# The Flapping Birds in the Pentagram Zoo

- Published in 2024
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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.

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- TheFlappingBirdsinthePentagramZoo
- type
- article
- date_added
- 2024-03-14
- date_published
- 2024-04-10

### 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 = {2024-04-10}, 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} }