A Topological Approach to Bend‐Twist Maps with Applications

Anna Pascoletti,Fabio Zanolin

doi:10.1155/2011/612041

Abstract

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.

Highlights

We obtain some results about the existence and multiplicity of fixed points which are related to the classical PoincareBirkhoff twist theorem for area-preserving maps of the annulus; 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
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

Summary

A Topological Approach to Bend-Twist Maps with Applications

We obtain some results about the existence and multiplicity of fixed points which are related to the classical PoincareBirkhoff twist theorem for area-preserving maps of the annulus; in our approach, like in Ding 2007 , we do not require measure-preserving conditions. Our goal is to extend Ding’s definition to a pure topological setting and obtain some fixed point theorems for continuous bend-twist maps. For a planarly embedded topological annulus, they could be chosen as the inner and the outer boundaries of the annulus In such a special case, the bounded component of R2 \ X turns out to be an open connected set D D X with ∂D Xi and cl D D ∪ Xi homeomorphic to the closed unit disc this can be proved by means of the Jordan-Schoenflies theorem 10.

Bend-Twist Maps

An Application