finite number of intervals. If, further, q(·) is monotone in a left neighborhood of any point s ∈]a, b] where q(s) = 0, then, for each t_{0} ∈ [a, b[ any solution of (1.1) is continuable to [t_{0}, b]. Respectively, if q(·) is monotone in a right neighborhood of any point s ∈ [a, b[ where q(s) = 0, then, for each t_{0} ∈]a, b] any solution of (1.1) is continuable to [a, t_{0}].

Proof. The result follows from [19], [14] (see also [50] for more details).

For completeness, we give a sketch of the proof in the simpler case when q(·) is absolutely continuous. This assumption can be relaxed to that of q(·) of bounded variation on [a, b] arguing like in [19].

Let [t_{0}, t_{1}] ⊂ [a, b] be such that q(t) > 0 on ]t_{0}, t_{1}[ and suppose that q(t_{1}) = 0 and q is monotone nonincreasing in a left neighborhood of t_{1} . By Peano’s theorem, we have that the solution x(·) is defined on [t_{0}, t_{2}] for some t_{2} ∈]t_{0}, t_{1}[ . Consider now V (t, x, ẋ) = \frac{1}{2}ẋ^{2} + \frac{1}{2}x^{2} + q(t)G(x) + 1, where G(x) = ∫_{0}^{x} g(ξ) dξ. Differentiation of v(t) := V (t, x(t), ẋ(t)) yields to

v̇(t) = xẋ + q̇(t)G(x) ≤ ζ(t)v(t),

for a.e. t ∈ [t_{2}, t_{1}] where x(·) is defined and where ζ is a suitably chosen function, summable on [t_{2}, t_{1}] and such that ζ(t)q(t) ≥ q̇(t) for a.e. t ∈ [t_{2}, t_{1}]. Hence, v(t) ≤ v(t_{2}) exp ∫_{t_{2}}^{t_{1}} ζ(s) ds := N holds for all t ∈ [t_{2}, t_{1}]. From this a priori bound and the fact that V (t, x, ẋ) → +∞ uniformly for t ∈ [a, b] as |x| + |ẋ| → +∞, we obtain the continuability of the solution up to t = t_{1} . By a finite number of steps, we can then obtain the continuability on [t_{0}, b].

• Let [b, c] be an interval such that q ≤ 0 on [b, c]. Then, property (2.10)

holds for each [t_{0}, t] ⊂ [b, c]. Proof. Let x(·) be a solution of (1.1). Multiplying the equation by x(s) gives ẍ(s)x(s) ≥ 0 for a.e. t ∈ [t_{0}, t], that is, for ℓ(t) = \frac{1}{2}x(t)^{2}, we have

\frac{d^{2}}{dt^{2}} ℓ(s) ≥ 0, for a.e. s ∈ [t_{0}, t].

Hence, it follows immediately that

|x(s)| ≤ max {|x(t_{0})|, |x(t)|} ≤ max {|x(t_{0})| + |ẋ(t_{0})|, |x(t)| + |ẋ(t)|} := L_{1} holds for all s ∈ [t_{0}, t]. Now, we easily conclude, using the equation.

In fact, |ẍ(s)| ≤ |q(s)| max_{|ξ|≤L_{1}}{|g(ξ)|} := μ(t), for a.e. s ∈ [t_{0}, t] and then |ẋ(s)| ≤ L_{1} + ∫_{t_{0}}^{t} μ(ξ) dξ := L_{2} . From the proof it is clear that L_{1} + L_{2} is a continuous function of ( |z(t_{0})|, |ż(t_{0})|).

As a consequence of the above results, we can give the following corollary.

• Let q(·) be a continuous and piece-wise monotone function defined in an interval I. Suppose that I = [t_{0}, t_{1}] ∪ [t_{1}, t_{2}] ∪ ⋯ ∪ [t_{k−1}, t_{k}] and either q ≥ 0 or q ≤ 0, on each of the sub-intervals I_{j} = [t_{j−1}, t_{j}]. Then, for any compact set K ⊂ ℝ^{2}, and for each t_{0}, t ∈ I, the map ϕ_{t_{0}}^{t} is proper on D(t_{0}, t) ∩ K.

Proof. It is sufficient to observe that the composition of maps fits with the property of properness on compact sets that we have defined.

Now we come back to the general setting considered at the beginning and prove the following fixed point theorem.

Theorem 1. Suppose that (D, ψ) is proper on compact sets and consider a conical shell W = [v̂0w]_{r_{1}}^{r_{2}} . Assume that any path σ contained in W and intersecting C_{r_{1}} and C_{r_{2}} , contains a sub-path σ′ ⊂ D such that

ψ(σ′) ⊂ v̂0w , and ψ(σ′) ∩ C_{r_{1}} ≠ ∅, ψ(σ′) ∩ C_{r_{2}} ≠ ∅.

Then, the map ψ has at least one fixed point z̃ ∈ D ∩ W.

Remark 2. In the above theorem we don’t assume ψ to be defined on the whole set W.

Proof. We consider the set

S = {z ∈ D ∩ W : ψ(z) ∈ W, |ψ(z)| = |z|}.

Since W is compact, by the properness of (D, ψ) on compact sets, we have that S is compact.

Consider now an arbitrary path σ contained in W and intersecting C_{r_{1}} and C_{r_{2}} . By the assumption, σ contains a path σ′, with σ′ ⊂ D such that ψ(σ′) intersects C_{r_{1}} and C_{r_{2}} . This implies that there are z_{1} , z_{2} ∈ σ′ such that |ψ(z_{1})| = r_{1} ≤ |z_{1}| and |ψ(z_{2})| = r_{2} ≥ |z_{2}|, and therefore, by the continuity of ψ and the connectedness of σ′, we can conclude that σ′ intersects S. Thus we have proved that the closed set S intersects any path σ contained in W and intersecting C_{r_{1}} and C_{r_{2}} .

At this point, using the elementary fact that W is homeomorphic to a rectangle, Lemma 4 guarantees the existence of a continuum C ⊂ S with C intersecting both ℓ_{v} and ℓ_{w} . By the assumptions, we also have that ψ is defined on C and ψ(C) ⊂ W with |ψ(z)| = |z| on C. Then, Corollary 2 implies the existence of a fixed point for ψ in C.

