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

We study eigenvalues along periodic cycles of post-critically finite endomorphisms of $\mathbb{CP}^n$ 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, Ergodic Theory and Dynamical Systems, pages 1-33, 2020] that the same result holds for every $n \geq 2$. In this article, we verify the conjecture for the class of weakly post-critically finite all the way down maps which was introduced in [Matthieu Astorg, Dynamics of post-critically finite maps in higher dimension, Ergodic Theory and Dynamical Systems, 40(2):289-308, 2020]. This class contains a well-known class of post-critically finite maps constructed in [Sarah Koch, Teichm\"uller theory and critically finite endomorphisms, Advances in Mathematics, 248:573-617, 2013]. As a consequence, we verify the conjecture for Koch maps.