Quantitative equidistribution of angles of multipliers

TL;DR

This paper investigates the distribution of angles of multipliers for repelling cycles in hyperbolic rational maps, showing that most small intervals contain angles with multipliers bounded polynomially in the interval size.

Contribution

It establishes quantitative equidistribution results for multiplier angles, providing bounds on the size of multipliers within small angular intervals.

Findings

Most small intervals contain a multiplier angle with polynomially bounded norm.

Quantitative bounds on the distribution of multiplier angles.

Enhanced understanding of the geometric distribution of repelling cycle multipliers.

Abstract

We study angles of multipliers of repelling cycles for hyperbolic rational maps in $\mathbb C(z)$. For a fixed $K \gg 1$, we show that almost all intervals of length $2\pi/K$ in $(-\pi,\pi]$ contain a multiplier angle with the property that the norm of the multiplier is bounded above by a polynomial in $K$.