Periodic points of weakly post-critically finite all the way down maps

Abstract

We study eigenvalues along periodic cycles of post-critically finite endomorphisms of $${\mathbb{CP}\mathbb{}}^n$$ in higher dimension. It is a classical result when $$n = 1$$ that those values are either 0 or of modulus strictly bigger than 1. It has been conjectured in [Van Tu Le. Periodic points of post-critically algebraic holomorphic endomorphisms. Le (Ergodic Theory Dyn Syst 1–33, 2020)] that the same result holds for every $$n \ge 2$$ . In this article, we verify the conjecture for the class of weakly post-critically finite all the way down maps which was introduced in Astorg (Ergodic Theory Dyn Syst, 40(2):289–308, 2020). This class contains a well-known class of post-critically finite maps constructed in [Sarah Koch. Teichmüller theory and critically finite endomorphisms. Koch (Adv Math 248:573–617, 2013)]. As a consequence, we verify the conjecture for Koch maps.