Manhattan curves in complex dynamics and asymptotic correlation of multiplier spectra

TL;DR

This paper extends the concept of Manhattan curves from hyperbolic geometry to complex dynamics, analyzing the growth of multiplier spectra and their correlations for pairs of holomorphic maps.

Contribution

It introduces the Manhattan curve in the context of complex dynamics and establishes a link between this curve and the asymptotic correlation of multiplier spectra.

Findings

Manhattan curve defined for pairs of hyperbolic rational maps and holomorphic endomorphisms.

Counting results for multiplier spectra in complex dynamics.

Relation between Manhattan curve and correlation number of multiplier spectra.

Abstract

The Manhattan curve for a pair of hyperbolic structures (possibly with cusps) on a given surface is a geometric object that encodes the growth rate of lengths of closed geodesics with respect to the two different hyperbolic metrics. It has been extensively studied as a way to understand geodesics on surfaces, the thermodynamic formalism of the geodesic flows and comparison of hyperbolic metrics. Via Sullivan's dictionary, in this paper, we define and study the Manhattan curve for a pair of hyperbolic rational maps on $\mathbb C\mathbb P^1$, and more generally of holomorphic endomorphisms of $\mathbb{C}\mathbb{P}^k$. We discuss several counting results for the multiplier spectrum and show that the Manhattan curve for two holomorphic endomorphisms is related to the correlation number of their multiplier spectra.