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Periodic points for piecewise monotonic transformations. (English) Zbl 0572.54036

The paper investigates periodic points of piecewise monotonic transformations T on an interval X. The main tool is an isomorphism of (X,T) with a topological Markov chain which is described by a finite or countable transition matrix M. In the first part it is shown that the inverse of the zeta-function can be considered as a kind of characteristic polynomial of M, which is a power series, if M is infinite. Further results in this direction can be found in the paper by the author and G. Keller [J. Reine Angew. Math. 352, 100-113 (1984; Zbl 0533.28011)]. In the second part of the paper the sets are determined which occur as \(\{\) \(n\in {\mathbb{N}}:\) there is an \(x\in X\) with \(T^ n(x)=x\) and \(T^ i(x)\neq x\) for \(0<x<n\}\), where T is a monotonic mod one transformation. Using different methods, this problem is also solved by M. Misiurewicz in Rotation interval for a class of maps of the real line into itself. Institut Mittag-Leffler (1984).

MSC:

54H20 Topological dynamics (MSC2010)
28D05 Measure-preserving transformations

Citations:

Zbl 0533.28011
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References:

[1] DOI: 10.1007/BF01394248 · Zbl 0475.58014
[2] Hofbauer, Studia Math. none pp none– (none)
[3] Hofbauer, Erg. Th. & Dynam. Sys. 1 pp 159– (1981)
[4] DOI: 10.1007/BFb0086977
[5] DOI: 10.1007/BF00538893 · Zbl 0415.28018
[6] DOI: 10.1007/BF02760884 · Zbl 0422.28015
[7] DOI: 10.1112/jlms/s2-23.1.92 · Zbl 0431.54025
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