Dynamics of post-critically finite maps in higher dimension

TL;DR

This paper investigates the dynamics of post-critically finite endomorphisms in higher-dimensional complex projective spaces, establishing properties of their post-critical sets, periodic points, and Fatou components under certain conditions.

Contribution

It extends known results from dimension two to higher dimensions, showing that under mild conditions, post-critically finite maps have specific dynamical behaviors and Fatou components.

Findings

Post-critically finite endomorphisms are always post-critically finite down to lower dimensions.

Eigenvalues of periodic points are characterized under these maps.

Only super-attracting basins can be Fatou components under certain hyperbolicity conditions.

Abstract

We study the dynamics of post-critically finite endomorphisms of P^k(C). We prove that post-critically finite endomorphisms are always post-critically finite all the way down under a mild regularity condition on the post-critical set. We study the eigenvalues of periodic points of post-critically finite endomorphisms. Then, under a weak transversality condition and assuming Kobayashi hyperbolicity of the complement of the post-critical set, we prove that the only possible Fatou components are super-attracting basins, thus partially extending to any dimension a result of Fornaess-Sibony and Rong holding in the case k = 2.