Glider automorphisms and a finitary Ryan’s theorem for transitive subshifts of finite type

For any mixing SFT X we construct a reversible shift-commuting continuous map (automorphism) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. As an application we prove a finitary Ryan’s theorem: the automorphism group Aut(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Aut}\,}}(X)$$\end{document} contains a two-element subset S whose centralizer consists only of shift maps. We also give an example which shows that a stronger finitary variant of Ryan’s theorem does not hold even for the binary full shift.


Introduction
Let X A Z be a one-dimensional subshift over a symbol set A. If w is a finite word over A, we may say that an element x 2 X is w-finite if it begins and ends with infinite repetitions of w. In this paper we consider the problem of constructing reversible shift-commuting continuous maps (automorphisms) on X which decompose all w-finite configurations into collections of gliders traveling into opposing directions. As a concrete example, consider the binary full shift X ¼ f0; 1g Z and the map g ¼ g 3 g 2 g 1 : X ! X defined as follows. In any x 2 X, g 1 replaces every occurrence of 0010 by 0110 and vice versa, g 2 replaces every occurrence of 0100 by 0110 and vice versa, and g 3 replaces every occurrence of 00101 by 00111 and vice versa. In Fig. 1 we have plotted the sequences x; gðxÞ; g 2 ðxÞ; . . . on consecutive rows for some 0-finite x 2 X. It can be seen that the sequence x eventually diffuses into two different ''fleets'', the one consisting of 1 s going to the left and the one consisting of 11 s going to the right.
It can be proved, along similar lines as in the proofs of Lemmas 6 and 7, that this diffusion happens eventually no matter which finite initial point x 2 X is chosen. In Sect. 4 we construct, on all nontrivial mixing SFTs, a function that we call a diffusive glider automorphism and that has the same diffusion property as the binary automorphism g above.
The existence of such a diffusive glider automorphism g on a subshift X is interesting, because g can be used to convert an arbitrary finite x 2 X into another sequence g t ðxÞ (for some t 2 N þ ) with a simpler structure, which nevertheless contains all the information concerning the original point x because g is invertible. Such maps have been successfully applied to other problems. We give some examples. The paper (Salo 2017) contains a construction of a finitely generated group G of automorphisms of A Z (when jAj ¼ 4) whose elements can implement any permutation on any finite collection of 0-finite non-constant configurations that belong to different shift orbits. An essential part of the construction is that one of the generators of G is a diffusive glider automorphism on A Z . Another example is the construction of a physically universal cellular automaton g on A Z (when jAj ¼ 16) in Salo and Törmä (2017). Also here it is essential that g is a diffusive glider automorphism (but g also implements certain additional collision rules for gliders).
We also consider a finitary version of Ryan's theorem. Let X be a mixing SFT and denote the set of its automorphisms by Aut ðXÞ, which we may consider as an abstract group. According to Ryan's theorem (Boyle et al. 1988;Patrick Ryan 1972) the center of the group Aut ðXÞ is generated by the shift map r. There may also be subsets S Aut ðXÞ whose centralizers C(S) are generated by r. Denote the minimal cardinality of such a finite set S by k(X). In Salo (2017) it was proved that kðXÞ 10 when X is the full shift over the four-letter alphabet. In the same paper it is noted that k(X) is an isomorphism invariant of Aut ðXÞ and therefore computing it could theoretically separate Aut ðXÞ and Aut ðYÞ for some mixing SFTs X and Y. Finding good isomorphism invariants of Aut ðXÞ is of great interest, and it is an open problem whether for example Aut ðf0; 1g Z Þ ffi Aut ðf0; 1; 2g Z Þ [Problem 22.1 in Boyle (2008)]. We show that kðXÞ ¼ 2 for all nontrivial mixing SFTs, the proof of which uses our diffusive glider automorphism construction and Lemma 1 (Main lemma). It is then a simple corollary that kðXÞ ¼ 2 for every transitive SFT X that does not consist of the orbit of a single periodic point. The diffusive glider automorphism construction and the proof of kðXÞ ¼ 2 was done for mixing SFTs containing a fixed point in the paper (Kopra 2018) published in the proceedings of AUTOMATA 2018.
Lemma 1 is a criterion saying essentially that if S is a collection of automorphisms that acts together with the diffusive glider automorphism g in a special way, then CðS [ fggÞ is not very complicated. We have formulated a reasonably general version of the lemma to allow its application in other contexts. To further showcase our Main lemma, we consider an alternative finitary variant of Ryan's theorem. In Sect. 7.3 of Salo (2017) the question was raised whether for a mixing SFT X and for every G Aut ðXÞ such that CðGÞ ¼ r h i there is a finite subset S G such that also CðSÞ ¼ r h i. In the same section it was noted that to construct a counterexample it would be sufficient to find a locally finite group G Aut ðXÞ whose centralizer is generated by r. We use a different strategy based on Lemma 1 to construct a counterexample in the case when X is the binary full shift.

