The Flapping Birds in the Pentagram Zoo

• Published in 2024
• Added on 2024-03-14

We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map $\Delta_k$. The map $\Delta_1$ is the pentagram map and $\Delta_k$ is a generalization. $\Delta_k$ does not preserve convexity, but we prove that $\Delta_k$ preserves a subset $B_k$ of certain star-shaped polygons which we call $k$-birds. The action of $\Delta_k$ on $B_k$ seems similar to the action of $\Delta_1$ on the space of convex polygons. We show that some classic geometric results about $\Delta_1$ generalize to this setting.

Links

• http://arxiv.org/abs/2403.05735v1
• http://arxiv.org/pdf/2403.05735v1

Other information

BibTeX entry

@article{TheFlappingBirdsinthePentagramZoo,
	key = {TheFlappingBirdsinthePentagramZoo},
	type = {article},
	title = {The Flapping Birds in the Pentagram Zoo},
	author = {Richard Evan Schwartz},
	abstract = {We study the {\$}(k+1,k){\$} diagonal map for {\$}k=2,3,4,...{\$}. We call this map
{\$}\Delta{\_}k{\$}. The map {\$}\Delta{\_}1{\$} is the pentagram map and {\$}\Delta{\_}k{\$} is a
generalization. {\$}\Delta{\_}k{\$} does not preserve convexity, but we prove that
{\$}\Delta{\_}k{\$} preserves a subset {\$}B{\_}k{\$} of certain star-shaped polygons which we
call {\$}k{\$}-birds. The action of {\$}\Delta{\_}k{\$} on {\$}B{\_}k{\$} seems similar to the action
of {\$}\Delta{\_}1{\$} on the space of convex polygons. We show that some classic
geometric results about {\$}\Delta{\_}1{\$} generalize to this setting.},
	comment = {},
	date_added = {2024-03-14},
	date_published = {2024-12-07},
	urls = {http://arxiv.org/abs/2403.05735v1,http://arxiv.org/pdf/2403.05735v1},
	collections = {animals},
	url = {http://arxiv.org/abs/2403.05735v1 http://arxiv.org/pdf/2403.05735v1},
	year = 2024,
	urldate = {2024-03-14},
	archivePrefix = {arXiv},
	eprint = {2403.05735},
	primaryClass = {math.DS}
}