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

TL;DR

This paper improves the understanding of how quickly preimages of points distribute evenly under rational maps on the Riemann sphere, providing near-optimal and optimal speed estimates for various classes of maps and observables.

Contribution

It establishes near-optimal equidistribution speeds of order O(nd^{-n}) in dimension one for non-super-attracting points and extends results to H"older continuous observables, also proving optimal speeds for geometrically finite maps.

Findings

Near-optimal equidistribution speed O(nd^{-n}) for non-super-attracting points.

Speed O(d^{-n}) for geometrically finite rational maps.

Applicability to H"older continuous d.s.h. observables.

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.