Inter-hierarchical gap between topological entropy and physical observability

Characterize the unresolved relationship between topological entropy, representing the total amount of dynamical possibility, and the physical measure, representing effective macroscopic observability, in dissipative dynamical systems.

Background

The paper distinguishes between transversal homoclinic intersections, which generate invariant chaotic sets with positive topological entropy, and the behavior observed in the steady state, which is governed by the physical measure and may have zero Kolmogorov–Sinai entropy when chaotic saddles are non-attracting. The numerical results also show that homoclinic intersections associated with a tracked primary saddle are neither sufficient nor universally necessary for the observation of macroscopic chaos.

The authors therefore identify an unresolved inter-hierarchical gap between microscopic topological possibilities and macroscopic physical observability. Resolving this gap would require determining how geometric structures, coexisting invariant sets, and attractor selection jointly control the emergence of observable chaotic behavior.