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In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section.

The investigation of twist maps defined on annular domains can be considered as a relevant topic in the study of dynamical systems in two-dimensional manifolds. Twist maps naturally appear in a broad number of situations, and thus they have been widely considered both from the theoretical point of view and for their significance in various applications which range from celestial mechanics to fluid dynamics.

One of the most classical examples of a fixed point theorem concerning twist maps on an annulus is the celebrated Poincaré-Birkhoff “twist theorem,” also known as the “Poincaré last geometric theorem.” It asserts the existence of at least two fixed points for an area-preserving homeomorphism

Our study for the present paper is motivated by a recent approach considered by Ding in [

We end this introduction with some definition and basic results which will be useful in the subsequent sections.

A topological space

Let

Let

In order to simplify the presentation, we write

Let

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

Let

For our applications to bend-twist maps we also need a more refined version of the above result which reads as follows.

Let

First of all we claim that there exists a closed set

In the sequel we denote by

In this section, we reconsider, in a purely topological framework, the concept of

Let

Let

In some applications (for instance, to some planar maps associated to ordinary differential equations), the number

Conversely, one can easily check that any fixed point

Looking for a solution of system (

We say that

If we prefer to express the twist condition directly on

The celebrated Poincaré-Birkhoff “twist” theorem, in its original formulation, considers the case of a standard annulus

Assume (

Boundary invariance:

Usually, the hypothesis that the homeomorphism

In order to introduce the concept of bend-twist maps we recall a (wrong) attempt of proving Theorem

We put in Italic the original words by Wilson. The notation is the original one and to make it compatible with that of the present paper we have to notice that the lifting of

The gap in this argument is not only on the fact that the set of points of the annulus where

A sketch of the problem in Wilson’s argument. We depict a sector of an annular domain in which there is a portion of a non-star-shaped curve

Of course, if we were able to prove that the radial displacement function

In [

In [

Let

We notice that, in Ding’s theorem, the assumptions that

Let

Our aim now is to reformulate the above results in a general topological setting in order to obtain a version of Theorem

Let

Let

Our claim is an immediate consequence of Lemma

Our result corresponds to [

Let

As a consequence of this definition, the following theorem, which is a version of Theorem

Let

The proof is an obvious consequence of the connectedness of

In general, and in contrast with Theorem

Let

A description of the geometry in Example

Perhaps the set

On the other hand, we are able to recover the existence of two fixed points, as in Corollary

Let

Our argument is reminiscent of a similar one in the proof of the bifurcation result in [

First of all, by the covering projection

With the same argument of the proof of Theorem

Let

Observe that Theorem

On the other hand, we remark that Theorem

Let

Up to now we have presented all our results in terms of liftings of planar maps given by the standard covering projection

It appears that the presence of bend-twist maps associated to planar differential equations is ubiquitous. This does not mean that proving their existence in concrete equations would be a simple task. It is a common belief that periodic solutions obtained for planar Hamiltonian systems via the Poincaré-Birkhoff fixed point theorem are not preserved by arbitrarily small perturbations which destroy the Hamiltonian structure of the equations. A typical example occurs when we add a small friction to a conservative system of the form

For (

In this setting we propose an application of the Poincaré-Birkhoff twist theorem and the bend-twist maps theorem to equations which are small perturbations of (

Just to start, we suppose that there exist

Consider the level line

The continuity of the map

As a next step, we consider a perturbation of (

Assume (

Theorem

We give a sketch of the proof of Theorem

If we denote by

Finally, using the fact that

A natural question that now can arise is whether such (nontrivial)

Assume (

In comparison to this result obtained via the Poincaré-Birkhoff fixed point theorem, using Corollary

Assume (

Without loss of generality (via a time shift leading to an equivalent equation), we can suppose that

To begin with the proof, we consider the Poincaré map

In order to check the validity of the condition on the map

We split now the map

Let us consider now a solution

Let

Finally, recalling (

An analysis of the proof and of inequality (

Clearly, the same result holds also for (

We have achieved our result for a very special form of the weight function. A natural question concerns which kind of shape for a

A theorem about the existence of four solutions in this setting appears rather unusual (with respect to Corollary

The authors are grateful to Professor Tongren Ding for providing them with a copy of his interesting book [