Abstract
Let Δn be the ball |x| < 1 in the complex vector space ℂn, let f: Δn → ℂn be a holomorphic mapping and let M be a positive integer. Assume that the origin 0 = (0, ..., 0) is an isolated fixed point of both f and the M-th iteration f M of f. Then the (local) Dold index P M (f, 0) at the origin is well defined, which can be interpreted to be the number of virtual periodic points of period M of f hidden at the origin: any holomorphic mapping f 1: Δn → ℂn sufficiently close to f has exactly P M (f, 0) distinct periodic points of period M near the origin, provided that all the fixed points of f M1 near the origin are simple. Therefore, the number O M (f, 0) = P M (f, 0)/M can be understood to be the number of virtual periodic orbits of period M hidden at the fixed point.
According to the works of Shub-Sullivan and Chow-Mallet-Paret-Yorke, a necessary condition so that there exists at least one virtual periodic orbit of period M hidden at the fixed point, i.e., O M (f, 0) ⩾ 1, is that the linear part of f at the origin has a periodic point of period M. It is proved by the author recently that the converse holds true.
In this paper, we will study the condition for the linear part of f at 0 so that O M (f, 0) ⩾ 2. For a 2 × 2 matrix A that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for A, such that O M (f, 0) ⩾ 2 for all holomorphic mappings f: Δ2 → ℂ2 such that f(0) = 0, Df(0) = A and that the origin 0 is an isolated fixed point of f M.
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Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday
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Zhang, G. The numbers of periodic orbits hidden at fixed points of 2-dimensional holomorphic mappings. Sci. China Math. 53, 863–886 (2010). https://doi.org/10.1007/s11425-010-0030-x
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DOI: https://doi.org/10.1007/s11425-010-0030-x