We consider now two conical shells

W_{0} := [{ \atop v_{0}0w_{0}}]_{r_{1}^{0}}^{r_{2}^{0}} , W_{1} := [{ \atop v_{1}0w_{1}}]_{r_{1}^{1}}^{r_{2}^{1}} ,

which we assume to be disjoint.

In the next result, we denote by ψ^{j} the j-th iterate of ψ, with the convention that ψ^{1} = ψ.

Theorem 2. Suppose that (D, ψ) is proper on compact sets. For W_{0} and W_{1} as above, let us suppose that the following property hold: (a_{0}) any path σ contained in W_{0} and intersecting C_{r_{1}^{0}} and C_{r_{2}^{0}} , contains

a sub-path σ′_{(0,0)} ⊂ D such that

ψ(σ′_{(0,0)}) ⊂ { \atop v_{0}0w_{0}} , ψ(σ′_{(0,0)}) ∩ C_{r_{1}^{0}} ≠ ∅, ψ(σ′_{(0,0)}) ∩ C_{r_{2}^{0}} ≠ ∅,

as well as a sub-path σ′_{(0,1)} ⊂ D such that

ψ(σ′_{(0,1)}) ⊂ { \atop v_{1}0w_{1}} , ψ(σ′_{(0,1)}) ∩ C_{r_{1}^{1}} ≠ ∅, ψ(σ′_{(0,1)}) ∩ C_{r_{2}^{1}} ≠ ∅;

any path σ contained in W_{1} and intersecting C_{r_{1}^{1}} and C_{r_{2}^{1}} , contains a sub-path σ′_{(1,0)} ⊂ D such that

ψ(σ′_{(1,0)}) ⊂ { \atop v_{0}0w_{0}} , ψ(σ′_{(1,0)}) ∩ C_{r_{1}^{0}} ≠ ∅, ψ(σ′_{(1,0)}) ∩ C_{r_{2}^{0}} ≠ ∅,

as well as a sub-path σ′_{(1,1)} ⊂ D such that

ψ(σ′_{(1,1)}) ⊂ { \atop v_{1}0w_{1}} , ψ(σ′_{(1,1)}) ∩ C_{r_{1}^{1}} ≠ ∅, ψ(σ′_{(1,1)}) ∩ C_{r_{2}^{1}} ≠ ∅.

Then, the following conclusions hold: (w_{1}) for each i = 0, 1, the map ψ has at least one fixed point z̃_{i} ∈ D ∩ W_{i}; (w_{2}) for each i = 0, 1 and for any finite sequence δ = (δ_{1}, …, δ_{k}), with k ≥ 1

and δ_{j} ∈ {0, 1} (for all j = 1, …, k), there are points z̃_{(i,δ)} ∈ D ∩ W_{i} which are fixed points of ψ^{k+1} and satisfy

ψ^{j}(z̃_{(i,δ)}) ∈ W_{δ_{j}} , ∀ j = 1, …, k;

(w_{3}) for each i = 0, 1 and for any sequence δ = (δ_{1}, δ_{2}, …), with δ_{j} ∈ {0, 1} (∀ j ∈ ℕ), there is a continuum Γ_{i}^{δ} ⊂ W_{i} , with Γ_{i}^{δ}∩ℓ_{v_{i}} ≠ ∅ and Γ_{i}^{δ}∩ℓ_{w_{i}} ≠ ∅ such that for each z ∈ Γ_{i}^{δ} , it follows that

ψ^{j}(z) ∈ W_{δ_{j}} , ∀ j ∈ ℕ;

(w_{4}) for any doubly-infinite sequence δ = (…, δ_{−2}, δ_{−1}, δ_{0}, δ_{1}, δ_{2}, …), with δ_{j} ∈ {0, 1} (∀ j ∈ ℕ), there is a double-sided sequence of points

w̃_{δ} = (…, w̃_{−2}, w̃_{−1}, w̃_{0}, w̃_{1}, w̃_{2}, …),

with w̃_{j} ∈ D ∩ W_{δ_{j}} and such that

ψ(w̃_{j}) = w̃_{j+1} , ∀ j ∈ ℤ.

Proof. Clearly, (w_{1}) is a straightforward consequence of Theorem 1, so that we can pass to the proof of the other properties.

Let us fix a finite sequence δ = (δ_{1}, …, δ_{k}) for some k ≥ 1 and with δ_{i} ∈ {0, 1} for all i = 1, …, k. Let γ : [0, 1] → W_{i} be a continuous map such that γ(0) ∈ C_{r_{1}^{i}} and γ(1) ∈ C_{r_{2}^{i}} . By assumption (a_{0}), we have that there is a closed interval I_{1} ⊂ [0, 1] such that ψ is defined in γ(I_{1}), ψ(γ(t)) ∈ W_{δ_{1}} , for all t ∈ I_{1} and

ψ(γ(t′_{1})) ∈ C_{r_{1}^{δ_{1}}} , ψ(γ(t′′_{1})) ∈ C_{r_{2}^{δ_{1}}} , for {t′_{1}, t′′_{1}} = ∂I_{1} .

We apply now the same argument to the continuous map γ_{1} : I_{1} → W_{δ_{1}} defined by γ_{1}(t) := ψ(γ(t)) and find a closed interval I_{2} ⊂ I_{1} , such that ψ is defined in γ_{1}(I_{2}), ψ(γ_{1}(t)) ∈ W_{δ_{2}} , for all t ∈ I_{2} and

ψ(γ_{1}(t′_{2})) ∈ C_{r_{1}^{δ_{2}}} , ψ(γ(t′′_{2})) ∈ C_{r_{2}^{δ_{2}}} , for {t′_{2}, t′′_{2}} = ∂I_{2} .

Continuing this way for a finite number of steps, we have that for each j = 1, …, k, k + 1 there is a closed interval I_{j} , with

[0, 1] ⊃ I_{1} ⊃ ⋯ ⊃ I_{j} ⊃ I_{j+1} ⊃ ⋯ ⊃ I_{k} ⊃ I_{k+1} ,

such that

ψ^{j}(γ(I_{k+1})) ⊂ W_{δ_{j}} for j = 1, …, k

and

ψ^{k+1}(γ(t′_{k+1})) ∈ C_{r_{1}^{i}} , ψ(γ(t′′_{k+1})) ∈ C_{r_{2}^{i}} , for {t′_{k+1}, t′′_{k+1}} = ∂I_{k+1} .

