Statistics of multipliers for hyperbolic rational maps

TL;DR

This paper studies the statistical distribution of multipliers for hyperbolic rational maps, establishing results akin to a local central limit theorem and equidistribution for orbit properties.

Contribution

It introduces a new statistical framework for analyzing multipliers of hyperbolic rational maps, extending previous work with local limit theorems and holonomy equidistribution.

Findings

Established a local central limit theorem for multipliers.

Proved equidistribution results for holonomies.

Analyzed the behavior of multipliers under varying constraints.

Abstract

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.