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A
characterization of the sets of periods within shifts of
finite type
Madeline Doering and Ronnie Pavlov
Vol. 12 (2019), No. 2, 203–220
DOI: 10.2140/involve.2019.12.203
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Abstract |
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We characterize precisely the possible sets of periods and least periods for the periodic points of a shift of finite type (SFT). We prove that a set is the set of least periods of some mixing SFT if and only if it is either or cofinite, and the set of periods of some mixing SFT if and only if it is cofinite and closed under multiplication by arbitrary natural numbers. We then use these results to derive similar characterizations for the class of irreducible SFTs and the class of all SFTs. Specifically, a set is the set of (least) periods for some irreducible SFT if and only if it can be written as a natural number times the set of (least) periods for some mixing SFT, and a set is the set of (least) periods for an SFT if and only if it can be written as the finite union of the sets of (least) periods for some irreducible SFTs. Finally, we prove that the possible sets of (least) periods of mixing sofic shifts are exactly the same as for mixing SFTs, and that the same is not true for the class of nonmixing sofic shifts. |
Keywords
periodic points, shifts of finite type, Sharkovsky's
theorem
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Mathematical Subject Classification 2010
Primary: 37B10
Secondary: 37E15
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Milestones
Received: 27 November 2016
Revised: 16 May 2018
Accepted: 22 July 2018
Published: 8 October 2018
Communicated by Kenneth S. Berenhaut
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Authors |
| Madeline Doering | |
| Department of Mathematics University of Denver Denver, CO United States |
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| Ronnie Pavlov | |
| Department of Mathematics University of Denver Denver, CO United States |
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