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...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Author: Van Tu Le
Format: Article
Language:English
Subjects:
Dynamical systems
Eigenvalues
Ergodic processes
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Summary: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.
ISSN:2331-8422