Conley index theory without index pairs. I: The point-set level theory

TL;DR

This paper introduces a novel framework for Conley index theory that eliminates the need for index pairs by using compactifiable subsets and index neighbourhoods, applicable to both discrete and continuous time systems.

Contribution

It presents a new approach to Conley index theory that simplifies the framework by removing index pairs, broadening the theoretical foundation.

Findings

Established basic results using the new framework

Unified treatment of discrete and continuous time cases

Provided foundational concepts for future research

Abstract

We propose a new framework for Conley index theory. The main feature of our approach is that we do not use the notion of index pairs. We introduce, instead, the notions of compactifiable subsets and index neighbourhoods, and formulate and prove basic results in Conley index theory using these notions. We treat both the discrete time case and the continuous time case.