Julia sets and geometrically finite maps over finite extensions of the $p$-adic field
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866909070113374208 |
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| author | Fan, Shilei Liao, Lingmin Nie, Hongmin Wang, Yuefei |
| author_facet | Fan, Shilei Liao, Lingmin Nie, Hongmin Wang, Yuefei |
| contents | Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers, and $ϕ\in K(z)$ be a rational map of degree at least $2$. We prove that the $K$-Julia set of $ϕ$ is the natural restriction of $\mathbb{C}_p$-Julia set, provided that the critical orbits are well-behaved. Moreover, under further assumption that $ϕ$ is geometrically finite, we prove that the dynamics on the $K$-Julia set of $ϕ$ is a countable state Markov shift. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_01579 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Julia sets and geometrically finite maps over finite extensions of the $p$-adic field Fan, Shilei Liao, Lingmin Nie, Hongmin Wang, Yuefei Dynamical Systems Complex Variables Primary 37P05, Secondary 11S82, 37B10 Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers, and $ϕ\in K(z)$ be a rational map of degree at least $2$. We prove that the $K$-Julia set of $ϕ$ is the natural restriction of $\mathbb{C}_p$-Julia set, provided that the critical orbits are well-behaved. Moreover, under further assumption that $ϕ$ is geometrically finite, we prove that the dynamics on the $K$-Julia set of $ϕ$ is a countable state Markov shift. |
| title | Julia sets and geometrically finite maps over finite extensions of the $p$-adic field |
| topic | Dynamical Systems Complex Variables Primary 37P05, Secondary 11S82, 37B10 |
| url | https://arxiv.org/abs/2111.01579 |