Preliminaries
A finite set A containing at least two elements (letters) is called an alphabet and the set A Z of bi-infinite sequences (configurations) over A is called a full shift. Formally any x 2 A Z is a function Z ! A and the value of x at i 2 Z is denoted by x[i]. It contains finite and one-directionally infinite subsequences denoted by x½i; j ¼ x½ix½i þ 1 Á Á Á x½j, x½i; 1 ¼ x½ix½i þ 1 Á Á Á and x½À1; i ¼ Á Á Á x½i À 1x½i. Occasionally we signify the symbol at position zero in a configuration x by a dot as follows: where i; j 2 Z, and we interpret the sequence to be empty if j\i. Any finite sequence w ¼ w½1w½2 Á Á Á w½n (also the empty sequence, which is denoted by k) where w½i 2 A is a word over A. The concatenation of a word or a left-infinite sequence u with a word or a right-infinite sequence v is denoted by uv. A word u is a prefix of a word or a rightinfinite sequence x if there is a word or a right-infinite sequence v such that x ¼ uv. Similarly, u is a suffix of a word or a left-infinite sequence x if there is a word or a leftinfinite sequence v such that x ¼ vu. The set of all words over A is denoted by A Ã , and the set of non-empty words is A þ ¼ A Ã nfkg. More generally, for any L A Ã , let i.e. L Ã is the set of all finite concatenations of elements of L. The set of words of length n is denoted by A n . For a word w 2 A Ã , jwj denotes its length, i.e. jwj ¼ n () w 2 A n . Given x 2 A Z and w 2 A þ we define the sets of left (resp. right) occurrences of w in x by occ ' ðx; wÞ ¼ fi 2 Z j x½i; i þ jwj À 1 ¼ wg (resp.) occ r ðx; wÞ ¼ fi 2 Z j x½i À jwj þ 1; i ¼ wg: Note that both of these sets contain the same information up to a shift in the sense that occ r ðx; wÞ ¼ occ ' ðx; wÞþ jwj À 1. Typically we refer to the left occurrences and we say that w 2 A n occurs in x 2 A Z at position i if i 2 occ ' ðx; wÞ. We define the shift map r A : The subscript A in r A is typically omitted. The set A Z is endowed with the product topology (with respect to the discrete topology on A), under which r is a homeomorphism on A Z . For any S A Z the collection of words appearing as factors of elements of S is the language of S, denoted by L(S). Any closed set X A Z such that rðXÞ ¼ X is called a subshift. The restriction of r to X may be denoted by r X , but typically the subscript X is omitted. The orbit of a point x 2 X is OðxÞ ¼ fr i ðxÞ j i 2 Zg. Fig. 1 The diffusion of x 2 X under the map g : X ! X. White and black squares correspond to digits 0 and 1 respectively For any word w 2 A þ we denote by 1 w and w 1 the leftand right-infinite sequences obtained by infinite repetitions of the word w. We denote by w Z 2 A Z the configuration defined by w Z ½in; ði þ 1Þn À 1 ¼ w (where n ¼ jwj) for every i 2 Z. We say that x 2 A Z is w-finite if x½À1; i ¼ 1 w and x½j; 1 ¼ w 1 for some i; j 2 Z.
We say that subshifts X A Z and Y B Z are conjugate if there is a continuous bijection (a conjugacy) w : X ! Y such that w r X ¼ r Y w.
Definition 1 A (directed) graph is a pair G ¼ ðV; EÞ where V is a finite set of vertices (or nodes or states) and E is a finite set of edges or arrows. Each edge e 2 E starts at an initial state denoted by iðeÞ 2 V and ends at a terminal state denoted by sðeÞ 2 V. We say that e 2 E is an outgoing edge of iðeÞ and an incoming edge of sðeÞ. For a state s 2 V, E s denotes the set of outgoing edges of s and E s denotes the set of incoming edges of s.
Although the notation for the set E s of incoming edges of s is similar to the notation for the set E n of words of length n over E, in practice the distinction should be clear from the context.
A sequence of edges e½1 Á Á Á e½n in a graph G ¼ ðV; EÞ is a path (of length n) if sðe½iÞ ¼ iðe½i þ 1Þ for 1 i\n, it is a cycle if in addition sðe½nÞ ¼ iðe½1Þ and it is a simple cycle if in addition iðe½iÞ for 1 i n are all distinct. We say that the path starts at iðe½1Þ and ends at sðe½nÞ. A graph G is irreducible if for every v 1 ; v 2 2 V there is a path starting at v 1 and ending at v 2 and it is primitive if there is n 2 N þ such that for every v 1 ; v 2 2 V there is a path of length n starting at v 1 and ending at v 2 . For any graph G ¼ ðV; EÞ we call the set fx 2 E Z j sðx½iÞ ¼ iðx½i þ 1Þ for all i 2 Zg (i.e. the set of bi-infinite paths on G) the edge subshift of G.
