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arXiv · 1906.04097

Periodic points of post-critically algebraic endomorphisms

Van Tu Le
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Abstract

A holomorphic endomorphism of $\mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map along a periodic cycle. When $n=1$, a well-known fact is that the eigenvalue along a periodic cycle of a post-critically finite rational map is either superattracting or repelling. We prove that when $n=2$ the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson. When $n\geq 2$ and the cycle is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one which was already obtained by Fornaess and Sibony under a hyperbolicity assumption on the complement of the post-critical set.

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Van Tu Le. 2019-06-10. Periodic points of post-critically algebraic endomorphisms. https://doi.org/10.1017/etds.2021.48

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