Generic collapse for twisted polygons (Conjecture)

Establish that for an algebraically closed, complete field k with a non-trivial valuation and a polygonal transformation T on projective space P^d(k) whose induced dynamics on the moduli space is discretely integrable, a generic twisted n-gon P=((p_1, …, p_n), M) has vertices that, under iteration of T, collapse in the past and future to fixed points of its monodromy M.

Background

The paper introduces polygonal dynamics on projective spaces, focusing on systems that admit a scaling symmetry and exhibit collapse phenomena. For twisted polygons, the monodromy M generically has distinct fixed points, suggesting where collapsing could occur. Motivated by numerical evidence and integrability structures, the authors formalize a conjecture asserting generic collapse under iteration to fixed points of M.

This conjecture underpins the broader program of understanding collapse in polygonal dynamics without requiring convexity or real field assumptions, and connects with classical systems such as the pentagram map and cross-ratio dynamics.