Translator Disclaimer
2019 A characterization of the sets of periods within shifts of finite type
Madeline Doering, Ronnie Pavlov
Involve 12(2): 203-220 (2019). DOI: 10.2140/involve.2019.12.203

Abstract

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 {1} 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.

Citation

Download Citation

Madeline Doering. Ronnie Pavlov. "A characterization of the sets of periods within shifts of finite type." Involve 12 (2) 203 - 220, 2019. https://doi.org/10.2140/involve.2019.12.203

Information

Received: 27 November 2016; Revised: 16 May 2018; Accepted: 22 July 2018; Published: 2019
First available in Project Euclid: 25 October 2018

zbMATH: 06980498
MathSciNet: MR3864214
Digital Object Identifier: 10.2140/involve.2019.12.203

Subjects:
Primary: 37B10
Secondary: 37E15

Keywords: Periodic points , Sharkovsky's theorem , shifts of finite type

Rights: Copyright © 2019 Mathematical Sciences Publishers

JOURNAL ARTICLE
18 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.12 • No. 2 • 2019
MSP
Back to Top