Vol. 12, No. 2, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12
Issue 2, 181–360
Issue 1, 1–180

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Ethics Statement
Editorial Login
Author Index
Coming Soon
Contacts
 
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
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
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.

Keywords
periodic points, shifts of finite type, Sharkovsky's theorem
Mathematical Subject Classification 2010
Primary: 37B10
Secondary: 37E15
Milestones
Received: 27 November 2016
Revised: 16 May 2018
Accepted: 22 July 2018
Published: 8 October 2018

Communicated by Kenneth S. Berenhaut
Authors
Madeline Doering
Department of Mathematics
University of Denver
Denver, CO
United States
Ronnie Pavlov
Department of Mathematics
University of Denver
Denver, CO
United States