Hyperbolic components of rational maps: Quantitative equidistribution and counting

TL;DR

This paper establishes exponential convergence of parameters with multiple neutral cycles towards bifurcation currents in the moduli space of rational maps, with implications for hyperbolic component counting.

Contribution

It proves a quantitative equidistribution result for neutral cycles and introduces a locally uniform approximation of Lyapunov exponents in this context.

Findings

Exponential speed of convergence towards bifurcation currents.

Quantitative approximation of Lyapunov exponents by multipliers.

Applications to counting hyperbolic components.

Abstract

Let $\Lambda$ be a quasi-projective variety and assume that, either $\Lambda$ is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, or $\Lambda$ parametrizes an algebraic family $(f_\lambda)_{\lambda\in\Lambda}$ of degree $d$ rational maps on $\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the $\log^+$ of the modulus of the multipliers of periodic points.