Morse hyper-graphs and abstract weak orbit spaces of semi-decompositions

TL;DR

This paper develops topological invariants like Morse hyper-graphs and abstract weak element spaces to analyze semi-decompositions in dynamical systems, linking them to known structures such as Reeb graphs and face posets.

Contribution

It introduces new topological invariants for semi-decompositions, connecting them to classical structures like Reeb graphs and face posets, enhancing dynamical systems analysis.

Findings

Morse hyper-graphs of sublevel sets correspond to Reeb graphs.

Abstract weak element spaces relate to face posets and directed multi-graphs.

New invariants provide tools for analyzing semi-decompositions in topology.

Abstract

We introduce topological invariants of semi-decompositions (e.g. filtrations, semi-group actions, multi-valued dynamical systems, combinatorial dynamical systems) on a topological space to analyze semi-decompositions from a dynamical systems point of view. In fact, we construct Morse hyper-graphs and abstract weak elements of semi-decompositions. Moreover, the Morse hyper-graphs of the set of sublevel sets of a Morse function of a compact manifold is the Reeb graph of such function as abstract multi-graphs. The abstract weak element space for a simplicial complex is the face poset. The abstract weak element spaces of positive orbits of acyclic directed graphs are their abstract directed multi-graphs.