Definition 2 A subshift X A Z is a subshift of finite type (SFT) if it is conjugate to some edge subshift. It is a transitive SFT if it is conjugate to the edge subshift of an irreducible graph G ¼ ðV; EÞ. It is a mixing SFT if G is primitive and it is a nontrivial mixing SFT if G contains at least two edges. We will mostly consider an SFT X as being equal to an edge subshift instead of just being conjugate (in which case E A).
Example 1 Let A ¼ f0; a; bg. The graph in Fig. 2 defines a mixing SFT X also known as the golden mean shift. A typical point of X looks like Á Á Á 000abab0ab00ab000 Á Á Á i.e. the letter b cannot occur immediately after 0 or b and every occurrence of a is followed by b.
Definition 3 An automorphism of a subshift X A Z is a continuous bijection f : X ! X such that r f ¼ f r. We say that f is a radius-r automorphism if f ðxÞ½0 ¼ f ðyÞ½0 for all x; y 2 X such that x½Àr; r ¼ y½Àr; r (such r always exists by continuity of f). The set of all automorphisms of X is a group denoted by Aut ðXÞ. (In the case X ¼ A Z automorphisms are also known as reversible cellular automata.) The centralizer of a set S Aut ðXÞ is CðSÞ ¼ ff 2 Aut ðXÞ j f g ¼ g f for every g 2 Sg and the subgroup generated by S Aut ðXÞ is denoted by S h i. The following definition is from Salo (2017): Definition 4 For a subshift X, let kðXÞ 2 N [ f1; ?g be the minimal cardinality of a set S Aut ðXÞ such that CðSÞ ¼ r h i if such a set S exists, and kðXÞ ¼? otherwise.
It is proven in Patrick Ryan (1972) and as Theorem 7.7 in Boyle et al. (1988) that kðXÞ 6 ¼? whenever X is a mixing SFT. The following observation is from Sect. 7.6 of Salo (2017).
Theorem 1 Let X be a subshift. The case kðXÞ ¼ 0 occurs if and only if Aut ðXÞ ¼ r h i. The case kðXÞ ¼ 1 cannot occur.
The statement kðXÞ ¼ 1 means that Cðff gÞ ¼ r h i for some f 2 Aut ðXÞ. Because f commutes with itself, it follows that f ¼ r i for some i 2 Z. But all g 2 Aut ðXÞ commute with r i and so Aut ðXÞ ¼ Cðff gÞ ¼ r h i and kðXÞ ¼ 0, a contradiction. h For conjugate subshifts X and Y it necessarily holds that kðXÞ ¼ kðYÞ.

Main lemma
In this section we prove as our main lemma a useful criterion which can be used to significantly restrict the kinds of automorphisms that can occur in C(G) when G Aut ðXÞ is chosen carefully. We state a reasonably general version of the lemma to make it applicable in many different contexts. A special case occurs as part of the proof of Theorem 14 in Kopra (2018).
Definition 5 Given a subshift X A Z , an abstract glider automorphism group is any tuple ðG; 0; I ; spd; 1; GFÞ (or just G when the rest of the tuple is clear from the context) where G Aut ðXÞ is a subgroup, I is an index set, 0 2 A þ and • spd : I ! Z is called a speed map and 1 : I ! G (image at i 2 I is denoted by 1 i ) is called a local shift map • GF is a map from I to subsets of X whose image at i 2 I is and is called a glider fleet set. Elements of GF i are called glider fleets.
This tuple is an abstract diffusive glider automorphism group if in addition • For every 0-finite x 2 X and every N 2 N there is a g 2 G such that for every i 2 Z, gðxÞ½i; i þ N 2 LðGF j Þ for some j 2 I.
If G is generated by a single automorphism g 2 Aut ðXÞ, we say that g is an abstract (diffusive) glider automorphism.
The idea of an abstract diffusive glider automorphism group is the following. For any 0-finite x 2 X there is a g 2 G that can be used to ''diffuse'' x into a point g(x) such that elements of OðgðxÞÞ locally look like elements of some GF i , and in practice GF i will be in some sense simpler subshifts than X. The local shift maps 1 i are used to dynamically distinguish the points in GF i nOð0 Z Þ. In the proof of our main lemma we will also require that the points of GF i consist of gliders in a more concrete sense. We encode this in the following definition.
Definition 6 Given a subshift X A Z , a (diffusive) glider automorphism group is any tuple ðG; 0; I ; $ ; spd; 1; GFÞ (or just G when the rest of the tuple is clear from the context) where ðG; 0; I ; spd; 1; GFÞ is an abstract (diffusive) glider automorphism group and • $ : I ! A þ is a map whose image at i 2 I is denoted by $ i and is called a glider • For every i 2 I there is some n 2 N such that GF i ¼ 1 0ð $ i 0 n 0 Ã Þ Ã 0 1 ; note that these configurations are 0-finite • For every i 2 I and x 2 GF i it holds that jj À kj ! j $ i j whenever j; k 2 occ ' ðx; $ i Þ are distinct, i.e. the occurrences of $ i do not overlap in any point of GF i . If G is generated by a single automorphism g 2 Aut ðXÞ, we say that g is a (diffusive) glider automorphism.
is the four-letter full shift. Any point x 2 X can be naturally identified with a point ðx 1 ; x 2 Þ 2 B Z Â B Z such that x½i ¼ ðx 1 ½i; x 2 ½iÞ for all i 2 Z. We define g 2 Aut ðXÞ by gðxÞ ¼ ðrðx 1 Þ; x 2 Þ. This map is a diffusive glider automorphism with an associated diffusive glider automorphism group ðG; 0; I ; $ ; spd; 1; x 2 ) contains no occurrences of the digit 1.
