Fixed points and Determining Sets for Holomorphic Self-Maps of a Hyperbolic Manifold, preprint

Abstract. We study fixed point sets for holomorphic automor-phisms (and endomorphisms) on complex manifolds. The main object of our interest is to determine the number and configura-tion of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a num-ber of papers for a bounded domain in Cn. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different. Let M be a complex manifold. H(M,M) is the set of holomorphic maps fromM toM, i.e., the set of endomorphisms ofM. A special case of endomorphisms are automorphisms of M, Aut(M) ⊂ H(M,M). Definition 0.1. A set K ⊂M is called a determining subset ofM wit