Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:63 https://doi.org/10.1007/s12044-018-0431-2 Set of periods of a subshift K ALI AKBAR1,∗ and V KANNAN2 1Department of Mathematics, Central University of Kerala, Kasaragod 671 316, India 2School of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046, India *Corresponding author. E-mail: [email protected]; [email protected] MS received 31 October 2016; revised 15 June 2017; accepted 12 October 2017; published online 25 October 2018 Abstract. In this article, subsets of N that can arise as sets of periods of the following subshifts are characterized: (i) subshifts of finite type, (ii) transitive subshifts of finite type, (iii) sofic shifts, (iv) transitive sofic shifts, and (v) arbitrary subshifts. Keywords. Subshift of finite type; transitive subshift; strongly connected digraph; sofic shift. 2010 Mathematics Subject Classification. Primary: 37B10; Secondary: 54H20. 1. Introduction and preliminaries The subshifts of finite type (SFT) form a very important class of dynamical systems. There are plenty of books that explain how their study would throw light on still larger classes of dynamical systems (see [7,8,11]). A dynamical system is a pair (X, f ), where X is a metric space and f is a continuous self map. For each dynamical system (X, f ), the set Per( f ) ={n ∈ N :∃x ∈ X such that f n(x) = x = f m(x) ∀m < n} consisting of the lengths of the cycles, is a subset of the set N of positive integers. Given a class of dynamical systems, we obtain in this way a family of subsets of N. There have been attempts to describe explicitly this family of subsets of N, for various classes of dynamical systems (see [2,3,5,6,8,9,12] and [13]). It is therefore natural to ask for a description of sets of periods for the class of all subshifts of finite type. In this paper, we answer this and two other related questions (see Theorems 1 and 3). Here, we concentrate on two-sided shifts. The case of one-sided shifts is similar. Let A be a non-empty finite set, called alphabet with discrete topology and consider the Z set A , which denotes the set of doubly-infinite sequences (xi )i∈Z where each xi ∈ A, with product topology. It is compact and metrizable. The shift is the homeomorphism Z Z Z σ : A → A given by σ(x)i = xi+1 for all i ∈ Z. The pair (A ,σ)forms a dynamical system called a full shift. We say that a point x ∈ AZ is periodic if σ n x = x for some n ∈ N, where σ n is the composition of σ with itself n times. The smallest such positive integer n is called the σ-period (simply period) of x. A subset A ⊂ AZ is called σ -invariant © Indian Academy of Sciences 1 63 Page 2 of 8 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:63 if σ(A) ⊂ A. A subshift is a σ-invariant nonempty closed subset X of a full shift, together with the restriction of σ to X. We denote the set of all periodic points of σ in X by P(X) and the set of periods of all periodic points of σ in X by Per(X).Awordw on A is a concatenation w1w2 ...wk, whenever each wi ∈ A, w¯ is defined by (w)¯ n = wr where Z n ≡ r(mod k). A subshift X is said to be a subshift of finite type if X = XF ={x ∈ A : no word in F appears in x} for some finite set of words F. A subshift is said to be sofic if there exists a continuous surjection from an SFT to it. A subshift (X,σ)is said to be transitive if for any nonempty open sets U, V in X, there exists n ∈ N such that σ n(U) ∩ V =∅. An infinite subshift is said to be Devaney chaotic (simply we say chaotic), if it is transitive and has a dense set of periodic points. Our main results are: (1) period set of an SFT (or sofic shift) is a non empty set of the form FNG, where F and G are finite subsets of N, (2) period set of a transitive SFT (or a transitive sofic shift) is either a singleton subset of N or a set of the form kN \ F for some finite subset F of N and for some k ∈ N, (3) for any S ⊂ N, there exists a subshift X such that Per(X) = S. 2. Main results 2.1 Subshifts of finite type