For X and 0 as above we let Aut ðX; 0Þ ¼ ff 2 Aut ðXÞ j f ðOð0 Z ÞÞ ¼ Oð0 Z Þg. For x; y 2 A Z and i 2 Z we denote by x i y 2 A Z the ''gluing'' of x and y at i, i.e. ðx i yÞ½À1; i À 1 ¼ x½À1; i À 1 and ðx i yÞ½i; 1 ¼ y½i; 1. Typically we perform gluings at the origin and we denote x y ¼ x 0 y.
In the next lemma we need the notion of a bipartite nondirected graph. By this we mean a pair B ¼ ðV; EÞ where V is the set of vertices with a nontrivial partition V ¼ V 1 [ V 2 and E V 1 Â V 2 is the set of edges, i.e. an edge cannot connect two vertices belonging in the same element of the partition. V and E are not necessarily finite. We say that B is connected if the equivalence relation on V generated by E is equal to V Â V, which is equivalent to saying that it is possible to traverse between any two vertices by a finite path in which edges can be crossed in both directions.
Lemma 1 (Main lemma) Let X A Z be a subshift with a diffusive glider automorphism group ðG; 0; I ; $ ; spd; 1; GFÞ such that 0-finite configurations are dense in X. Let I 1 [ I 2 ¼ I be a nontrivial partition and let B ¼ ðI ; EÞ be a bipartite non-directed graph with an edge from i 2 I 1 to j 2 I 2 if and only if there are d; e 2 N þ , a strictly increasing sequence ðN m Þ m2N 2 N N and ðg m Þ m2N 2 G N such that for any Before the proof we continue our previous example and show how this lemma can be applied to it.
The map f has two important properties. First, it replaces any occurrence of 00 by 0 0. Second, if x 2 X is a configuration containing only gliders and and every occurrence of is sufficiently far from every occurrence of , then f ðxÞ ¼ x. We use the lemma to show that CðG 0 Þ \ Aut ðX; 0Þ ¼ r h i. The bipartite graph B in the statement of the lemma has in this case the set of vertices f0; 1g with the partition I 1 ¼ f0g and I 2 ¼ f1g, so it suffices to show that there is an edge between 0 and 1. Still using the same notation as in the statement of the lemma, let g Ã 0 1 , 1 0 y 2 GF 1 ¼ 1 0f0; g Ã 0 1 be arbitrary. Fix some m 2 N. Since X is a full shift, it is clear that x :0 N m y 2 X and it is easy to verify that It follows that there is an edge between 0 and 1, so CðG 0 Þ \ Aut ðX; 0Þ ¼ r h i. In other words, if h 2 Aut ðXÞ has 0 Z as a fixed point and if it commutes with both f and g, then h ¼ r i for some i 2 Z.
In our example the construction of a nontrivial diffusive glider automorphism g was simple because of the existence of a decomposition A Z ¼ B Z Â B Z . On more general subshifts we cannot rely on such decompositions. In the example we also augmented G by an automorphism f and got a group G 0 satisfying the assuptions of Lemma 1. The construction of such a map f will be essentially the same in all our later applications of the lemma. To gain a better understanding of Main Lemma, it may be helpful to consider how the following proof would go in the case of the previous example.
Proof of Lemma 1 Assume that f 2 CðGÞ \ Aut ðX; 0Þ is a radius-r automorphism whose inverse is also a radiusr automorphism. Since we aim to prove that f 2 r h i, we lose no generality by transforming f throughout the proof by taking inverses and composing it with some shift. We start by noting that without loss of generality (by composing f with a suitable power of r if necessary) 0 Z is a fixed point of f.
We have that f ðGF i Þ GF i for i 2 I. To see this, assume to the contrary that x 2 GF i but f ðxÞ 6 2 GF i . Then f ð1 i ðxÞÞ ¼ f ðr spdðiÞ ðxÞÞ ¼ r spdðiÞ ðf ðxÞÞ 6 ¼ 1 i ðf ðxÞÞ, contradicting the assumption f 2 CðGÞ.