We can consider now the set

S_{i}^{k} = {z ∈ D ∩ W_{i} : ψ^{j}(z) ∈ W_{δ_{j}}, ∀ j = 1 …, k, |ψ^{k+1}(z)| = |z|}

for i = 0, 1. By the properness of (D, ψ) on the compact sets, we have that the set S_{i}^{k} is a compact subset of W_{i} and, arguing like in the proof of Theorem 1 (using Lemma 4), we have that S_{i}^{k} contains a continuum C_{i}^{k} which intersects both ℓ_{v_{i}} and ℓ_{w_{i}} . Then, Corollary 2 ensures the existence of a point z̃ ∈ W_{i} which is a fixed point for ψ^{k+1} and such that ψ^{j}(z̃) ∈ W_{δ_{j}} . Thus (w_{2}) is proved.

To prove (w_{3}), let us fix i ∈ {0, 1} and choose a sequence δ = (δ_{1}, δ_{2}, …) with δ_{j} ∈ {0, 1}. Repeating the first part of the proof of (w_{2}), we have that for any continuous map γ : [0, 1] → W_{i} such that γ(0) ∈ C_{r_{1}^{i}} and γ(1) ∈ C_{r_{2}^{i}} , and for each j ∈ ℕ, there is a compact interval I_{j} ⊂ [0, 1] such that

[0, 1] ⊃ I_{1} ⊃ I_{2} ⊃ ⋯ ⊃ I_{j} ⊃ I_{j+1} ⊃ …,

moreover, ψ^{j}(γ(t)) ⊂ W_{δ_{j}} for all t ∈ I_{j} and

ψ^{j}(γ(t′_{j})) ∈ C_{r_{1}^{δ_{j}}} , ψ(γ(t′′_{j})) ∈ C_{r_{2}^{δ_{j}}} , for {t′_{j}, t′′_{j}} = ∂I_{j} .

We can consider now the set

S_{i} = {z ∈ D ∩ W_{i} : ψ^{j}(z) ∈ W_{δ_{j}}, ∀ j ∈ ℕ}

for i = 0, 1 which is compact by the assumption of properness of (D, ψ). If we take s ∈ ∩_{j=1}^{∞}I_{j} , we have that γ(s) ∈ S_{i} and thus we have proved that σ ∩ S_{i} ≠ ∅ for each path σ ⊂ W_{i} with σ intersecting both C_{r_{1}^{i}} and C_{r_{2}^{i}} . Hence, Lemma 4 ensures the existence of a continuum C_{i} ⊂ S_{i} such that C_{i} ∩ ℓ_{v_{i}} ≠ ∅ and C_{i} ∩ ℓ_{w_{i}} ≠ ∅. From the definition of S_{i} we have that, for each z ∈ C_{i} , ψ^{j}(z) ∈ W_{δ_{j}} for all j ∈ ℕ. Therefore, (w_{3}) is proved.

To prove (w_{4}), let us fix a double sequence δ = (…, δ_{−1}, δ_{0}, δ_{1}, …) with δ_{j} ∈ {0, 1} for each j ∈ ℤ. By (w_{3}) we have that for each n = 1, 2, …, there is a continuum Γ_{n} ⊂ W_{δ_{−n}} such that ψ^{j}(Γ_{n}) ⊂ W_{δ_{j−n}} . We take a point y_{j,n} ∈ ψ^{j+n}(Γ_{n}) ⊂ W_{δ_{j}} , for each j ≥ −n, in order to form the infinite matrix

y_{−1,1} y_{0,1} y_{1,1} … y_{j,1} …
y_{−2,2} y_{−1,2} y_{0,2} y_{1,2} … y_{j,2} …
...
y_{−n,n} … y_{−2,2} y_{−1,n} y_{0,n} y_{1,n} … y_{j,n} …
...

where, for each n and j, we have that ψ(y_{j,n}) = y_{j+1,n} . At this point, a standard compactness argument allows to pass to the limit on each “column” along a common subsequence of indexes in order to find, for each j ∈ ℤ a point w̃_{j} ∈ W_{δ_{j}} and the continuity of ψ implies also that ψ(w̃_{j}) = w̃_{j+1} , ∀ j ∈ ℤ.

3. Tagged Maps

In order to apply the results of the preceding section to the Poincaré operator associated to a second order ordinary differential equation with a changing sign weight, it may be convenient to take advantage also of the information about the rotation number that we obtain from the analysis of the solutions in the time-intervals in which the weight is positive. With this respect, we are led to attach a “tag” to the points of the domain of ψ. Such a tag, in the applications, will be somehow linked to the rotation number associated to the solution departing from the given initial point. Accordingly, we define any continuous map η : D_{ψ} \ {0} → ℝ, as a tag associated to ψ. The pair (ψ, η) will be called a tagged map. The choice of excluding the origin from the domain of η is purely conventional and it is made here only in order to remember the fact that the number of turns around the origin is defined only for those solutions not passing through it.

Now, a “tagged” version of Theorem 1 reads as follows.

Theorem 3. Let D ⊂ D_{ψ} and let (ψ, η) be a tagged map with (D, ψ) proper on the compact sets. Let W = [v̂0w]_{r_{1}}^{r_{2}} be a conical shell. Suppose that there is a (non-empty) set of indexes N ⊂ ℕ such that, for each k ∈ N , there is a compact interval Λ^{k} ⊂ ℝ, with Λ^{k} ∩Λ^{j} = ∅ for k ≠ j and the following property holds: (e_{1}) for each k ∈ N and any path σ contained in W and intersecting C_{r_{1}} and

C_{r_{2}} , there is a sub-path σ^{k} ⊂ D such that

ψ(σ^{k}) ⊂ v̂0w , η(σ^{k}) ⊂ Λ^{k} ,

and

ψ(σ^{k}) ∩ C_{r_{1}} ≠ ∅, ψ(σ^{k}) ∩ C_{r_{2}} ≠ ∅.

Then, for each k ∈ N , the map ψ has at least one fixed point z̃_{k} ∈ D ∩ W with η(z̃_{k}) ∈ Λ^{k} . Hence, z̃_{k} ≠ z̃_{j} for k ≠ j and therefore, ψ has at least #(N) distinct fixed points in W ∩ D.

