Collapsing in polygonal dynamics

TL;DR

This paper investigates collapsing phenomena in polygonal dynamical systems, providing formulas for limit points, generalizing operators, and introducing new dynamics, with implications for projective space systems.

Contribution

It proves collapsing behavior in some polygonal dynamics, generalizes Glick's operator, and introduces a new staircase cross-ratio dynamics.

Findings

Collapse occurs in most polygonal dynamics systems.

A formula for limit points using roots of degree d+1 polynomials.

Introduction of staircase cross-ratio dynamics.

Abstract

We define polygonal dynamics as a family of dynamical systems acting on points in projective spaces. The most famous example is the pentagram map. Similar collapsing phenomena seem to occur in most of these systems. We prove it in some case, and conjecture that it almost always happens. Moreover, we give a formula for the limit point in term of roots of $d+1$ degree polynomials (where $d$ is the dimension of the projective space). We do so by generalizing Glick's operator, interpreted as an infinitesimal monodromy. This answers questions about its reappearance in many systems, together with preserved quantities. We apply these results to several polygonal dynamics in $\mathbb{P}^1$ and introduce a new one called ``staircase'' cross-ratio dynamics, for which we study particular configurations.