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The behavior of the index of periodic points under iterations of a mapping. (English. Russian original) Zbl 0742.58027
Math. USSR, Izv. 38, No. 1, 1-26 (1992); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 1, 3-31 (1991).
In this paper, the strengthening of A. Dold’s theorem on the algebraic properties of Lefschetz numbers of fixed points of iterated maps is given. This result is obtained on the basis of detailed investigation of the arithmetical and analytical properties of the Lefschetz $$\zeta$$- function. For $$C^ 1$$-mapping $$f: M\to M$$ of the smooth manifold $$M$$ into itself, the lower estimates for the number of periodic orbits of $$f$$, with period $$\leq k$$, is derived. At the end of the paper, the bifurcations of the periodic points and periodic trajectories of a vector field, are discussed. To this aim the useful bifurcation function is defined. This important paper is well written and well organized.
Reviewer: A.Klíč (Praha)

##### MSC:
 37B99 Topological dynamics 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 37G99 Local and nonlocal bifurcation theory for dynamical systems
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