The Flapping Birds in the Pentagram Zoo

Abstract

We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map $\hat{I}{”}_k$. The map $\hat{I}{”}_1$ is the pentagram map and $\hat{I}{”}_k$ is a generalization. $\hat{I}{”}_k$ does not preserve convexity, but we prove that $\hat{I}{”}_k$ preserves a subset $B_k$ of certain star-shaped polygons which we call $k$-birds. The action of $\hat{I}{”}_k$ on $B_k$ seems similar to the action of $\hat{I}{”}_1$ on the space of convex polygons. We show that some classic geometric results about $\hat{I}{”}_1$ generalize to this setting.

60 pages, computer experiment inspired but mostly traditional math. Similar to previous version except that I cleaned it up in many places, as a prelude to publishing it