# zbMATH — the first resource for mathematics

Fixed point indices and periodic points of holomorphic mappings. (English) Zbl 1110.32007
Let $$f$$ be a germ of holomorphic self-maps of $$\mathbb C^n$$ fixing the origin $$O$$. If $$O$$ is an isolated fixed point for $$f$$ then one can define the A. Dold index [Invent. Math. 74, No. 3, 419–435 (1983; Zbl 0583.55001)] $$P_M(f,O)$$ for each integer $$M$$ such that $$O$$ is an isolated fixed point for $$f^{\circ M}$$. If $$g$$ is a holomorphic map close to $$f$$ then it has $$P_M(f,O)$$ (counting multiplicity) periodic points of period $$M$$ near $$O$$.
In this paper the author proves that $$P_M(f,O)>0$$ if and only if $$df_O$$ has a periodic point of period $$M$$.
The necessity of the condition on $$df_O$$ has already been proved for smooth maps by M. Shub and D. Sullivan [Topology 13, 189–191 (1974; Zbl 0291.58014)], but the condition is known to be not sufficient if the map is not holomorphic.
The strategy of the proof is as follows. The author explicitly computes the Dold index in some special cases (proving the sufficiency of the condition on $$df_O$$ in such cases). Then show that the general case can be reduced to the previous one by means of small perturbations and changes of coordinates. He also reproves the necessity of the condition in the holomorphic setting.

##### MSC:
 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
Full Text:
##### References:
 [1] Arrowsmith D.K., Place C.M. (1990). An introduction to Dynamical Systems. Cambridge University Press, Cambridge · Zbl 0702.58002 [2] Buzzard G.T. (1998). Kupka–Smale theorem for automorphisms of $$\mathbb{C}^{n}$$ . Duke Math. J. 93(3): 487–503 (MR 99g:32046) · Zbl 0946.32012 [3] Chow, S-N., Mallet-Paret, J., Yorke, J.A.: A periodic orbit index which is a bifurcation invariant. Geometric dynamics (Rio de Janeiro, 1981). Lecture Notes in Mathematics, vol. 1007, pp. 109–131 Springer, Berlin Heidelberg New York (1983) (MR 85d:58058) [4] Chirka, E.M.: Complex analytic sets. Translated from the Russian by R.A.M. Hoksbergen, Mathematics and its Applications (Soviet Series), vol. 46. Kluwer Dordrecht (1989) · Zbl 0683.32002 [5] Cronin J. (1953). Analytic functional mappings. Ann. Math. 58(2): 175–181 (MR 15,234a) · Zbl 0050.34301 [6] Dold A. (1983). Fixed point indices of iterated maps. Invent. Math. 74(3):419–435 (MR 85c:54077) · Zbl 0583.55001 [7] Fagella N., Llibre J. (2000). Periodic points of holomorphic maps via Lefschetz numbers. Trans. Am. Math. Soc. 352(10):4711–4730 (MR2001b:55003) · Zbl 0947.55001 [8] Hardy G.H., Wright W.M. (1979). An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford · Zbl 0423.10001 [9] Lloyd, N.G.: Degree theory, Cambridge Tracts in Mathematics, No. 73. Cambridge University Press, Cambridge-New York-Melbourne (1978) (MR 58 #12558) [10] Milnor J. (1999). Dynamics in One Complex Variable: Introductory Lectures. Friedr. Vieweg & Sohn, Braunschweig · Zbl 0946.30013 [11] Shub M., Sullivan D. (1974). A remark on the Lefschetz fixed point formula for differentiable maps. Topology 13:189–191 (MR 50 #3274) · Zbl 0291.58014 [12] Steinlein H. (1972). Ein Satz über den Leray-Schauderschen Abbildungsgrad (German). Math. Z. 126: 176–208 (MR 47#5667) · Zbl 0223.47023 [13] Zabreĭko, P.P., Krasnosel’skiĭ, M.A.: Iterations of operators, and fixed points (Russian). Dokl. Akad. Nauk SSSR 196, 1006–1009 (1971); Soviet Math. Dokl. 12, 294–298 (1971). (MR 47 #4082) [14] Zhang G.Y. (1999). Bifurcations of periodic points of holomorphic maps from $$\mathbb{C}^{2}$$ into $$\mathbb{C}^{2}$$ . Proc. London Math. Soc. 79(2):353–380 (MR 2000f:32027) · Zbl 0974.32011 [15] Zhang G.Y. (2001). Multiplicities of fixed points of holomorphic maps in several complex variables. Sci. China Ser. A 44(3):330–337 (MR 2002b:37026) · Zbl 1019.37010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.