Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces

Area of Science:

• Computational Mathematics
• Topology
• Dynamical Systems

Background:

• The Conley-Morse-Forman theory provides a framework for analyzing dynamical systems using combinatorial methods.
• Previous work by Mrozek (2017) introduced this theory for combinatorial multivector fields but had limitations.

Purpose of the Study:

• To generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields.
• To adapt the theory to a more general setting of finite topological spaces.
• To provide a more flexible framework for studying combinatorial topological dynamics.

Main Methods:

• Generalizing the theory by removing the unique maximal element assumption for multivectors.
• Defining the induced dynamical system in a less restrictive manner.
• Shifting the theoretical setting from Lefschetz complexes to finite topological spaces.

Main Results:

• Definition of isolated invariant sets, isolating neighborhoods, Conley index, and Morse decompositions within the new framework.
• Establishment of the additivity property of the Conley index.
• Derivation of the Morse inequalities.

Conclusions:

• The generalized theory offers a more flexible and broadly applicable approach to Conley-Morse-Forman theory.
• The use of finite topological spaces enhances the understanding of combinatorial topological dynamics.
• The extended framework supports further research in computational mathematics and dynamical systems analysis.