Proof. Let us fix k ∈ N and consider the set

S_{k} = {z ∈ D ∩ W : |ψ(z)| = |z| , η(z) ∈ Λ^{k}}.

Take a sequence (z_{n})_{n} ∈ S_{k} , with (z_{n})_{n} → z̄ as n → ∞. Since |ψ(z_{n})| ≤ r_{2} for all n, by the properness of (D, ψ), we have that z̄ ∈ D. Moreover, z̄ ≠ 0, as z̄ ∈ W. From the continuity of ψ and η and the fact that Λ^{k} is closed, it follows that z̄ ∈ S_{k} and thus we have proved that S_{k} is closed.

Consider now an arbitrary path σ contained in W and intersecting C_{r_{1}} and C_{r_{2}} . By (e_{1}) and repeating the same argument in the proof Theorem 1, we find that the sub-path σ^{k}, of σ intersects S_{k} . At this point, Lemma 4 guarantees the existence of a continuum C_{k} ⊂ S_{k} with C_{k} intersecting both ℓ_{v} and ℓ_{w} . By the assumptions, we also have that ψ is defined on C_{k} , ψ(C_{k}) ⊂ v̂0w and η(C_{k}) ⊂ Λ^{k} .

Hence, Corollary 2 implies the existence of a fixed point z̃_{k} for ψ in C_{k} .

It may be interesting to observe that Theorem 3 could be directly obtained from Theorem 1, just applying Theorem 1 on ψ restricted to the domain D ∩ η^{−1}(Λ_{k}).

We can also write the following generalized version of Theorem 2.

Theorem 4. Let D ⊂ D_{ψ} and let (ψ, η) be a tagged map with (D, ψ) proper on the compact sets. For W_{0} and W_{1} as in Theorem 2, let us suppose that the following properties hold:

• there is a (non-empty) set of indexes N_{(0,0)} ⊂ ℕ such that, for each k ∈ N_{(0,0)} there is a compact interval Λ^{k} = Λ_{(0,0)}^{k} ⊂ ℝ, with Λ^{k} ∩ Λ^{j} = ∅ for k ≠ j, and such that any path σ contained in W_{0} and intersecting C_{r_{1}^{0}} and C_{r_{2}^{0}} , contains a sub-path σ_{(0,0)}^{k} ⊂ D such that

ψ(σ_{(0,0)}^{k}) ⊂ { \atop v_{0}0w_{0}} , η(σ_{(0,0)}^{k}) ⊂ Λ^{k}

and

ψ(σ_{(0,0)}^{k}) ∩ C_{r_{1}^{0}} ≠ ∅, ψ(σ_{(0,0)}^{k}) ∩ C_{r_{2}^{0}} ≠ ∅;

• there is a (non-empty) set of indexes N_{(0,1)} ⊂ ℕ such that, for each k ∈ N_{(0,1)} there is a compact interval Λ^{k} = Λ_{(0,1)}^{k} ⊂ ℝ, with Λ^{k} ∩ Λ^{j} = ∅ for k ≠ j, and such that any path σ contained in W_{0} and intersecting C_{r_{1}^{0}} and C_{r_{2}^{0}} , contains a sub-path σ_{(0,1)}^{k} ⊂ D such that

ψ(σ_{(0,1)}^{k}) ⊂ { \atop v_{1}0w_{1}} , η(σ_{(0,1)}^{k}) ⊂ Λ^{k}

and

ψ(σ_{(0,1)}^{k}) ∩ C_{r_{1}^{1}} ≠ ∅, ψ(σ_{(0,1)}^{k}) ∩ C_{r_{2}^{1}} ≠ ∅;

• there is a (non-empty) set of indexes N_{(1,0)} ⊂ ℕ such that, for each k ∈ N_{(1,0)} there is a compact interval Λ^{k} = Λ_{(1,0)}^{k} ⊂ ℝ, with Λ^{k} ∩ Λ^{j} = ∅ for k ≠ j, and such that any path σ contained in W_{1} and intersecting C_{r_{1}^{1}} and C_{r_{2}^{1}} , contains a sub-path σ_{(1,0)}^{k} ⊂ D such that

ψ(σ_{(1,0)}^{k}) ⊂ { \atop v_{0}0w_{0}} , η(σ_{(1,0)}^{k}) ⊂ Λ^{k}

and

ψ(σ_{(1,0)}^{k}) ∩ C_{r_{1}^{0}} ≠ ∅, ψ(σ_{(1,0)}^{k}) ∩ C_{r_{2}^{0}} ≠ ∅;

• there is a (non-empty) set of indexes N_{(1,1)} ⊂ ℕ such that, for each k ∈ N_{(1,1)} there is a compact interval Λ^{k} = Λ_{(1,1)}^{k} ⊂ ℝ, with Λ^{k} ∩ Λ^{j} = ∅ for k ≠ j, and such that any path σ contained in W_{1} and intersecting C_{r_{1}^{1}} and C_{r_{2}^{1}} , contains a sub-path σ_{(1,1)}^{k} ⊂ D such that

ψ(σ_{(1,1)}^{k}) ⊂ { \atop v_{1}0w_{1}} , η(σ_{(1,1)}^{k}) ⊂ Λ^{k}

and

ψ(σ_{(1,1)}^{k}) ∩ C_{r_{1}^{1}} ≠ ∅, ψ(σ_{(1,1)}^{k}) ∩ C_{r_{2}^{1}} ≠ ∅.

