Collapsing in polygonal dynamics
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917489533779968 |
|---|---|
| author | Stiegler, Jean-Baptiste |
| author_facet | Stiegler, Jean-Baptiste |
| contents | We define polygonal dynamics as a family of dynamical systems acting on points in projective spaces. The most famous example is the pentagram map. Similar collapsing phenomena seem to occur in most of these systems. We prove it in some case, and conjecture that it almost always happens. Moreover, we give a formula for the limit point in term of roots of $d+1$ degree polynomials (where $d$ is the dimension of the projective space). We do so by generalizing Glick's operator, interpreted as an infinitesimal monodromy. This answers questions about its reappearance in many systems, together with preserved quantities. We apply these results to several polygonal dynamics in $\mathbb{P}^1$ and introduce a new one called ``staircase'' cross-ratio dynamics, for which we study particular configurations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Collapsing in polygonal dynamics Stiegler, Jean-Baptiste Dynamical Systems Exactly Solvable and Integrable Systems We define polygonal dynamics as a family of dynamical systems acting on points in projective spaces. The most famous example is the pentagram map. Similar collapsing phenomena seem to occur in most of these systems. We prove it in some case, and conjecture that it almost always happens. Moreover, we give a formula for the limit point in term of roots of $d+1$ degree polynomials (where $d$ is the dimension of the projective space). We do so by generalizing Glick's operator, interpreted as an infinitesimal monodromy. This answers questions about its reappearance in many systems, together with preserved quantities. We apply these results to several polygonal dynamics in $\mathbb{P}^1$ and introduce a new one called ``staircase'' cross-ratio dynamics, for which we study particular configurations. |
| title | Collapsing in polygonal dynamics |
| topic | Dynamical Systems Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2507.16432 |