Methodological gap for quasi-periodic motion

Develop a rigorous framework to prove collapse for quasi-periodic motion on the moduli space by showing that appropriate subsequences of iterates asymptotically emulate iterations of a projective transformation, thereby yielding collapse to fixed points of the monodromy for twisted polygons.

Background

The authors provide a partial proof of collapse for periodic motion on the moduli space but note that the generic case is quasi-periodic. They suggest mimicking the periodic case by identifying subsequences of iterates that behave asymptotically like projective transformations, potentially using generalized Schwarzian derivatives, but report no success.

Bridging this methodological gap is crucial to establishing the twisted-polygon collapse conjecture in full generality.