Then, the following conclusions hold:

• for each i = 0, 1 and k_{i} ∈ N_{(i,i)} , the map ψ has at least one fixed point z̃_{k_{i}} ∈ D ∩ W_{i} with η(z̃_{k_{i}}) ∈ Λ_{(i,i)}^{k_{i}} . In particular, z̃_{k_{i}} ≠ z̃_{m_{j}} for k_{i} ≠ m_{j} , where i, j ∈ {0, 1} and k_{i} ∈ N_{(i,i)} , m_{j} ∈ N_{(j,j)} . In particular, for i = 0, 1, the map ψ has at least #(N_{(i,i)}) distinct fixed points in W_{i};

• for each i = 0, 1, for any finite sequence δ = (δ_{1}, …, δ_{m}), with m ≥ 1 and δ_{j} ∈ {0, 1} (for each j = 1, …, m) and for any finite sequence κ = (k_{1}, …, k_{m}, k_{m+1}), with

k_{1} ∈ N_{(i,δ_{1})} , k_{2} ∈ N_{(δ_{1},δ_{2})} , …, k_{m} ∈ N_{(δ_{m−1},δ_{m})} , k_{m+1} ∈ N_{(δ_{m},i)} ,

there are points z̃_{(i,δ,κ)} ∈ D ∩ W_{i} which are fixed points of ψ^{m+1} and satisfy

ψ^{j}(z̃_{(i,δ,κ)}) ∈ W_{δ_{j}} , η(ψ^{j}(z̃_{(i,δ,κ)})) ∈ Λ_{(δ_{j−1},δ_{j})}^{k_{j}} , ∀ j = 1, …, m + 1

(we use here the convention that δ_{0} = δ_{m+1} = i);

• for each i = 0, 1, for any sequence δ = (δ_{1}, δ_{2}, …), with δ_{j} ∈ {0, 1} (∀ j ∈ ℕ) and for any sequence κ = (k_{1}, k_{2}, …), with

k_{1} ∈ N_{(i,δ_{1})} , k_{2} ∈ N_{(δ_{1},δ_{2})} , …, k_{m} ∈ N_{(δ_{m−1},δ_{m})} , …,

there is a continuum Γ_{i}^{δ,κ} ⊂ W_{i} , with Γ_{i}^{δ,κ} ∩ ℓ_{v_{i}} ≠ ∅ and Γ_{i}^{δ,κ} ∩ ℓ_{w_{i}} ≠ ∅ such that for each z ∈ Γ_{i}^{δ,κ} , it follows that

ψ^{j}(z) ∈ W_{δ_{j}} , η(ψ^{j}(z)) ∈ Λ_{(δ_{j−1},δ_{j})}^{k_{j}} , ∀ j ∈ ℕ;

• for any doubly-infinite sequence δ = (…, δ_{−2}, δ_{−1}, δ_{0}, δ_{1}, δ_{2}, …), with δ_{j} ∈ {0, 1} (∀ j ∈ ℤ) and for any κ = (…, k_{−2}, k_{−1}, k_{0}, k_{1}, k_{2}, …), with

k_{m} ∈ N_{(δ_{m−1},δ_{m})} , (∀ m ∈ ℤ),

there is a double-sided sequence of points

w̃_{(δ,κ)} = (…, w̃_{−2}, w̃_{−1}, w̃_{0}, w̃_{1}, w̃_{2}, …),

with w̃_{j} ∈ D ∩ W_{δ_{j}} such that

ψ(w̃_{j}) = w̃_{j+1} , η(ψ^{j}(w̃_{j})) ∈ Λ_{(δ_{j−1},δ_{j})}^{k_{j}} , ∀ j ∈ ℤ.

The proof is omitted, as it is just a repetition of that of Theorem 2. An application of Theorem 4 to the nonlinear Hill’s equation is the following,

where we give a new proof to some of the results contained in [16], [51]. We warn that we don’t claim that all the solutions of equation (1.1) are oscillatory under the mild sign conditions on q(t) considered here. What we prove is that there exist solutions with such a complicated oscillatory behavior like those depicted in the next Theorem 5.

Theorem 5. Let g : ℝ → ℝ be a locally Lipschitz continuous function satisfying (g_{0}), (g_{mon}) and (g_{∞}). Let q : ℝ → ℝ be a continuous, T -periodic and piece-wise monotone function. Suppose that there is a finite sequence of points t_{0} < t_{1} < ⋯ < t_{2k+1} < t_{2k+2} = t_{0} + T such that for j = 0, …, k,

q ≥ 0 and q ≢ 0 on [t_{2j}, t_{2j+1}], q ≤ 0 and q ≢ 0 on [t_{2j+1}, t_{2j+2}].

Then, there are positive integers m_{0}^{∗}, m_{1}^{∗} …, m_{k}^{∗} , such that the following conclusion hold:

• For each n ≥ 1 and for each finite sequence of integers

(m_{0}^{1}, …, m_{k}^{1}, m_{0}^{2}, …, m_{k}^{2}, …, m_{0}^{n}, …, m_{k}^{n})

with m_{j}^{s} ≥ m_{j}^{∗} and for each finite sequence

(δ_{0}^{1}, …, δ_{k}^{1}, δ_{0}^{2}, …, δ_{k}^{2}, …, δ_{0}^{n}, …, δ_{k}^{n}),

with δ_{j}^{s} ∈ {0, 1}, (for each s = 1, … n and j = 0, …, k) and such that

∑_{{s=1,...,n \atop j=0,...k}} m_{j}^{s} + δ_{j}^{s} = even,

there are at least two nontrivial nT -periodic solutions x(·) of equation (1.1) such that x has exactly m_{j}^{s} zeros in the interval ]t_{2j}+(s−1)T, t_{2j+1}+ (s − 1)T [ and exactly δ_{j}^{s} zeros and 1 − δ_{j}^{s} changes of sign of ẋ in the interval ]t_{2j+1} + (s − 1)T, t_{2j+2} + (s − 1)T [ . These two solutions are such that x(t_{0}) ≥ 0, ẋ(t_{0}) ≥ 0 and x(t_{0}) ≤ 0, ẋ(t_{0}) ≤ 0, respectively.