For all i 2 I 1 , j 2 I 2 and all x 1 2 GF i and x 2 2 GF j not in Oð0 Z Þ we define the right and left offsets We claim that off ' ðx 2 Þ À off r ðx 1 Þ ¼ 0. To see this, assume to the contrary that this does not hold. Since B is connected, there is a path from i to j, along which there is an edge from i 0 2 I 1 to j 0 2 I 2 and some x 0 Then we can assume without loss of generality that off r ðx 0 1 and x 0 2 suitably). Then consider x ¼ x 0 1 x 0 2 and note that f ðxÞ ¼ f ðx 0 1 Þ f ðx 0 2 Þ by the choice of N m . By our assumption on offsets and the map g m it follows that and thus g m f 6 ¼ f g m , contradicting the assumption f 2 CðGÞ. In the following we may therefore assume that off ' ðx 2 Þ ¼ off r ðx 1 Þ ¼ 0 for all i 2 I 1 , j 2 I 2 and all x 1 2 GF i and x 2 2 GF j not in Oð0 Z Þ. If x 2 GF i is a configuration containing exactly one occurrence of $ i , then f ðxÞ ¼ x. To see this, assume to the contrary (without loss of generality), that f(x) contains at least two occurrences of $ i , that i 2 I 1 (the case i 2 I 2 being similar), that y 2 GF j is a configuration containing a single $ j for j such that there is an edge from i to j in B and that minfocc ' ðy; $ j Þg ¼ N m , maxfocc r ðx; $ i Þg ¼ À1 with m 2 N such that N m ! 2r þ 1 (by shifting x and y suitably). Then consider z ¼ x y and note that g m ðzÞ ¼ z but g m ðf ðzÞÞ 6 ¼ f ðzÞ because g m at least shifts the leftmost glider in f(z). Thus f ðg m ðzÞÞ ¼ f ðzÞ 6 ¼ g m ðf ðzÞÞ, contradicting the assumption f 2 CðGÞ.
Now let us prove that if x 2 GF i , then f ðxÞ ¼ x. To see this, assume to the contrary that f ðxÞ 6 ¼ x, that i 2 I 1 (the case i 2 I 2 being similar), that x contains a minimal number of occurrences of $ i (at least two by the previous paragraph) and that the distance from the rightmost $ i to the second-to-rightmost $ i in x is maximal. Let y 2 GF j be a configuration containing a single $ j for j such that there is an edge from i to j in B and assume that minfocc ' ðy; $ j Þg ¼ N m , maxfocc r ðx; $ i Þg ¼ À1 with m 2 N such that N m ! 2r þ 1 (by shifting x and y suitably). Then x½À1; À1; f ðxÞ½À1; À1 are of the form contradicting the maximal distance between the two rightmost occurrences of $ i in x.
If x is a 0-finite configuration, then f ðxÞ ¼ x. Namely, let N ! 2r þ 1, and because G is a diffusive glider automorphism group of X, there exists g 2 G such that for every i 2 Z, gðxÞ½i; i þ N 2 LðGF j Þ for some j 2 I. Because F acts like the identity on all GF j , it follows that f ðgðxÞÞ ¼ gðxÞ. By using the assumption f 2 CðGÞ it follows that Finally, because f c the identity map on the dense set of 0-finite configurations, it follows that f is the identity map and in particular f 2 r h i. h

Diffusive glider automorphisms for mixing SFTs
In this section we construct for an arbitrary nontrivial mixing SFT X (with a distinguished periodic point 0 Z ) an automorphism g which breaks every 0-finite point of X into a collection of gliders traveling in opposite directions. More precisely, we will construct a diffusive glider automorphism g : X 0 ! X 0 for a subshift X 0 which is conjugate to X (via some conjugacy / : X ! X 0 ) but has a graph presentation that makes our constructions simpler. Then the map / À1 g / is an abstract diffusive glider automorphism on X.
To begin, consider a nontrivial mixing SFT X defined by a graph G ¼ ðV; EÞ and let 0 ¼ 0 1 Á Á Á 0 p 2 E p be some fixed simple cycle in G. We will want, among other things, that occurrences of the letters 0 i can only occur within occurrences of the word 0 in points of X. We start with some auxiliary definitions.
Definition 7 Given a graph G ¼ ðV; EÞ, we say that a path w 2 E þ has a unique successor in G (resp. a unique predecessor) if wa (resp. aw) is a path for a unique a 2 E. Then we say that a is the unique successor (resp. the unique predecessor) of w.
Definition 8 Let G ¼ ðV; EÞ be a graph and let w ¼ w½1 Á Á Á w½n be a path. If w[i] have unique successors for 1 i\n, we say that w is future deterministic in G and if w[j] have unique predecessors for 1\j n, we say that w is past deterministic in G. If w is both future and past deterministic in G, we say that w is deterministic in G.
We emphasize that if w is a deterministic path, we do not require that w[1] has a unique predecessor or that w[n] has a unique successor.
Lemma 2 Let X 1 be a nontrivial mixing SFT defined by the graph G ¼ ðV; EÞ and let 0 ¼ 0 1 Á Á Á 0 p 2 E p be a simple cycle in G. Then X 1 is conjugate to a subshift X 2 defined by a graph H ¼ ðV 0 ; E 0 Þ such that 0 is a past deterministic simple cycle in H.