Let G = (V, E) be any directed graph(digraph) with vertex set V and edge set E.A subgraph G = (V , E ) of G is said to be a full subgraph if E = E ∩{(v1,v2) : v1,v2 ∈ V }. A digraph is said to be simple if from every vertex v to a vertex w there is atmost one edge and it is said to be strongly connected if between any two vertices there exists a directed path. It is to be noted that a connected digraph may not be strongly connected. For every subshift of finite type, there is an associated digraph and an associated matrix as described in [7]. This may or may not be strongly connected. But, the period set of a strongly connected simple digraph can be described easily. The subshift of finite type associated with a simple digraph G is denoted as XG . Note that Per(XG ) = Per(XG ), G is a finite union of such strongly connected simple digraphs. A cycle C is said to be simple if XC contains a periodic point whose period is exactly the length of C. For a finite subset A of N, we denote A ⊂⊂ N. PROPOSITION 1 (see [7], p. 58) Every SFT is isomorphic to one in which every forbidden word has length 2. Lemma 1[see 4]. Let a, b ∈ N. (i) If gcd(a, b) = 1, then for every n > ab there exist positive integers x and y such that n = ax + by. (ii) If gcd(a, b) = 1 and n = ab, then there are no positive integers x and y such that n = ax + by. Proof. If gcd(a, b) = 1, there exist positive integers x and y such that ax − by = 1. For every n > ab, write n = m(ab) + k,1≤ k ≤ ab. Hence the lemma follows. Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:63 Page 3 of 8 63 The following remarks easily follows from Lemma 1. Remark 1. If ai ∈ N, i = 1, 2,...,n and gcd(a1,...,an) = 1, then for every positive integer n > a1a2 ···an there exist positive integers, xi , i = 1, 2,...,n such that n = a1x1 + a2x2 +···+an xn. Remark 2. If ai ∈ N, i = 1, 2,...,n and (a1, a2,...,an) = k, then for every n > ... a1a2 an = , ,..., = + + kn there exist positive integers xi , i 1 2 n such that kn a1x1 a2x2 ···+an xn. Theorem 1. The following are equivalent for a subset S of N: (1) S = Per(XG ) for some strongly connected simple digraph G, (2) either S is singleton or S = kN \ Fforsomek∈ N and for some F ⊂⊂ N. Proof. 1 ⇒ 2. Let S be a nonempty, non-singleton subset of N such that S = Per(XG ). Consider k = gcd(S). Claim. kN \ S is finite. Case 1. When k ∈ S.Ifm and l are distinct and are lengths two cycles having one vertex common, then al + bm ∈ S for all a, b ∈ N.IfG is a strongly connected digraph, then for all m ∈ S, there exists l ∈ S such that al + bm ∈ S for all a, b ≥ 1. Let k = gcd(l, m). Then all but finitely many elements of k N are in S and hence mN\ S is finite for all m ∈ S. In particular, kN \ S is finite. Case 2. When k is not in S. Weclaim that there exists a finite subset S of S such that k is equal to the gcd of all its elements. Note that k < s for all s ∈ S.Lets ∈ S. Since k = gcd(S), there exists t1 ∈ S such that k ≤ gcd(t1, s)<s. If gcd(t1, s) = k, then take S ={t1, s}. Else there exists t2 ∈ S such that k ≤ gcd(t1, t2, s)<gcd(t1, s).Ifk = gcd(t1, t2, s), then take S ={t1, t2, s}. After a finite number of steps we will get S ,say{s1, s2,...,sr },as we claimed. Let Ci be a simple cycle having length si and xi ∈ Ci for i = 1, 2,...,r. Then there exists a path from xi(mod r) to xi+1(mod r) which does not contain any cycle. Let C be a simple cycle of length s obtained from the cycles Ci s and the above paths. Then si N + sN ⊂ S for i = 1, 2,...,r. Hence s1N + sN + s2N + sN +···+sr N + sN ⊂ S. ie., s1N + s2N +···+snN + rsN ⊂ S.Buts1N + s2N +···+snN + rsN ⊂ S contains all but finitely many elements of kN. Therefore kN \ S is finite. Hence from Case 1 and Case 2, we have S = kN \ F, where F is the finite set kN \ S. 2 ⇒ 1. Now, let S be a subset of N such that it is either singleton or it is equal to kN \ F for some k ∈ N and F ⊂⊂ N. Case 1. When S ={k} for some k ∈ N. Consider a strongly connected simple digraph G with exactly one cycle of length k.