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.
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Acknowledgements
The author would like to thank Valentin Huguin for his comments on an early version of this article and for communicating Lemma 2.3. The author would also like to thank his advisors, Xavier Buff and Jasmin Raissy, for introducing him the subject. The author was supported by MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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Le, V.T. Periodic points of weakly post-critically finite all the way down maps. Annali di Matematica 202, 1187–1195 (2023). https://doi.org/10.1007/s10231-022-01275-x
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DOI: https://doi.org/10.1007/s10231-022-01275-x