An asymptotic for sums of Lyapunov exponents in families

TL;DR

This paper studies the asymptotic behavior of the sum of Lyapunov exponents in families of complex endomorphisms, providing explicit error estimates under certain conditions.

Contribution

It extends Favre's result by giving an explicit error term for the asymptotic of Lyapunov sums in specific meromorphic families.

Findings

Derived explicit error bounds for Lyapunov sum asymptotics.

Confirmed the asymptotic formula with refined estimates.

Analyzed conditions under which the error term applies.

Abstract

Let f_t be a meromorphic family of endomorphisms of P^N_C of degree at least 2, and let L(f_t) be the sum of Lyapunov exponents associated to f_t. Favre showed that L(f_t)=L(f)\log|t^{-1}|+o(\log|t^{-1}|) as t -> 0, where L(f) is the sum of Lyapunov exponents on the generic fibre, interpreted as an endomorphism of some projective Berkovich space. Under some additional constraints on the family, we provide an explicit error term.