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Periodic points of holomorphic maps via Lefschetz numbers. (English) Zbl 0947.55001
This is a nicely written article in which the authors study the sets of periods of holomorphic maps on complex manifolds using A. Dold’s periodic Lefschetz number [Invent. Math. 74, 419-435 (1983; Zbl 0583.55001)]. If $$f$$ is a self-map of a sufficiently nice space and $$m\in\mathbb{N}$$ the Lefschetz number of period $$m$$ is defined as $$l(f^m)=\sum_{r|m}\mu(r)L(f^{m/r})$$ where $$\mu$$ is the Möbius function and $$L$$ is the ordinary Lefschetz number. Let $$M$$ be a compact complex manifold and $$f:M\to M$$ a nonconstant holomorphic map. The authors show that for sufficiently large primes $$p$$ we have that $$l(f^p)\not=0$$ if and only if $$f$$ has a periodic orbit of period $$p$$. If $$f$$ is transversal then $$l(f^m)$$ equals the number of periodic orbits of period $$m$$ for all $$m$$. In particular, for a holomorphic $$f:M\to M$$ such that all periodic points are isolated there are positive constants $$C$$ and $$\lambda$$ such that the number of fixed points of $$f^m$$ is bounded by $$C\lambda^m$$. As an application, the authors show that a holomorphic map of degree $$d>2$$ on complex projective space $$\mathbb{C}P(n)$$ always has infinitely many periodic orbits and that all sufficiently large primes occur as periods. In fact, for transversal $$f$$ the periods equal the set of positive integers. Finally, the authors show that the theory of periodic Lefschetz numbers can be used to give a proof of I. N. Baker’s theorem [J. Lond. Math. Soc. 39, 615-622 (1964; Zbl 0138.05503)] which completely characterizes the set of periods of a holomorphic map on the Riemann sphere.

##### MSC:
 55M20 Fixed points and coincidences in algebraic topology 58C30 Fixed-point theorems on manifolds 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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##### References:
 [1] V. I. Arnold, Aspects des systèmes dynamiques, Preprint Ecole Polytechnique, Journées X-UPS 1994. Also in Topological Methods in Nonlinear Analysis, Torum, 1994. [2] I. K. Babenko and S. A. Bogatyĭ, Behavior of the index of periodic points under iterations of a mapping, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 1, 3 – 31 (Russian); English transl., Math. USSR-Izv. 38 (1992), no. 1, 1 – 26. [3] I. N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc. 39 (1964), 615 – 622. · Zbl 0138.05503 [4] Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. · Zbl 0742.30002 [5] Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman and Co., Glenview, Ill.-London, 1971. · Zbl 0216.19601 [6] R. Grimshaw, Nonlinear ordinary differential equations, Applied Mathematics and Engineering Science Texts, CRC Press, Boca Raton, FL, 1993. · Zbl 0743.34002 [7] A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), no. 3, 419 – 435. · Zbl 0583.55001 [8] John M. Franks, Homology and dynamical systems, CBMS Regional Conference Series in Mathematics, vol. 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C.; by the American Mathematical Society, Providence, R. I., 1982. · Zbl 0497.58018 [9] D. Fried, Periodic points of holomorphic maps, Topology 25 (1986), 429-441. · Zbl 0633.58028 [10] John Erik Fornæss and Nessim Sibony, Complex dynamics in higher dimension. I, Astérisque 222 (1994), 5, 201 – 231. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). · Zbl 0761.32015 [11] John Guaschi and Jaume Llibre, Periodic points of \?\textonesuperior maps and the asymptotic Lefschetz number, Proceedings of the Conference ”Thirty Years after Sharkovskiĭ’s Theorem: New Perspectives” (Murcia, 1994), 1995, pp. 1369 – 1373. · Zbl 0886.58094 [12] A. Guillamon, X. Jarque, J. Llibre, J. Ortega, and J. Torregrosa, Periods for transversal maps via Lefschetz numbers for periodic points, Trans. Amer. Math. Soc. 347 (1995), no. 12, 4779 – 4806. · Zbl 0846.58045 [13] Antoni A. Kosinski, Differential manifolds, Pure and Applied Mathematics, vol. 138, Academic Press, Inc., Boston, MA, 1993. · Zbl 0767.57001 [14] Solomon Lefschetz, Differential equations: Geometric theory, Second edition. Pure and Applied Mathematics, Vol. VI, Interscience Publishers, a division of John Wiley & Sons, New York-Lond on, 1963. · Zbl 0080.06401 [15] Jaume Llibre, Lefschetz numbers for periodic points, Nielsen theory and dynamical systems (South Hadley, MA, 1992) Contemp. Math., vol. 152, Amer. Math. Soc., Providence, RI, 1993, pp. 215 – 227. · Zbl 0793.58030 [16] Takashi Matsuoka, The number of periodic points of smooth maps, Ergodic Theory Dynam. Systems 9 (1989), no. 1, 153 – 163. · Zbl 0656.58024 [17] Ivan Niven and Herbert S. Zuckerman, An introduction to the theory of numbers, 4th ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. · Zbl 0431.10001 [18] Norbert Steinmetz, Rational iteration, De Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1993. Complex analytic dynamical systems. · Zbl 0773.58010 [19] M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189 – 191. · Zbl 0291.58014 [20] James W. Vick, Homology theory, 2nd ed., Graduate Texts in Mathematics, vol. 145, Springer-Verlag, New York, 1994. An introduction to algebraic topology. · Zbl 0789.55004 [21] Joachim Wehler, Versal deformation of Hopf surfaces, J. Reine Angew. Math. 328 (1981), 22 – 32. · Zbl 0459.32009 [22] Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. · Zbl 0265.32008
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