The author gives a clear and systematic answer to certain basic questions in a study of surface homomorphisms. He considers an isolated fix point of a homomorphism and its Lefschetz index. In this setting, the author proves analogues of classical results from complex and hyperbolic dynamics. In particular, if the index n of the fixed point is bigger than one, he proves an analogue of the Leau-Fatou flower theorem (i.e., he is giving a construction of a family of p attractive and p repulsive petals, where p=n-1). If the index n of the fixed point is smaller than one, he describes an analogue of the stable manifold theorem (i.e., he constructs a family of p stable and p unstable local branches, where p=1-n). He also gives a new proof of the fact (due to M. Brown) that the index of a fixed point x is the same for a homeomorphism f and for its power f n. The methods used are coming from a study of global dynamics on the sphere.