Lyapunov exponents and bifurcation current for polynomial-like maps

TL;DR

This paper investigates how Lyapunov exponents vary in polynomial-like maps and introduces bifurcation currents to understand stability and changes in dynamical systems as parameters vary.

Contribution

It establishes that sums of Lyapunov exponents are plurisubharmonic functions and introduces the bifurcation locus as the support of bifurcation currents.

Findings

Partial sums of Lyapunov exponents are plurisubharmonic in parameters.

Continuity properties of Lyapunov exponents are analyzed.

Bifurcation locus is characterized as the support of bifurcation currents.

Abstract

We study holomorphic families of polynomial-like maps depending on a parameter s. We prove that the partial sums of largest Lyapunov exponents are plurisubharmonic functions of s. We also study their continuity and introduce the bifurcation locus as the support of bifurcation currents.