Collapsing in polygonal dynamics

TL;DR

This work unifies polygonal dynamics by introducing a general framework for twisted $n$-gons with monodromy and a scaling symmetry, and it formalizes a collapse phenomenon whereby generic orbits converge to fixed points of a related monodromy. A central tool is the infinitesimal monodromy $M'$, which, together with the scaling action, yields computable predictions for collapse points—particularly for closed polygons where collapse points are fixed points of $M'$, lying in low-degree field extensions. In the $P^1$ setting, the authors derive an explicit formula for $M'$ in terms of cross-ratio data and provide a quadratic invariant $ ext{χ}_P(X,Y)$ that governs collapse directions, applicable to the leapfrog map, flat cross-ratio dynamics, and the newly defined staircase cross-ratio dynamics. The staircase dynamics are organized by flips forming an affine symmetric group, with a scaling symmetry yielding a tractable infinitesimal monodromy and a clear path toward algebraic integrability. Overall, the paper connects well-known dynamics (e.g., the pentagram map) to a broader theory, yielding computable collapse points and highlighting deep ties between monodromy, cross-ratio structures, and discrete integrability.

Abstract

Polygonal dynamics is a family of dynamical systems containing many studied systems, like the famous pentagram map. Similar collapsing phenomena seem to occur in most of these systems. We give a unifying, general definition of polygonal dynamics, and conjecture that a generic orbit collapses towards a predictable point. We manage to prove it in some setting. For the special case of ``closed polygons'', we show as a corollary that the collapse point depends algebraicly on the vertices of the starting polygon, using tools called scaling symmetry and infinitesimal monodromy. This generalizes previous results about the pentagram map. Then, we investigate the case of polygonal dynamics in $\mathbb{P}^1$ for which we give an explicit polynomial equation satisfied by the collapse point. Based on previous works, we define a new dynamical system, the ``staircase'' cross-ratio dynamics, for which we study particular configurations.

Figures (5)

• Figure 1: The convex pentagon $P$, lying on the real plane, is sent to $P'$ by the pentagram map.
• Figure 2: Visualisation of the action of the flip $\phi_j$ at index $j$, read from up to down. It it similar to a braid on a cylinder, but it is important not to forget that it acts also on the polygon $(p_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$; indeed, the point $p_j$ is changed in $\tilde{p}_j=h^{\mu_j/\mu_{j-1}}_{p_{j+1},p_{j-1}}(p_{j})$.
• Figure 3: A visualization of 3D consistency, meaning that one can "flip around the cube consistently". The red vertices and edges are the ones taken originally.
• Figure 4: Python simulation of the evolution of one randomly chosen closed $5$-gon in $\mathbb{P}^1(\mathbb{C})$ with randomly chosen discrete curvature, under the iteration of the sequence of flips $\phi_5\circ\dots\circ\phi_1$ (applied 25 times). The initial vertices are represented with red crosses. A new edge is drawn each time a flip is done, and the colour indicates time (from blue to red). There is a rapid spiraling towards one of the collapse points determined by corollary \ref{['coro:collapse_closed']}.
• Figure 5: Special "staircase" of the cross-ratio dynamic, for $n=3$. On the left, this makes appear a parabolic Möbius transformation, and on the right an elliptic or loxodromic (depending on $|q|$).

Theorems & Definitions (66)

• Definition 1.1
• Remark 1
• Remark 2
• Definition 1.2
• Conjecture 1.3: \ref{['conj:asympt']}
• Theorem 1.4: \ref{['thm:asympt_periodic']}
• Corollary 1.5: \ref{['coro:collapse_closed']}
• Corollary 1.6: \ref{['cor:field_ext']}
• Theorem 1.7: \ref{['thm:monod_closed']}
• proof