Continuity of Lyapunov exponents for polynomial automorphisms of $\mathbb{C}^2$

Abstract

Abstract. We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of C 2 . The first continuity result holds for any family of polynomial automorphisms of constant dynamical degree. The second result is the continuity of the upper exponent for families degenerating to a 1-dimensional map.

Bifurcation currents and equidistribution in parameter space

“…Notice that a polynomial automorphism has constant jacobian, so χ + (f ) + χ − (f ) = log |Jac(f )| is a pluriharmonic function on parameter space. It is not difficult to see that the function f → χ + (f ) is psh (in particular upper semi-continuous), and it was shown in [Du1] that is actually continuous (even for families degenerating to a one-dimensional map).…”