• For each N > max{m_{i}^{∗} : i = 0, …, k} and for each sequence of integers

(m_{0}, m_{1}, …, m_{j}, …)

with m_{j} ∈ [m_{i}^{∗}, N] for j ≡ i (mod k) and for each sequence

(δ_{0}, δ_{1}, …, δ_{j}, …)

with δ_{j} ∈ {0, 1}, there are two closed connected sets C^{+} ⊂ Ā_{1} \ {0} and C^{−} ⊂ Ā_{3} \ {0}, with C^{+} and C^{−} intersecting both the x and the y axes such that for each (a, b) ∈ C^{+} ∪ C^{−}, there is at least one solution x(·) of equation (1.1) with x(t_{0}) = a, ẋ(t_{0}) = b, such that, if j = sk + i, then x has exactly m_{j} zeros in the interval ]t_{2i} + (s − 1)T, t_{2i+1} + (s − 1)T [ and exactly δ_{j} zeros and 1 − δ_{j} changes of sign of ẋ in the interval ]t_{2i+1} + (s − 1)T, t_{2i+2} + (s − 1)T [ ;

• For each N > max{m_{i}^{∗} : i = 0, …, k} and for each double-sided sequence of integers

(…, m_{−1}, m_{0}, m_{1}, …, m_{j}, …)

with m_{j} ∈ [m_{i}^{∗}, N] for j ≡ i (mod k) and for each sequence

(…, δ_{−1}, δ_{0}, δ_{1}, …, δ_{j}, …)

with δ_{j} ∈ {0, 1}, there is at least one solution x(·) of equation (1.1) such that x has exactly m_{j} zeros in the interval ]t_{2i} + (s − 1)T, t_{2i+1} + (s − 1)T [ and exactly δ_{j} zeros and 1 − δ_{j} changes of sign of ẋ in the interval ]t_{2i+1} + (s − 1)T, t_{2i+2} + (s − 1)T [ , when j = sk + i.

Proof. We give a proof in the simpler case in which k = 0. We also assume t_{0} = 0 (there is no loss of generality in this latter assumption), so that we can split the interval [0, T] as the union of two intervals [0, τ] and [τ, T] such that

q ≥ 0 and q ≢ 0 on [0, τ], q ≤ 0 and q ≢ 0 on [τ, T].

Now we take as ψ the Poincaré map ϕ_{0}^{T} associated to the planar system (1.4). We note that, due to the superlinear growth of g at infinity, we have that some solutions of (1.1) blow up in the intervals where q ≤ 0 (cf. [11] ), so that D_{ψ} is a proper open subset of ℝ^{2} which may possess a complicated structure [15]. In any case, thanks to Example 3, we have that ψ is proper on the compact sets. Then we define

W_{0} = Ā_{1} ∩ A[r_{1}, r_{2}], W_{1} = Ā_{3} ∩ A[r_{1}, r_{2}],

where r_{1} and r_{2} are chosen as follows: from Proposition 1, we first fix r_{1} > R^{∗} . This determines a number n_{r_{1}}^{∗} and hence we can define m_{0}^{∗} = n_{r_{1}}^{∗} + 1. At this point, we can choose any number N > m_{0}^{∗} and take r_{2} > max{R_{j} : j ∈ [m_{0}^{∗}, N]} for the R_{j}’s as in Proposition 1. After these definitions, and using the arc expansion property ensured by Proposition 1, we can easily enter into the frame of Theorem 4 and arrive to the conclusion.

For the case k ≥ 1, one should use a more general version of Theorem 4, using a decomposition of ψ of the form

ψ = ϕ_{t_{2k}}^{t_{2k+2}} ∘ ⋯ ∘ ϕ_{t_{0}}^{t_{2}}

and repeating a similar argument like that of the proof of Theorem 4, but taking into account the the whole information about the oscillation of the solutions in the single intervals [t_{2j}, t_{2j+2}]. For a different proof of these results, see [51] for the periodic problem and [16] for the “chaotic solutions”.

Remark 3. The upper bound N for the numbers of zeros in the intervals of positivity of the function q(t) is assumed in Theorem 5 only for the purpose to have a direct application of Theorem 4. Using more carefully the properties given in Lemma 1 and Lemma 2 (instead of putting a bound for the oscillations like in Proposition 1) one could obtain a more general result in which the request of upper bound N can be avoided. Results in this direction have been obtained by Terracini and Verzini in [68] and by Capietto, Dambrosio and Papini in [16].

We also point out that the same result of Theorem 5 can be obtained for the damped nonlinear Hill’s equation

ẍ + cẋ + q(t)g(x) = 0,  (3.11)

with c ∈ ℝ and q(t) and g(x) satisfying the same conditions of Theorem 5 (see also [16] and [51]). In fact, Lemma 1 and Lemma 2 and hence Proposition 1 are true for equation (3.11), too.

4. Appendix 1: Looking for an Underlying Geometric Structure

Here, we show how the results of the previous sections can be generalized to the case of arbitrary two-dimensional cells. In doing this, we try to make more explicit the geometric structure which allows to obtain results like Theorem 1 and Theorem 2.

Results about chaotic dynamical systems are nowadays available in various excellent books and articles. Probably, the most famous model for the chaotic dynamics is given by the celebrated Smale Horseshoe [58], [59]. General results about the horseshoe, as well applications to various examples of dynamical systems can be found, for instance, in [27], [47], [70], [55]. In some recent years, various authors have developed topological approaches in order to prove the existence of chaotic dynamics either for some specific examples of differential equations ([30], [32], [67]) or for dynamical systems satisfying suitable assumptions in order to apply some kind of index theory [17], [18], [45], [60], [61], [62], [63], [64], [66], [71], [73], [74], [75], [80]. Other approaches based on variational methods have been applied to differential equations as well (see, e.g., [2], [3],

[56] for results in that direction).

The aim of this section is just to show how the arguments described in Theorem 1 and Theorem 2 can be easily extended to the more general setting of two-dimensional cells (the case of one-dimensional cells is much more simple and is only sketched in Remark 6). In some recent articles, [34], [35], a general theory for expanding maps yielding to chaotic dynamics has been developed. The settings of [34] and [35] appear to be more general than ours. On the other hand, in our case, as a by-product of the arc expansion property, it is possible to obtain a fixed point theorem which, in turns, allows to prove, in an elementary manner, the existence of periodic orbits (and periodic solutions for the nonlinear Hill’s equation) too.

First of all, we introduce some definitions.

Definition 1. We define an oriented cell R̃ as a pair (R, R^{−}), where R ⊂ ℝ^{2} is a two-dimensional cell (i.e., a subset of the plane homeomorphic to the unit square Q = [−1, 1]^{2} ) and R^{−} ⊂ ∂R is the union of two disjoint compact arcs, too. The two components of R^{−} will be denoted by R_{l}^{−} and R_{r}^{−} and conventionally called the left and the right sides of R. The order in which we make the choice of naming R_{l}^{−} and R_{r}^{−} is not important in what follows.

