Lyapunov exponents, bifurcation currents and laminations in bifurcation loci

TL;DR

This paper investigates the structure of bifurcation loci in the moduli space of degree d rational maps, establishing equidistribution properties and lamination structures using potential theory and Lyapunov functions.

Contribution

It introduces new approximation formulas for the Lyapunov function and demonstrates lamination structures in bifurcation loci for degree 2 maps.

Findings

Equidistribution of hypersurfaces in bifurcation loci

Lamination structure in degree 2 bifurcation regions

Holomorphic motions in attracting basins

Abstract

Bifurcation loci in the moduli space of degree $d$ rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier 0 or $e^{i\theta}$. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree $d=2$, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin of fixed period exists.