Proof
The proof is by induction. We assume that 0 1 Á Á Á 0 iÀ1 is past deterministic for some 1\i p in G and we will construct a conjugate subshift Y defined by H ¼ ðV 0 ; E 0 Þ such that 0 1 Á Á Á 0 i is past deterministic in H. The induction can be started because 0 1 is vacuously past deterministic in G, and the claim will follow by repeating the argument for increasing i.
Denote s j ¼ ið0 j Þ for 1 j p. Let us assume that 0 1 Á Á Á 0 i is not past deterministic in G, because otherwise we could choose H ¼ G. Then E s i ¼ f0 iÀ1 ; a 1 ; . . .; a k g for some k ! 1 and a 1 ; . . .; a k 2 E. We denote by b 1 ; . . .; b ' the outgoing edges of s i different from 0 i (some may be equal to an edge a j ) and construct an in-split graph ' g with the starting and ending nodes of e 2 E the same as in G with the exception of sða j Þ ¼ s 0 and for b j distinct from any a j 0 , sðb 0 j Þ ¼ sðb j Þ (see Fig. 3). The edge subshift of H is conjugate to X 1 [see Sect. 2.4 of Lind and Marcus (1995)], H contains the cycle 0 1 Á Á Á 0 p with ið0 i Þ ¼ s i , and all the states s 2 ; . . .; s i have only one incoming edge so 0 1 Á Á Á 0 i is past deterministic. h Lemma 3 Let X 2 be a nontrivial mixing SFT defined by the graph G ¼ ðV; EÞ and let 0 ¼ 0 1 Á Á Á 0 p 2 E p be a past deterministic simple cycle in G. Then X 2 is conjugate to a subshift X 3 defined by a graph H ¼ ðV 0 ; E 0 Þ such that 0 is a deterministic simple cycle in H.
Proof The proof is by induction. We assume that 0 i Á Á Á 0 p is future deterministic for some 1\i p in G and we will construct a conjugate subshift X 3 defined by H ¼ ðV 0 ; E 0 Þ such that 0 iÀ1 Á Á Á 0 p is future deterministic and 0 is still past deterministic in H. Denote s j ¼ ið0 j Þ for 1 j p and assume that 0 iÀ1 Á Á Á 0 p is not future deterministic in G. Then E s i ¼ f0 i ; a 1 ; . . .; a k g for some k ! 1 and a 1 ; . . .; a k 2 E and it would be possible to construct an out-split graph H ¼ iÀ1 g with the starting and ending nodes of e 2 E the same as in G with the exception of iða j Þ ¼ s 0 Fig. 4). The edge subshift of H is conjugate to X 2 [see again Sect. 2.4 of Lind and Marcus (1995)], G contains the cycle 0 1 Á Á Á 0 p with ið0 i Þ ¼ s i , and all the states s i ; . . .; s p have only one outgoing edge. h Lemma 4 Let X 3 be a nontrivial mixing SFT defined by the graph G ¼ ðV; EÞ and let 0 ¼ 0 1 Á Á Á 0 p 2 E p be a deterministic simple cycle in G. Then X 3 is conjugate to a subshift X defined by a graph G ¼ ðV; EÞ such that 0 is a deterministic simple cycle in G and the graph G 0 ¼ ðV 0 ; E 0 Þ ¼ ðVnfs i j 1\i pg; Enf0 i j 1 i pgÞ gained by removing the cycle 0 from G is primitive (here we denote s i ¼ ið0 i Þ). Furthermore, G 0 contains a cycle 1 ¼ a 1 Á Á Á a q 2 E 0q with p and q coprime such that iða 1 Þ ¼ sða q Þ ¼ s 1 and iða i Þ 6 ¼ s 1 for 1\i q.
