Prime number theorems and holonomies for hyperbolic rational maps

TL;DR

This paper establishes prime number theorem analogues for hyperbolic rational maps, providing counting estimates and holonomy equidistribution with power savings error terms, based on dynamical zeta functions twisted by characters.

Contribution

It introduces new counting and equidistribution results for hyperbolic rational maps using twisted dynamical zeta functions and non-vanishing properties.

Findings

Counting estimates for primitive periodic orbits with power savings

Equidistribution of holonomies with power savings

Non-vanishing of twisted zeta functions on a half-plane

Abstract

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted by characters of $S^1$. We will show that these zeta functions are non-vanishing on a half plane $\Re(s) > \delta - \epsilon$, where $\delta$ is the Hausdorff dimension of the Julia set of f.