We present a topological result, named

In 1883-1884, Poincaré [

Let

A heuristic proof of this result,

In order to prove the existence of such continua, one can observe that the set

Important theorems, where some forms of these crossing properties are considered, appear in dimension theory with the results of Hurewicz and Wallman [

In some applications, typically the set

When the operators whose fixed points correspond to the element of

In the present paper we move from the framework of generalized rectangles already discussed in previous articles [

We conclude this introduction with some preliminary results which will be used in what follows. Throughout the paper, by a

Slightly modifying an analogous definition of Berarducci et al. [

Let

In order to simplify the statements of the next results, we write

Let

The proof is a standard application of Zorn’s lemma. Let

In [

Let

In general, a set

A topological space

For a generalized rectangle

An

Given an oriented rectangle,

The next (classical) result guarantees the fact that continua connecting opposite sides of an oriented rectangle must cross each other. It can be proved as a consequence of the Jordan curve theorem, and the proof is omitted (see [

Let

The connectedness of

We present now some results about sets separating the opposite faces of an oriented rectangle. We have shown their role in the proof of the existence of fixed points and periodic points for continuous maps in [

The following version of Alexander’s lemma will be used in our next results. The proof requires only an elementary modification of the standard one [

Let

The original version of Alexander’s lemma is a variant of Lemma

The next result is perhaps one of the most classical and useful consequences of the previous lemma (see [

Let

By Lemma

The cutting property for

Let

If

Lemma

The combination of Lemma

In this section, we are moving our attention from generalized rectangles to planar annuli, trying to develop analogous results of Section

A closed planar annulus (of inner radius

A topological space

Let

In the sequel, we use the standard covering projection of

Our aim now is to reconsider the results obtained for topological rectangles in Section

Let

This result is an elementary variant of the version of Alexander’s lemma in [

We say that a set

The next result is a corollary of the Borsuk separation theorem [

Let

Up to a homeomorphism defining the annulus

Suppose that

Suppose now that

Passing to the covering space

A variant of the above proof has been suggested by the referee, and we wish to report it here, since it may be more interesting from a geometrical point of view. The argument is as follows.

Having obtained (as above)

A version of Lemma

Let

The proof follows the same argument of the one of Lemma

The result in Lemma

Let

Without loss of generality, we can assume that

We observe that the existence of two solutions is not guaranteed if the minimality of the set

In the case of topological rectangles, there is a complete symmetry between the fact that a set cuts the paths between a given pair of opposite sides or it cuts the paths connecting a complementary pair of opposite sides. Thus, once we have achieved a result like Lemma

Let

Without loss of generality (up to a homeomorphism), we suppose that

Suppose, by contradiction, that

Our last result in this section can be seen as a continuous version of the no-tie theorem for the Hex game on the annulus. See [

Let

Assume, by contradiction, that

Let

Let

Let

Our goal now is to develop a result analog of Lemma

In order to fix the ideas and before moving to the class of generalized and oriented rectangles, we consider for a moment a planar rectangle

Let

Observe that in Example

Let

By Lemma

In order to have the connectedness of the set

Let

Assume that there exists a compact set

As a first step, we apply Lemmas

In order to obtain an invariant set, we define a sequence of continua by naming

Since we know that

Observe that the same result holds true (with an obvious modification in the proof) if we replace condition (

By the assumption

Let

Notice that in Example

A trivial case of a continuous map which is monotone nondecreasing along the horizontal lines is the identity. In such a case, Corollary

A useful property of the continua connecting two compact disjoint sets is the minimality, often named as irreducibility [

Then, as a next step, we look for the existence of irreducible invariant continua in this new setting. Such a result, indeed, is of general nature and independent of the fact that in the present paper we consider only the case of planar continua.

Let

Let

Given any chain

Assume, by contradiction, that

We claim that there exists an index

To conclude, we observe that

We conclude this paper by presenting a possible application of the above-described classical separation results to an example which is inspired by a model arising from the theory of fluid mixing previously considered by Kennedy and Yorke in [

In [

We are going to prove a result which, although not so sharp like that in [

Let

Plainly speaking,

There exist

According to the above hypotheses, the map

Under the previous assumptions on

Let

Let us denote by

Suppose that

Let us consider now the intersection of

We are now in a position to apply Lemma

We end the proof with a final comment about the “stability” of the existence result with respect to small perturbations. By this assertion, we mean that the existence of at least four fixed points is preserved if, instead

Following the proof, it is clear that if

For a different application of these results to the existence of fixed points and periodic points to planar maps arising from ordinary differential equations, we refer also to [

The authors are sincerely grateful and indebted to the referee of their paper who checked very carefully their work and gave them extremely useful suggestions on how to correct some bugs in the original proofs and enhance the presentation.