Proof We denote . . .; d k g, E s 1 ¼ f0 1 ; e 1 ; . . .; e ' g and construct the graph with the other initial and terminal vertices remaining the same as in G (see Fig. 5). If X is the edge subshift of G, then it is easy to see that the map U : X ! X 3 defined by is a conjugacy. Since X is mixing, there is a large enough prime number p 0 [ p such that G contains a path of length p 0 from s 1 to s 1 . If all cycles 0 are removed from this path, we get a path wc p 0 Ànp ¼ c 1 Á Á Á c p 0 Ànp 2 E p 0 Ànp from s 1 to s 1 , where n is the number of removed 0-cycles. In particular, the length of wc p 0 Ànp is coprime with p. Then 1 ¼ 0 0 1 Á Á Á 0 0 p wc 0 p 0 Ànp is a path in G which visits s 1 only at the beginning and ending and j1j is coprime with p. Moreover, the graph G 0 is primitive, because it contains cycles wc p 0 Ànp and 1 of coprime length. h Now let X be a nontrivial mixing SFT. By applying the three previous lemmas consecutively, we may assume up to conjugacy that X is defined by a graph G ¼ ðV; EÞ such that G 0 , 0, 1, etc. are as in the conclusion of the previous lemma. In the rest of this section we will construct a diffusive glider automorphism g : X ! X with the associated diffusive glider automorphism group ð g h i; 0; I ; $ ; spd; 1; GFÞ. Let I ¼ f'; rg, spdð'Þ ¼ pq and spdðrÞ ¼ Àpq, which reflect the fact that we will have left-and rightbound gliders. The gliders will be note that these are of equal length ðp þ 1Þq. We define the glider fleet sets GF ' ¼ 1 0ð 00 Ã Þ Ã 0 1 GF r ¼ 1 0ð0 Ã 0 ! Þ Ã 0 1 and languages Since G 0 is primitive, it has a mixing constant n ! j1j pþ2 , i.e. a number such that for every n 0 ! n and s; s 0 2 V 0 there is a path of length n 0 in G 0 from s to s 0 . Denote N ¼ 2n and for each a 2 E 0 [ f0 1 g let W 0 a ¼ fw a;1 ; . . .; w a;k a g E 0N be the set of all those words over E 0 of length N such that w a;i does not have prefix 1 pþ2 and 0 p w a;i a 2 LðXÞ for 1 i k a , let w a 2 E 0N be some single word with prefix 1 pþ2 such that 0 p w a a 2 LðXÞ (such a word w a exists by the choice of the mixing constant n), and denote W a ¼ W 0 a [ fw a g. For each j 2 f1; . . .; pg let u 0 j ¼ 1 pþ1þj and let U 0 j ¼ fu 0 j;1 ; . . .; u 0 j;n j g E 0þ be all the cycles from s 1 to s 1 (which may visit s 1 several times) of length at most N À 1 such that ju 0 j;i j ju 0 j jðmod pÞ and u 0 j;i does not have prefix 1 pþ2 , with the additional restriction that 1; 1 pþ1 6 2 U 0 p . Finally, these words are padded to constant length; let u j ¼ 0 c j u 0 j and u j;i ¼ 0 c j;i u 0 j;i , where c j ; c j;i ! 100N are chosen in such a way that all u j , u j;i are of the same length for any fixed j. The words in W a and U 0 j have been chosen so as to allow the following structural definition.
Definition 9 Let x 6 2 GF ' be a 0-finite element of X not in Oð0 Z Þ. Then there is a maximal i 2 Z such that x½À1; i À 1 2 1 0L ' ; and there is a unique word w 2 f10g [ f1 pþ1 0g[ f1 pþ2 g [ ð S p j¼1 U 0 j 0Þ [ ð S a2E 0 [f0 1 g W 0 a aÞ such that w is a prefix of x½i; 1. If w ¼ 1 pþ1 0 or w 2 U 0 j 0, let k ¼ i þ jwj À 1 and otherwise let k ¼ i þ j10j À 1. We say that x is of left bound type (w, k) and that it has left bound k (note that k [ i).
We outline a deterministic method to narrow down the word w of the previous definition in a way that clarifies its existence and uniqueness. First, by the maximality of i it follows that x½i 2 E 0 . If x½i; i þ N À 1 2 E 0N , then w 2 W 0 x½iþN x½i þ N directly by the definition of the sets W 0 a unless x½i; 1 has prefix 1 pþ2 , in which case w ¼ 1 pþ2 is the only option. Otherwise x½i; i þ N À 1 6 2 E 0N and there is a minimal m\N such that x½i; i þ m À 1 2 E 0m and x½i þ m; i þ m þ p À 1 ¼ 0. Then x½i; i þ m À 1 is a cycle of length m\N from s 1 to s 1 and w 2 U 0 j 0 for some j 2 f1; . . .; pg unless we have specifically excluded x½i; i þ m À 1 from all the sets U 0 j . But this happens precisely if x½i; i þ m À 1 2 f1; 1 pþ1 g or x½i; i þ m À 1 has prefix 1 pþ2 . In these cases w 2 f10; 1 pþ1 0; 1 pþ2 g.
The point of this definition is that if x is of left bound type (w, k), then the diffusive glider automorphism g defined later will eventually create a new leftbound glider at position k and break it off from the rest of the configuration. (A possible exception to this is if w ¼ 1 pþ1 0 ¼ ! 0, in which case it might happen that the rightbound glider just travels to the right.) We define four automorphisms g 1 ; g 2 ; g 3 ; g 4 : X ! X as follows. In any x 2 X, • g 1 replaces every occurrence of 0ð0 q 1Þ0 by 0ð1 pþ1 Þ0 and vice versa. • g 2 replaces every occurrence of 0ð1 pþ1 Þ0 by 0ð10 q Þ0 and vice versa. • g 3 replaces every occurrence of 0 qþ1 ð1 pþ2 Þ by 0 qþ1 ð10 q 1Þ and vice versa. • g 4 replaces every occurrence of 0w a a, 0w a;i a and 0w a;k a a by 0w a;1 a, 0w a;iþ1 a and 0w a a respectively (for a 2 E 0 [ f0 1 g and 1 i\k a ) and every occurrence of 0u j 0, 0u j;i 0 and 0u j;n j 0 by 0u j;1 0, 0u j;iþ1 0 and 0u j 0 respectively (for j 2 f1; . . .; pg and 1 i\n j ).
