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

We study the dynamical properties of endomorphisms $f$ of ${\mathbb{P}}^k$ of algebraic degree $d\ge 2$. We investigate the relationships between the Green current $T$ of $f$, the equilibrium measure $\mu =T^k$, and the Lyapunov exponents $\lambda \mathrm{\_}1\ge \cdots \ge \lambda \mathrm{\_}k$ of $\mu$. The latter are bounded below by $\frac{1}{2}\mathrm{L}\mathrm{o}\mathrm{g}d$. Dujardin proved in \cite{Duj12} that if $\mu \ll T^r\wedge \omega \mathrm{\_}{{\mathbb{P}}^k}^{k-r}$ for some $1\le r\le k-1$, then $\lambda \mathrm{\_}r+1=\cdots =\lambda \mathrm{\_}k=\frac{1}{2}\mathrm{L}\mathrm{o}\mathrm{g}d$. In this article we prove that, conversely, if $\lambda \mathrm{\_}r>\lambda \mathrm{\_}r+1=\cdots =\lambda \mathrm{\_}k=\frac{1}{2}\mathrm{L}\mathrm{o}\mathrm{g}d$, then $\mu \ll T^r\wedge \omega \mathrm{\_}{{\mathbb{P}}^k}^{k-r}$, 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.