Periodic points of post-critically algebraic holomorphic endomorphisms

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