Equidistribution speed of iterated preimages for rational maps on the Riemann sphere
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910036466335744 |
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| author | Hao, Mai Ji, Zhuchao |
| author_facet | Hao, Mai Ji, Zhuchao |
| contents | The exponential equidistribution speed of iterated preimages for holomorphic endomorphisms on $\mathbb{P}^k$ was established by Drasin-Okuyama for $k=1$, and by Dinh-Sibony for arbitrary $k$. In this paper, we obtain a near-optimal equidistribution speed with order $O(nd^{-n})$ in dimension one for points that are not super-attracting periodic. Moreover, the equidistribution speed order $O(nd^{-n})$ holds not only for $C^2$ observables but also for Hölder continuous d.s.h. observables. For geometrically finite rational maps (including all hyperbolic rational maps), we prove that the equidistribution speed order is $O(d^{-n})$ for $C^2$ observables and points that are not super-attracting, attracting, or parabolic periodic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13950 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equidistribution speed of iterated preimages for rational maps on the Riemann sphere Hao, Mai Ji, Zhuchao Dynamical Systems Complex Variables The exponential equidistribution speed of iterated preimages for holomorphic endomorphisms on $\mathbb{P}^k$ was established by Drasin-Okuyama for $k=1$, and by Dinh-Sibony for arbitrary $k$. In this paper, we obtain a near-optimal equidistribution speed with order $O(nd^{-n})$ in dimension one for points that are not super-attracting periodic. Moreover, the equidistribution speed order $O(nd^{-n})$ holds not only for $C^2$ observables but also for Hölder continuous d.s.h. observables. For geometrically finite rational maps (including all hyperbolic rational maps), we prove that the equidistribution speed order is $O(d^{-n})$ for $C^2$ observables and points that are not super-attracting, attracting, or parabolic periodic. |
| title | Equidistribution speed of iterated preimages for rational maps on the Riemann sphere |
| topic | Dynamical Systems Complex Variables |
| url | https://arxiv.org/abs/2602.13950 |