A remark on tame dynamics in compact complex manifolds. (English) Zbl 1161.32007

This article deals with the dynamics of holomorphic self-maps of compact complex manifolds \(f : M \to M\). Assume that there is at least one subsequence of \((f^n)_n\) which converges uniformly on \(M\). Then the number of possible (exact) periods for periodic points is finite, and the set of periodic points is a complex submanifold of \(M\). Moreover, if that set is discrete, then its cardinal coincides with the Euler characteristic of \(M\).


32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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[1] M. Abate, Iteration theory of holomorphic maps on taut manifolds , Mediterranean, Rende, 1989. · Zbl 0747.32002
[2] S. Bochner and D. Montgomery, Groups on analytic manifolds, Ann. of Math. (2) 48 (1947), 659-669. · Zbl 0030.07501
[3] P. Griffiths and J. Harris, Principles of algebraic geometry , Reprint of the 1978 original, Wiley, New York, 1994. · Zbl 0836.14001
[4] K. Maegawa, On Fatou maps into compact complex manifolds, Ergodic Theory Dynam. Systems 25 (2005), no. 5, 1551-1560. · Zbl 1110.37039
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