Bifurcation currents and equidistribution on parameter space

TL;DR

This paper reviews how positive currents are used to analyze parameter spaces in one-dimensional holomorphic dynamics, focusing on bifurcation currents, their supports, and equidistribution of special subvarieties.

Contribution

It provides a comprehensive overview of the construction and properties of bifurcation currents and their role in understanding parameter space dynamics.

Findings

Bifurcation currents characterize stability regions.

Supports of bifurcation currents identify bifurcation loci.

Dynamically defined subvarieties equidistribute in parameter space.

Abstract

In this paper we review the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P^1 or subgroups of the Moebius group PSL(2,C)). The topics covered include: the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties on parameter space.