Weak index pairs and the Conley index for discrete multivalued dynamical systems

TL;DR

This paper revisits the Conley index theory for discrete multivalued dynamical systems, introducing weak index pairs to address limitations of traditional index pairs under less restrictive isolating neighborhoods.

Contribution

It proposes a new, less restrictive definition of isolating neighborhoods and introduces weak index pairs to extend the applicability of the Conley index theory.

Findings

New definition of isolating neighborhoods

Introduction of weak index pairs

Extended applicability of Conley index theory

Abstract

Motivated by the problem of reconstructing dynamics from samples we revisit the Conley index theory for discrete multivalued dynamical systems. We introduce a new, less restrictive definition of the isolating neighbourhood. It turns out that then the main tool for the construction of the index, i.e. the index pair, is no longer useful. In order to overcome this obstacle we use the concept of weak index pairs.