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)

To make \chi behave more like evolving probability distributions (i.e., a Wigner function), we could have instead defined it as a time-periodic measure over X which satisfies the Poisson or Moyal equation (as a distribution). This is currently incompatible with Floer theory, since it remains unclear how to define the action functional for this case.

— Hamiltonian Floer Theory for Quantum Electrodynamics up to First Order in $\hbar$  (2608.19996 - Fabert et al., 20 Aug 2026) in Remark immediately following Definition in Section 3, “Periodic Stochastic Orbits in the Nonmagnetic Limit”

Can we derive a version of Floer homology directly from the semiflow on sutured manifold?

— Pseudo-Anosov flow and dynamics on guts  (2608.30973 - Huang, 31 Aug 2026) in Section 6, subsection “Floer homology, volume, entropy and the guts”