On chain recurrence classes of endomorphisms of $\mathbb P^k$

TL;DR

This paper investigates the structure of chain recurrence classes in holomorphic endomorphisms of complex projective space, revealing finiteness properties and topological constraints derived from current actions.

Contribution

It establishes that minimal chain recurrence classes have finitely many connected components and provides new insights into the topological dynamics via current actions.

Findings

Minimal chain recurrence classes have finitely many connected components.

Results extend to arbitrary classes, imposing topological constraints.

Dynamics are analyzed through the action on a space of currents.

Abstract

We prove that the minimal chain recurrence classes of a holomorphic endomorphism of $\mathbb P^k$ have finitely many connected components. We also obtain results on arbitrary classes. These strong constraints on the topological dynamics in the phase space are all deduced from the associated action on a space of currents.