FIXED POINTS OF POST-CRITICALLY ALGEBRAIC HOLOMORPHIC ENDOMORPHISMS IN HIGHER DIMENSION

27 pages, 4 figuresA holomorphic endomorphism of CP n is post-critically algebraic if its critical hyper-surfaces 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 at its fixed points. When n = 1, a well-known fact is that the eigenvalue at a fixed point is either superattracting or repelling. We prove that when n = 2 the eigenvalues are still either superat-tracting or repelling. This is an improvement of a result by Mattias Jonsson. When n ≥ 2 and the fixed point 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