2020/04/20 by Jianyong Qiao, Qiao, Jianyong, Hongyu Qu +1
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2004.09016
openalex publication_date 2020/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f:(ℂn,0)↦(ℂn,0) be a germ of an n-dimensional holomorphic map. Assume that the origin is an isolated fixed point of each iterate of f. Then \Nq(f)\q=1∞, the sequence of the maximal number of periodic orbits of period q that can be born from the fixed point zero under a small perturbation of f, is well defined. According to Shub-Sullivan, Chow-Mallet-Paret-Yorke and G. Y. Zhang, the linear part of the holomorphic germ f determines some natural restrictions on the sequence(cf. Theorem 1.1). Later, I. Gorbovickis proves that when the linear part of f is contained in a certain large class of diagonal matrices, it has no other restrictions on the sequence only when the dimension n≤2 (cf. Theorem 1.3). In this paper for the general case we obtain a sufficient and necessary condition that the linear part of f has no other restrictions on the sequence \Nq(f)\q=1∞, except the ones given by Theorem 1.1.