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On the stability of holomorphic families of endomorphisms of ${\mathbb P}^k$

François Berteloot, Xavier Buff

TL;DR

This work analyzes the stability of holomorphic families of degree $d\ge 2$ endomorphisms of $\mathbb{P}^k$ parameterized by a simply connected complex manifold $M$. It introduces and interrelates stability notions through equilibrium webs and laminations, culminating in an equivalence result that unifies $\mu$-stability with weak and partial forms of stability. The approach hinges on the Lyapunov function $L(\lambda)=\int_{\mathbb{P}^k} \ln|{\rm Jac} f_\lambda|\, d\mu_\lambda$, its $dd^c L$-formula, and an approximation formula by repelling cycles, tying stability to critical dynamics and the distribution of repellers. The main theorem shows that, for simply connected $M$, the various stability notions are equivalent, extending prior work and clarifying the behavior of repelling-cycle bifurcations across the family. These results provide a rigorous framework for understanding stability propagation in higher-dimensional holomorphic dynamics.

Abstract

In the context of holomorphic families of ${\mathbb P}^k$ endomorphisms, we show that various notions of stability are equivalent. This allows us to both extend and simplify the architecture of the proof of certain results of [BBD]

On the stability of holomorphic families of endomorphisms of ${\mathbb P}^k$

TL;DR

This work analyzes the stability of holomorphic families of degree endomorphisms of parameterized by a simply connected complex manifold . It introduces and interrelates stability notions through equilibrium webs and laminations, culminating in an equivalence result that unifies -stability with weak and partial forms of stability. The approach hinges on the Lyapunov function , its -formula, and an approximation formula by repelling cycles, tying stability to critical dynamics and the distribution of repellers. The main theorem shows that, for simply connected , the various stability notions are equivalent, extending prior work and clarifying the behavior of repelling-cycle bifurcations across the family. These results provide a rigorous framework for understanding stability propagation in higher-dimensional holomorphic dynamics.

Abstract

In the context of holomorphic families of endomorphisms, we show that various notions of stability are equivalent. This allows us to both extend and simplify the architecture of the proof of certain results of [BBD]
Paper Structure (4 sections, 4 theorems, 15 equations)

This paper contains 4 sections, 4 theorems, 15 equations.

Table of Contents

  1. Introduction and results
  2. Equilibrium webs, laminations, and $\mu$-stability
  3. Proof of Proposition \ref{['PropMT']}
  4. Proof of Theorem \ref{['ThmEquiv']}

Key Result

Proposition 1.2

Let $M$ be a connected complex manifold and $f:M\times \mathbb{P}^k \to M\times \mathbb{P}^k$ be a degree $d\ge 2$ holomorphic family of endomorphisms of $\mathbb{P}^k$. If $f$ is $\mu$-stable then $f$ is weakly stable.

Theorems & Definitions (6)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • Theorem 4.1