Space spanned by characteristic exponents
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914419172179968 |
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| author | Ji, Zhuchao Xie, Junyi Zhang, Geng-Rui |
| author_facet | Ji, Zhuchao Xie, Junyi Zhang, Geng-Rui |
| contents | We prove several rigidity results on multiplier spectrum and length spectrum. For example, we show that for every non-exceptional rational map $f:\mathbb{P}^1(\mathbb{C})\to\mathbb{P}^1(\mathbb{C})$ of degree $d\geq2$, the $\mathbb{Q}$-vector space generated by all the (finite) characteristic exponents of periodic points of $f$ has infinite dimension. This answers a stronger version of a question of Levy and Tucker. Our result can also be seen as a generalization of recent results of Ji-Xie and of Huguin which proved Milnor's conjecture about rational maps having integer multipliers. We also get a characterization of postcritically finite maps by using its length spectra. Finally as an application of our result, we get a new proof of the Zariski-dense orbit conjecture for endomorphisms on $(\mathbb{P}^1)^N, N\geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00289 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Space spanned by characteristic exponents Ji, Zhuchao Xie, Junyi Zhang, Geng-Rui Dynamical Systems Algebraic Geometry Number Theory Primary 37P35, Secondary 37F10, 11G35 We prove several rigidity results on multiplier spectrum and length spectrum. For example, we show that for every non-exceptional rational map $f:\mathbb{P}^1(\mathbb{C})\to\mathbb{P}^1(\mathbb{C})$ of degree $d\geq2$, the $\mathbb{Q}$-vector space generated by all the (finite) characteristic exponents of periodic points of $f$ has infinite dimension. This answers a stronger version of a question of Levy and Tucker. Our result can also be seen as a generalization of recent results of Ji-Xie and of Huguin which proved Milnor's conjecture about rational maps having integer multipliers. We also get a characterization of postcritically finite maps by using its length spectra. Finally as an application of our result, we get a new proof of the Zariski-dense orbit conjecture for endomorphisms on $(\mathbb{P}^1)^N, N\geq 1$. |
| title | Space spanned by characteristic exponents |
| topic | Dynamical Systems Algebraic Geometry Number Theory Primary 37P35, Secondary 37F10, 11G35 |
| url | https://arxiv.org/abs/2308.00289 |