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

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 $O(n{d}^{-n})$ in dimension one for points that are not super-attracting periodic. Moreover, the equidistribution speed order $O(n{d}^{-n})$ 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.