Degeneration of endomorphisms of the complex projective space in the hybrid space

TL;DR

This paper studies how the maximal entropy measure of a family of complex projective space endomorphisms behaves as the system degenerates, showing convergence to a non-Archimedean equilibrium measure in the hybrid space.

Contribution

It proves the convergence of measures in the hybrid space for degenerating endomorphisms and provides estimates for Lyapunov exponent blow-up near poles.

Findings

Maximal entropy measures converge to non-Archimedean equilibrium measures during degeneration.

The convergence is established within the hybrid space framework.

An estimate for Lyapunov exponent blow-up near poles is derived.

Abstract

We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk.We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holdsin the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.