Normal forms, Lyapunov exponents, and pluripotential theory on $\mathbb{P}^k(\mathbb{C})$

TL;DR

This paper explores the relationship between Lyapunov exponents, Green currents, and equilibrium measures in complex dynamics on projective space, providing new characterizations and proofs using pluripotential and ergodic theories.

Contribution

It establishes a converse to Dujardin's result, linking Lyapunov exponent inequalities to measure regularity, and introduces normal forms for inverse branches to facilitate proofs.

Findings

Proves that specific Lyapunov exponent inequalities imply measure regularity.

Provides an alternative proof of Dujardin's result using normal forms.

Enhances understanding of dynamical properties of endomorphisms on projective space.

Abstract

We study the dynamical properties of endomorphisms $f$ of $\mathbb{P}^k$ of algebraic degree $d \geq 2$. We investigate the relationships between the Green current $T$ of $f$, the equilibrium measure $\mu = T^k$, and the Lyapunov exponents $\lambda\_1 \geq \cdots \geq \lambda\_k$ of $\mu$. The latter are bounded below by $\frac{1}{2} \mathrm{Log} \ d$. Dujardin proved in \cite{Duj12} that if $\mu \ll T^r \wedge \omega\_{\mathbb{P}^k}^{k-r}$ for some $1 \leq r \leq k-1$, then $\lambda\_{r+1} = \cdots = \lambda\_k = \frac{1}{2} \mathrm{Log} \ d$. In this article we prove that, conversely, if $\lambda\_r>\lambda\_{r+1} = \cdots = \lambda\_k = \frac{1}{2} \mathrm{Log} \ d$, then $\mu\ll T^r\wedge\omega\_{\mathbb{P}^k}^{k-r}$, answering a question asked by Dujardin. Our arguments rely on pluripotential theory, ergodic theory, and normal forms for the inverse branches of the endomorphism. We also use normal forms to provide another proof of Dujardin's result.