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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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