Generic collapse for closed polygons (Conjecture)

Establish that for an algebraically closed, complete field k with a non-trivial valuation and a polygonal transformation T in P^d(k) that admits a scaling symmetry and is discretely integrable on the moduli space, a generic closed polygon P has vertices that, under iteration of T, collapse in the past and future toward two fixed points q_- and q_+ of its infinitesimal monodromy M'.

Background

For closed polygons, the ordinary monodromy is the identity, but scaling symmetries allow one to define the infinitesimal monodromy M' whose eigenlines predict possible collapse points. Building on this, the authors conjecture actual collapsing to two fixed points of M', one attracting and one repelling, under discrete integrability assumptions.

This conjecture extends the twisted-polygon conjecture to closed polygons and aims to explain observed numerical behavior across various polygonal dynamics, including pentagram-type maps and cross-ratio systems.