Extending Morse–Floer-type constructions to broader dynamical systems

Extend Morse–Floer-type boundary operator and homology constructions to broader classes of dynamical systems beyond the currently treated Morse–Smale settings, ensuring the chain complex is well-defined (∂∘∂=0) and captures the topology of the underlying space.

Background

The paper develops Floer-type boundary operators counting flow lines in settings that include non-degenerate fixed points and non-degenerate periodic or homoclinic orbits, under suitable transversality and nondegeneracy conditions.

Expanding these constructions to more general dynamical systems would broaden the applicability of Morse–Floer-type homological tools and require addressing analytical and topological obstacles to preserve homological invariance and boundary squaring to zero.

References

Other open problems include defining topologically informative Laplacian-based random walks on simplicial complexes that their limiting behavior could be easily analyzed, or extending Morse–Floer-type constructions to broader classes of dynamical systems.

— From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions (2512.05319 - Eidi et al., 4 Dec 2025) in Section 6 (Conclusion and Outlook)