The numbers of periodic orbits hidden at fixed points of holomorphic maps

Let f be an n-dimensional holomorphic map defined in a neighborhood of the origin such that the origin is an isolated fixed point of all of its iterates, and let N-M (f ) denote the number of periodic orbits of f of period M hidden at the origin. Gorbovickis gives an efficient way of computing N-M (f ) for a large class of holomorphic maps. Inspired by Gorbovickis' work, we establish a similar method for computing N-M (f ) for a much larger class of holomorphic germs, in particular, having arbitrary Jordan matrices as their linear parts. Moreover, we also give another proof of the result of Gorbovickis [On multi-dimensional Fatou bifurcation. Bull. Sci. Math. 138(3)(2014) 356-375] using our method.