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Periodic points of a map of a system of intervals. (English. Russian original) Zbl 0658.58031

Math. Notes 43, No. 3, 210-219 (1988); translation from Mat. Zametki 43, No. 3, 365-381 (1988).
The authors prove the following main result. There exists a function \(g(n)\), \(n\) natural, having the following property. If \(I_ 1,...,I_ n\) are closed intervals in \({\mathbb R}\) and if \(f\) is a continuous map \({\mathbb R}\to {\mathbb R}\) such that \(f(I_ 1\cup...\cup I_ n)\supset I_ 1\cup...\cup I_ n,\) then there exists a periodic point of \(f\) in \(I_ 1\cup...\cup I_ n\) having period \(\leq g(n)\). If \(n\leq 4\) then \(g(n)=n\). The authors also discuss the multidimensional case.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

Keywords:

periodic point
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References:

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