Since ∂R is a simple closed curve, we have that R^{+} := \overline{∂R \ R^{−}} is the union of two disjoint compact arcs, too. The two components of R^{+} will be denoted by R_{b}^{+} and R_{t}^{+} (the base and the top of R); again, the order is not important.

To consider the unit square as an oriented cell, a natural choice will be that of Q_{l}^{−} = {−1} × [−1, 1], Q_{r}^{−} = {1} × [−1, 1], Q_{b}^{+} = [−1, 1] × {−1} and Q_{t}^{+} = [−1, 1] × {+1}.

As a consequence of the Shoenflies theorem [31] or [46], it is not difficult to see that, given an oriented cell R̃ = (R, R^{−}), there is a homeomorphism h of the plane onto itself such that h(Q) = R and h(Q_{l}^{−}) = R_{l}^{−} , h(Q_{r}^{−}) = R_{r}^{−} , h(Q_{b}^{+} ∪ Q_{t}^{+}) = R^{+}.

Let ψ : ℝ^{2} ⊃ D_{ψ} → ℝ^{2} be a (continuous) map that we consider to be defined on a set D_{ψ} . Let D be a subset of D_{ψ} and let also K ⊂ ℝ^{2} be compact set.

Definition 2. We say that (D, ψ) is proper on K if ψ is continuous on D and, for each sequence z_{n} ∈ D ∩ K such that ψ(z_{n}) is bounded, there is a subsequence z_{j_{n}} converging to a point of D.

This, clearly, is equivalent to the requirement that if there is any w ∈ (∂D \

D) ∩ K, then lim_{{z→w \atop z∈D∩K}} |ψ(z)| = ∞.

Remark 4. a) In the above definition D is not necessarily the whole domain of ψ. b) If (D, ψ) is proper on K, then (D, ψ) is proper on any closed subset of K, that is (D ∩ C, ψ) is proper on K, for each closed set C ⊂ ℝ^{2}. In particular, if D ∩ K is closed (for instance, if K ⊂ D ), then, any continuous map ψ is such that (D, ψ) is proper on K. c) If (D_{1}, ψ_{1}) is proper on K_{1} and (D_{2}, ψ_{2}) is proper on K_{2} then, (D_{1,2}, ψ_{2} ∘ ψ_{1}) is proper on K_{1} for

D_{1,2} := D_{1} ∩ ψ_{1}^{−1}(D_{2} ∩ K_{2}).

The set D_{1,2} will be considered as the natural domain for the composition of two proper maps. By induction, for the composition of ψ_{1}, …, ψ_{n} , with (D_{i}, ψ_{i}) proper on K_{i} for i = 1, …, n, the corresponding “natural domain” will be the set

D_{1,...,n} = {z ∈ D_{1} : ψ_{1}(z) ∈ D_{2,...,n} ∩ K_{2}}.

Definition 3. Let Ã = (A, A^{−}) and B̃ = (B, B^{−}) be two oriented cells and let ψ : ℝ^{2} ⊃ D_{ψ} → ℝ^{2} be a map. Consider also a set D ⊂ D_{ψ} . We say that (D, ψ) stretches Ã to B̃ and write

(D, ψ) : Ã ◁⇝ B̃

if • (D, ψ) is proper on A,

• for any path Γ ⊂ A such that Γ ∩ A_{l}^{−} ≠ ∅ and Γ ∩ A_{r}^{−} ≠ ∅, there is a path Γ′ ⊂ Γ ∩ D such that

ψ(Γ′) ⊂ B, ψ(Γ′) ∩ B_{l}^{−} ≠ ∅, ψ(Γ′) ∩ B_{r}^{−} ≠ ∅.

Lemma 5. Suppose that (D_{1}, ψ_{1}) : Ã ◁⇝ B̃ and (D_{2}, ψ_{2}) : B̃ ◁⇝ C̃. Then (D_{1,2}, ψ_{2} ∘ ψ_{1}) : Ã ◁⇝ C̃.

Proof. The result easily follows from the definition.

Theorem 6. Suppose that there is an oriented cell R̃ = (R, R^{−}), a map ψ : ℝ^{2} ⊃ D_{ψ} → ℝ^{2} and a set D ⊂ ℝ^{2}, such that (D, ψ) : R̃ ◁⇝ R̃. Then, there is at least one z^{∗} ∈ R ∩ D such that ψ(z^{∗}) = z^{∗}.

Proof. Let h : ℝ^{2} → ℝ^{2} be a homeomorphism such that h(Q) = R, h(Q_{l}^{−}) = R_{l}^{−} and h(Q_{r}^{−}) = R_{r}^{−} and consider the map

ϕ(x) := h^{−1}(ψ(h(x)), x ∈ E := h^{−1}(D).

By the assumptions on (D, ψ) and observing also that h and h^{−1} take bounded sets to bounded sets, is is straightforward to check that (E, ϕ) : (Q, Q^{−}) ◁⇝ (Q, Q^{−}).

For ϕ = (ϕ_{1}, ϕ_{2}) and x = (x_{1}, x_{2}), let us consider the set

S = {x ∈ E ∩ Q : ϕ_{1}(x) = x_{1} , |ϕ_{2}(x)| ≤ 1}.

The properness of (E, ϕ) on Q implies that S is closed.

Let γ : [0, 1] → Q be a continuous map such that γ(0) ∈ Q_{l}^{−} and γ(1) ∈ Q_{r}^{−} . By the “stretching” hypothesis, there are t_{1} , t_{2} ∈ [0, 1] such that ϕ(γ(t_{1})) ∈ Q_{l}^{−} and ϕ(γ(t_{2})) ∈ Q_{r}^{−} . Moreover, if we denote by I ⊂ [0, 1] the interval determined by t_{1} and t_{2} , we can also assume that γ(I) ⊂ E and ϕ(γ(I)) ⊂ Q.

Hence, ϕ_{1}(γ(t_{1})) ≤ γ_{1}(t_{1}) and ϕ_{1}(γ(t_{2})) ≥ γ_{1}(t_{2}) and therefore there is some t_{0} ∈ I such that ϕ_{1}(γ(t_{0})) = γ_{1}(t_{0}). As ϕ(γ(t_{0})) ∈ Q, we also have that |ϕ_{2}(γ(t_{0}))| ≤ 1.

