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In this article, we consider a counting problem for orbits of hyperbolicrational maps on the Riemann sphere, where constraints are placed on themultipliers of orbits. Using arguments from work of Dolgopyat, we considervarying and potentially shrinking intervals, and obtain a result whichresembles a local central limit theorem for the logarithm of the absolute valueof the multiplier and an equidistribution theorem for the holonomies.

Statistics of multipliers for hyperbolic rational maps