  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.
