The Morse equation in the Conley index theory for discrete multivalued dynamical systems

TL;DR

This paper extends the Conley index theory to discrete multivalued dynamical systems, establishing Morse equations and inequalities to analyze their attractor-repeller structures, with implications for data-driven dynamics.

Contribution

It introduces Morse equations and inequalities for multivalued systems without continuous selectors, advancing the theoretical framework of Conley index in this context.

Findings

Proved Morse equations for multivalued dynamical systems

Established Morse inequalities in this setting

Enhanced understanding of attractor-repeller pairs

Abstract

A recent generalization of the Conley index to discrete multivalued dynamical systems without a continuous selector is motivated by applications to data-driven dynamics. In the present paper we continue the program by studying attractor-repeller pairs and Morse decompositions in this setting. In particular, we prove Morse equation and Morse inequalities.