Degeneration of quadratic polynomial endomorphisms to a H\'enon map

TL;DR

This paper investigates how quadratic polynomial endomorphisms degenerate into Hénon maps, analyzing the bifurcation current, Lyapunov exponents, and the accumulation of Hénon maps in the parameter space.

Contribution

It provides an explicit analysis of the bifurcation current's potential extension and computes the non-archimedean Lyapunov exponent during degeneration.

Findings

Potential of bifurcation current extends harmonically across degeneration point.

Explicit computation of non-archimedean Lyapunov exponent for degenerating family.

Hénon maps are accumulation points of bifurcation locus in quadratic endomorphisms.

Abstract

For an algebraic family $(f_t)$ of regular quadratic polynomial endomorphisms of $\mathbb{C}^2$ parametrized by $\mathbb{D}^*$ and degenerating to a H\'enon map at $t=0$, we study the continuous (and indeed harmonic) extendibility across $t=0$ of a potential of the bifurcation current on $\mathbb{D}^*$ with the explicit computation of the non-archimedean Lyapunov exponent associated to $(f_t)$. The individual Lyapunov exponents of $f_t$ are also investigated near $t=0$. Using $(f_t)$, we also see that any H\'enon map is accumulated by the bifurcation locus in the space of quadratic holomorphic endomorphisms of $\mathbb {P}^2$.