  In this article, we consider a counting problem for orbits of hyperbolic
rational maps on the Riemann sphere, where constraints are placed on the
multipliers of orbits. Using arguments from work of Dolgopyat, we consider
varying and potentially shrinking intervals, and obtain a result which
resembles a local central limit theorem for the logarithm of the absolute value
of the multiplier and an equidistribution theorem for the holonomies.
