##
**A characterization of the sets of periods within shifts of finite type.**
*(English)*
Zbl 1404.37019

Summary: 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.

### MSC:

37B50 | Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) |

37B10 | Symbolic dynamics |

37E15 | Combinatorial dynamics (types of periodic orbits) |

PDF
BibTeX
XML
Cite

\textit{M. Doering} and \textit{R. Pavlov}, Involve 12, No. 2, 203--220 (2019; Zbl 1404.37019)

### References:

[1] | ; Bowen, Global Analysis. Proc. Symp. Pure Math., 14, 43 (1970) |

[2] | 10.1007/978-3-642-14455-4_23 · Zbl 1250.68109 |

[3] | 10.1090/S0894-0347-00-00342-8 · Zbl 0968.15005 |

[4] | 10.1090/S0273-0979-1983-15162-5 · Zbl 0524.58034 |

[5] | 10.1017/CBO9780511626302 · Zbl 1106.37301 |

[6] | ; Sharkovskiĭ, Ukrain. Mat. Zh., 16, 61 (1964) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.