A characterization of the sets of periods within shifts of finite type

Abstract

In this paper, 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 iff it is either {1} or cofinite, and the set of periods of some mixing SFT iff 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 iff 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 iff it can be written as the finite union of the sets of (least) periods for some irreducible SFTs. Sharkovsky's Theorem ([6]). For any interval I in R, if f : I → I is continuous and has a point of least period k, then there exist points of all least periods less than k in the Sharkovsky ordering, where the ordering is as follows: