On the number of periodic trajectories for analytic diffeomorphisms of the circle. (English. Russian original) Zbl 0658.58010
Funct. Anal. Appl. 19, 79-80 (1985); translation from Funkts. Anal. Prilozh. 19, No. 1, 91-92 (1985).
An orientation-preserving diffeomorphism of the circle can be represented in the form \(x\to x+f(x)\), where \(x\in \mathbb R\), \(f(x+2\pi)=f(x)\). Considering the case when \(f(x)\) is a trigonometric polynomial of degree \(n>0\) with real coefficients the author proves that the number of periodic cycles cannot exceed \(2n\). The estimate is sharp for \(x\to x+\epsilon \sin nx\). The problem was raised by V. I. Arnol’d [Usp. Mat. Nauk 38, No. 4(232), 189–203 (1983; Zbl 0541.34035)].
Reviewer: I.N.Baker
MSC:
37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |
37C27 | Periodic orbits of vector fields and flows |
37E10 | Dynamical systems involving maps of the circle |
37F15 | Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems |
30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Citations:
Zbl 0541.34035References:
[1] | V. I. Arnol’d, Usp. Mat. Nauk,38, No. 4, 189-203 (1983). |
[2] | P. Fatou, Bull. Soc. Math. France,47 (1919),48 (1920). |
[3] | P. Montel, Normal Families of Analytic Functions [Russian translation], ONTI, Moscow (1936). |
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