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 [Some properties of 2-critically finite holomorphic maps of P2. Ergod. Th. & Dynam. Sys.18(1) (1998), 171–187]. When $n\geq 2$ and the cycle is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one obtained by Fornæss and Sibony [Complex dynamics in higher dimension. II. Modern Methods in Complex Analysis (Princeton, NJ, 1992) (Annals of Mathematics Studies, 137). Ed. T. Bloom, D. W. Catlin, J. P. D’Angelo and Y.-T. Siu, Princeton University Press, 1995, pp. 135–182] under a hyperbolicity assumption on the complement of the post-critical set.

${{\mathbb {CP}}}^n$ 的全纯自同态 是后临界代数的，如果它的临界超曲面是周期的或前周期的。这个概念概括了第一维后临界有限有理映射的概念。我们将研究这种映射沿周期性循环的微分特征值。当 $n=1$ 时，一个众所周知的事实是，沿后临界有限有理映射的周期循环的特征值要么是超吸引的，要么是排斥的。我们证明，当 $n=2$ 时，特征值仍然是超吸引或排斥的。这是对 Mattias Jonsson [P2 的 2-临界有限全纯映射的一些性质。埃尔戈德。钍。&动态。系统。18(1) (1998), 171–187]。当 $n\geq 2$ 并且循环在后临界集之外时，我们证明特征值是排斥的。这一结果改进了由 Fornæss 和 Sibony [高维复杂动力学获得的结果。二、现代复杂分析方法（普林斯顿，新泽西州，1992 年）（数学研究年鉴，137）。埃德。T. Bloom、DW Catlin、JP D'Angelo 和 Y.-T。Siu, Princeton University Press, 1995, pp. 135–182] 在后临界集补集的双曲假设下。