Propagation of equilibrium states in stable families of endomorphisms of $\mathbb P^k(\mathbb C)$

TL;DR

This paper demonstrates that in holomorphic families of endomorphisms of complex projective space, the equilibrium states associated with certain weight functions are preserved under measurable holomorphic motions linked to dynamical stability.

Contribution

It establishes the invariance of equilibrium states within stable families of endomorphisms of complex projective spaces for specific weight functions.

Findings

Equilibrium states are preserved under holomorphic motions in stable families.

The result applies to all dimensions $k \\geq 1$ and degrees $d \\geq 2$.

The class of weight functions with bounded oscillation is considered.

Abstract

We prove that, within any holomorphic family of endomorphisms of $\mathbb P^k(\mathbb C)$ in any dimension $k \geq 1$ and algebraic degree $d \geq 2$, the measurable holomorphic motion associated to dynamical stability in the sense of Berteloot-Bianchi-Dupont preserves the class of equilibrium states associated with weight functions $\psi$ satisfying $\sup\psi - \inf \psi < \log d$.