The Flapping Birds in the Pentagram Zoo

TL;DR

This work generalizes the classical pentagram map to the $(k+1,k)$-diagonal map $Δ_k$ acting on planar polygons and introduces the $k$-bird class $B_{k,n}$. It proves that for $n>3k$, $Δ_k$ preserves $B_{k,n}$ and that every polygon in this class is strictly star-shaped with a well-defined collapse point, with the orbit remaining embedded and converging to the interior point under iteration; the collapse point serves as a natural center of the dynamics. A key device is the $k$-energy ${χ_k}$, an invariant built from cross-ratios of the $k$-diagonals, which remains constant under $D_k$, $D_{k+1}$, and hence under $Δ_k=D_k\circ D_{k+1}$. The paper develops a geometric framework around the soul and feathers of a bird, a degeneration lemma that controls limiting behavior, a persistent triangulation $τ_P$ of the annulus between $P$ and $Δ_k(P)$, and a duality-based structure that yields nesting properties and equalities $\,Δ_k(B_{k,n})=B_{k,n}$. Together, these results extend convex-case phenomena seen for $Δ_1$ to the broader setting, providing a geometric interpretation of the pentagram zoo and a robust toolkit for limit analysis via affine normalization and duality.

Abstract

We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map $Δ_k$. The map $Δ_1$ is the pentagram map and $Δ_k$ is a generalization. $Δ_k$ does not preserve convexity, but we prove that $Δ_k$ preserves a subset $B_k$ of certain star-shaped polygons which we call $k$-birds. The action of $Δ_k$ on $B_k$ seems similar to the action of $Δ_1$ on the space of convex polygons. We show that some classic geometric results about $Δ_1$ generalize to this setting.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: Factor I
• Lemma 2.4
• Theorem 3.1
• Lemma 3.2