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arXiv · 2602.13950

Equidistribution speed of iterated preimages for rational maps on the Riemann sphere

Mai Hao·Zhuchao Ji
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Abstract

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 H\"older 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.

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Mai Hao, Zhuchao Ji. 2026-02-15. Equidistribution speed of iterated preimages for rational maps on the Riemann sphere. https://arxiv.org/abs/2602.13950

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