On some problems in the theory of dynamical systems
The paper is a very interesting collection of a great variety of open problems in the theory of dynamical systems. These problems originate from different branches of natural sciences and mathematics. For example from physics -- like the second problem which deals with pseudoperiodic topology; from celestial mechanics -- like the seventh which handles with problems on the representation of resonance in analytic dynamical systems: which for example takes place in Siegel's theorem on rotation numbers in complex analytic dynamics on the corresponding domains nowadays usually called Siegel discs; or in mathematics -- like the sixth on some variation of Hilbert's 16th problem and abelian integrals. The paragraphs 3 until 5 are dealing mostly with questions on counting the number of periodic points and limit cycles in different kinds of dynamical systems. So -- for example -- in \S 3 the author considers analytic diffeomorphisms \(A: M\to M\) where \(M\) is a compact analytic manifold. A natural question is that on the number of periodic points, especially whether the number of periodic points of period \(n\) is majorized by some exponential function of the period. (For a special case such a bound has been given by \textit{M. Artin} and \textit{B. Mazur} [Ann. Math., II. Ser. 81, 82-99 (1965; Zbl 0127.13401)]).
scientific article; zbMATH DE number 828282
scientific article