Conley index for multivalued maps on finite topological spaces

TL;DR

This paper extends Conley's topological dynamical systems theory to multivalued maps on finite topological spaces, defining key concepts like invariant sets and the Conley index, and verifying fundamental properties.

Contribution

It introduces a new framework for Conley index theory applicable to multivalued maps on finite topological spaces, including definitions and fundamental properties.

Findings

Defined isolated invariant sets and index pairs for multivalued maps.

Established the well-defined Conley index in this setting.

Verified properties like Wazewski property and continuation.

Abstract

We develop Conley's theory for multivalued maps on finite topological spaces. More precisely, for discrete-time dynamical systems generated by the iteration of a multivalued map which satisfies appropriate regularity conditions, we establish the notions of isolated invariant sets and index pairs, and use them to introduce a well-defined Conley index. In addition, we verify some of its fundamental properties such as the Wazewski property and continuation.