On the centralizers of endomorphisms of the projective line
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908713055420416 |
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| author | Beaumont, Alonso |
| author_facet | Beaumont, Alonso |
| contents | Let $f$ be a dominant endomorphism of the projective line, which is not conjugate to a power map $z\mapsto z^{\pm d}$. We consider the centralizers of the iterates of $f$, $C(f^{n}):=\{\textrm{dominant}\;g:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}\;|\; g\circ f^{n}=f^{n}\circ g\}$, $n\geq1$, and prove that their union is equal to $C(f^{N})$ for some $N\geq1$. This solves a conjecture of F. Pakovich. As an application, we obtain a Tits alternative for cancellative semigroups of endomorphisms of the projective line, without an assumption of finite generation, extending the results of J.P. Bell, K. Huang, W. Peng and T.J. Tucker. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13523 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the centralizers of endomorphisms of the projective line Beaumont, Alonso Dynamical Systems Group Theory Number Theory 20M05, 14H37 Let $f$ be a dominant endomorphism of the projective line, which is not conjugate to a power map $z\mapsto z^{\pm d}$. We consider the centralizers of the iterates of $f$, $C(f^{n}):=\{\textrm{dominant}\;g:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}\;|\; g\circ f^{n}=f^{n}\circ g\}$, $n\geq1$, and prove that their union is equal to $C(f^{N})$ for some $N\geq1$. This solves a conjecture of F. Pakovich. As an application, we obtain a Tits alternative for cancellative semigroups of endomorphisms of the projective line, without an assumption of finite generation, extending the results of J.P. Bell, K. Huang, W. Peng and T.J. Tucker. |
| title | On the centralizers of endomorphisms of the projective line |
| topic | Dynamical Systems Group Theory Number Theory 20M05, 14H37 |
| url | https://arxiv.org/abs/2512.13523 |