Degree growth of skew pentagram maps
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914193786011648 |
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| author | Weinreich, Max |
| author_facet | Weinreich, Max |
| contents | Skew pentagram maps act on polygons by intersecting diagonals of different lengths. They were introduced by Khesin-Soloviev in 2015 as conjecturally non-integrable generalizations of the pentagram map, a well-known integrable system. In this paper, we show that certain skew pentagram maps have exponential degree growth and no preserved fibration. To formalize this, we introduce a general notion of first dynamical degree for lattice maps, or shift-invariant self-maps of $(\mathbb{P}^N)^\mathbb{Z}$. We show that the dynamical degree of any equal-length pentagram map is 1, but that there are infinitely many skew pentagram maps with dynamical degree 4. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10062 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degree growth of skew pentagram maps Weinreich, Max Dynamical Systems 37K60, 37L60, 14E05, 37F80 Skew pentagram maps act on polygons by intersecting diagonals of different lengths. They were introduced by Khesin-Soloviev in 2015 as conjecturally non-integrable generalizations of the pentagram map, a well-known integrable system. In this paper, we show that certain skew pentagram maps have exponential degree growth and no preserved fibration. To formalize this, we introduce a general notion of first dynamical degree for lattice maps, or shift-invariant self-maps of $(\mathbb{P}^N)^\mathbb{Z}$. We show that the dynamical degree of any equal-length pentagram map is 1, but that there are infinitely many skew pentagram maps with dynamical degree 4. |
| title | Degree growth of skew pentagram maps |
| topic | Dynamical Systems 37K60, 37L60, 14E05, 37F80 |
| url | https://arxiv.org/abs/2512.10062 |