In this manner, we have proved that any path contained in Q and joining Q_{l}^{−} with Q_{r}^{−} intersects the set S. By Lemma 4, it follows that S contains a continuum C joining Q_{b}^{+} with Q_{t}^{+} . From the definition of S it also follows that

ϕ_{2}(C) ⊂ [−1, 1]

and hence x_{2}−ϕ_{2}(x_{1}, x_{2}) ≤ 0 for (x_{1}, x_{2}) ∈ C ∩Q_{b}^{+} , as well as x_{2}−ϕ_{2}(x_{1}, x_{2}) ≥ 0 for (x_{1}, x_{2}) ∈ C ∩ Q_{t}^{+} . Then, the Bolzano theorem guarantees the existence of a w^{∗} = (w_{1}^{∗}, w_{2}^{∗}) ∈ C such that ϕ_{2}(w^{∗}) = w_{2}^{∗} . From the definition of S we can conclude that

ϕ(w^{∗}) = w^{∗} .

Clearly, z^{∗} := h(w^{∗}) ∈ D ∩R is a fixed point of ψ. This concludes the proof.

Theorem 7. Suppose that there are two oriented cells R̃_{0} = (R_{0}, R_{0}^{−}), and R̃_{1} = (R_{1}, R_{1}^{−}), with R_{0} ∩ R_{1} = ∅. Let ψ : ℝ^{2} ⊃ D_{ψ} → ℝ^{2} be a continuous map and assume that there is a set D ⊂ ℝ^{2}, such that

(D, ψ) : R̃_{i} ◁⇝ R̃_{j}

for each i, j ∈ {0, 1}. Then, the following conclusions hold: (r_{1}) for each i = 0, 1, the map ψ has at least one fixed point z̃_{i} ∈ D ∩ R_{i}; (r_{2}) for each i = 0, 1 and for any finite sequence δ = (δ_{1}, …, δ_{k}), with k ≥ 1

and δ_{j} ∈ {0, 1} (for all j = 1, …, k), there are points z̃_{(i,δ)} ∈ D ∩ R_{i} which are fixed points of ψ^{k+1} and satisfy

ψ^{j}(z̃_{(i,δ)}) ∈ R_{δ_{j}} , ∀ j = 1, …, k;

(r_{3}) for each i = 0, 1 and for any sequence δ = (δ_{1}, δ_{2}, …), with δ_{j} ∈ {0, 1} (∀ j ∈ ℕ), there is a continuum Γ_{i}^{δ} ⊂ R_{i} , with Γ_{i}^{δ} intersecting both the

components of R_{i}^{+} , such that for each z ∈ Γ_{i}^{δ} , it follows that
ψ^{j}(z) ∈ R_{δ_{j}} , ∀ j ∈ ℕ;

(r_{4}) for any doubly-infinite sequence δ = (…, δ_{−2}, δ_{−1}, δ_{0}, δ_{1}, δ_{2}, …), with δ_{j} ∈ {0, 1} (∀ j ∈ ℕ), there is a double-sided sequence of points

z̃_{δ} = (…, z̃_{−2}, z̃_{−1}, z̃_{0}, z̃_{1}, z̃_{2}, …),

with z̃_{j} ∈ D ∩ R_{δ_{j}} such that

ψ(z̃_{j}) = z̃_{j+1} , ∀ j ∈ ℤ.

Proof. The proof of (r_{1}) and (r_{2}) is a direct consequence of Theorem 6. For the proof of (r_{3}) and (r_{4}) we have only to repeat that of Theorem 2 with R_{i} playing the same role of W_{i} .

Remark 5. We notice that the hypothesis that R_{0} ∩ R_{1} = ∅ is assumed here only in order to avoid the possibility of “trivial” cases for the sequence (z̃)_{j} as well as to have different fixed points for the map ψ and its iterates. The condition of non-intersection for the cells can be avoided if we have some control on the sequence of points which enter in R_{0} ∩ R_{1} after some step. Sometimes, this will be not a difficult task, for instance, when R_{0} ∩R_{1} is a sufficiently small set (e.g., a singleton).

The same remark clearly applies to Theorem 2 and Theorem 4 with respect to W_{0} and W_{1} .

Remark 6. The argument we have used in the proof of Theorem 2 can be applied in order to show the well known fact that an expanding self-map of the unit interval produces a very complicated dynamics through its iterates [42].

Let f : [0, 1] → [0, +∞) be a continuous function such that f(0) = f(1) = 0 and suppose that f(c) ≥ 1 for some c ∈]0, 1[ . Let us set W_{1} = [0, a] and W_{2} = [b, 1], where a = min{t ∈ [0, 1] : f(t) = 1} and b = max{t ∈ [0, 1] : f(t) = 1}. Then it is easy to see that an expansion arc property holds and all the consequences of Theorem 7 hold in this one-dimensional case. (see [34], [38] for the same situation). To be more precise, if a < b (which is surely true when f(c) > 1), we enter in the setting of disjoint cells. Otherwise, if f(c) = 1 and a = c = b, we could apply Theorem 7 with Remark 5 and produce any sequence of symbols “left-right” (where “left” means “in the interval [0, a]” and “right” means “in the interval [b, 1]”) as we know that the only sequences which have a term z_{i} = c, must be identically equal to zero since then.

Acknowledgements

This work has been performed under the auspices of GNAMPA, Gruppo Nazio-nale per l’Analisi Matematica, la Probabilità e le loro Applicazioni and supported by MIUR-Cofin2001, Project “Sistemi dinamici non lineari ed applicazioni fisiche”.

A summary of the main results of this paper and of [51], [52] was presented during some lectures at the “Third Turin Fortnight on Nonlinear Analysis”, September 24-28, 2001. The authors thank Prof. Anna Capietto and the organizers for the pleasant hospitality.

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(Received 12.01.2002)

Authors’ addresses:

Duccio Papini Dipartimento dell’Ingegneria dell’Informazione Universita’ degli Studi di Siena via Roma 56, 53100 Siena Italy E-mail: ducciopapini@tiscali.it

Fabio Zanolin Dipartimento di Matematica e Informatica Università degli Studi di Udine via delle Scienze 206, 33100 Udine Italy E-mail: zanolin@dimi.uniud.it