Degeneration of quadratic polynomial endomorphisms to a Hénon map

Abstract

For an algebraic family $(f_t)$ of regular quadratic polynomial endomorphisms of ${\mathbb{C}}^2$ parametrized by ${\mathbb{D}}^{\ast }$ and degenerating to a Hénon map at $t=0$, we study the continuous (and indeed harmonic) extendibility across $t=0$ of a potential of the bifurcation current on ${\mathbb{D}}^{\ast }$ 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 Hénon map is accumulated by the bifurcation locus in the space of quadratic holomorphic endomorphisms of ${\mathbb{P}}^2$.