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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.

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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