It is easy to see that these maps are well-defined automorphisms of X. The automorphism g : X ! X is defined as the composition g 4 g 3 g 2 g 1 . We commence arguing that g is a diffusive glider automorphism with respect to ð g h i; 0; I ; $ ; spd; 1; GFÞ, where we choose 1 ' ¼ 1 r ¼ g.
Lemma 5 If x 2 GF ' (resp. x 2 GF r ), then gðxÞ ¼ r pq ðxÞ (resp. gðxÞ ¼ r Àpq ðxÞÞ. Proof Assume that x 2 GF ' (the proof for x 2 GF r is similar) and assume that i 2 Z is some position in x where occurs. Then x½i À p; i þ ðpq þ qÞ þ p À 1 ¼ 0 0 ¼ 0ð0 q 1Þ0 g 1 ðxÞ½i À p; i þ ðpq þ qÞ þ p À 1 ¼ 0ð1 pþ1 Þ0 g 2 ðg 1 ðxÞÞ½i À p À pq; i þ q þ p À 1 ¼ 0 q 0ð10Þ ¼ 0 0 gðxÞ ¼ g 4 ðg 3 ðg 2 ðg 1 ðxÞÞÞÞ ¼ g 2 ðg 1 ðxÞÞÞ; so every glider has been shifted by distance pq to the left and gðxÞ ¼ r pq ðxÞ. h We first give a heuristic argument showing that g could be a diffusive glider automorphism. It is easier to convince oneself that with the choices g 0 ¼ g 2 g 1 , G ¼ g 0 ; g 3 ; g 4 h i and 1 0 ' ¼ 1 0 r ¼ g 0 the tuple ðG; 0; I ; $ ; spd; 1 0 ; GFÞ is a diffusive glider automorphism group. Namely, the previous lemma would hold even if g were replaced by g 0 , and it also seems reasonable that GF ' , GF r are glider fleet sets with respect to 1 0 ' , 1 0 r in the sense of Definitions 5 and 6, so g 0 is a glider automorphism. It remains to show diffusiveness. If x 2 X is 0-finite, then x 1 ¼ g 0i ðxÞ for large i 2 N contains gliders very far from the origin going to opposing directions and possibly there is an occurrence of a word 0 M w (with large M 2 N) that does not look like a glider near the origin. Then for some j, the occurrence of this word is replaced in x 2 ¼ g j 4 ðx 1 Þ by 0 M 0 1 pþ2 , and then g 3 separates an occurrence of a glider from this pattern; x 3 ¼ g 3 ðx 2 Þ contains near the origin which can be shifted away by sufficiently many applications of g 0 . By repeating this argument we find an element g 2 G such that gðxÞ contains only leftbound gliders far to the left and rightbound gliders far to the right, so in particular the last item in Definition 5 is satisfied.
The reason why g ¼ g 4 g 3 g 0 could also have the diffusion property is that the words in points x 2 X on which g 0 , g 3 and g 4 can act nontrivially are for the most part distinct, e.g. g 3 can change occurrences of the word 0 qþ1 ð1 pþ2 Þ but in the definition of g 4 this occurs as a subword only in 0 qþ1 w a a and 0u j 0. Therefore, whenever one component in the map g does something conductive to the diffusion of x, it is unlikely that this effect is immediately reversed by some other component. We proceed with the actual proof that g is a diffusive glider automorphism.
Lemma 6 If x 2 X has left bound k, then there exists t 2 N þ such that the left bound of g t ðxÞ is strictly greater than k. Moreover, the left bound of g t 0 ðxÞ is at least k for all t 0 2 N.
aÞ. The gliders to the left of the occurrence of w near k move to the left at constant speed pq under action of g without being affected by the remaining part of the configuration. We show by case analysis that the left bound of g t ðxÞ increases for sufficiently large t. The cases from 1 to 5 correspond to different left bound types and Case 3.1 can be reached as a subcase from Case 1 and Case 3. From each case it is possible to proceed only to a case with a higher numbering, which prevents the possibility of circular arguments. The fact that the left bound never decreases can be extracted from the case analysis.
h Definition 10 Let x 6 2 GF r be a 0-finite element of X not in Oð0 Z Þ. Then there is a minimal k 2 Z such that x½k þ 1; 1 2 L r 0 1 and we say that x has right bound k.
Lemma 7 If x 2 X has right bound k, then there exists t 2 N þ such that the right bound of g t ðxÞ is strictly less than k. Moreover, the right bound of g t 0 ðxÞ for 0 t 0 t is at most k þ C for some C that does not depend on x, k or t.
Proof We prove that the right bound of g t ðxÞ eventually decreases by case analysis and that we can choose C ¼ pq.
The constant C does not play any role in the first case but it can be extracted from the second case and its subcases.
Case 1. Assume that the right bound of g t ðxÞ is at most k for every t 2 N þ . By the previous lemma the left bound of g t ðxÞ tends to 1 as t tends to 1, which means that for some t 2 N þ g t ðxÞ contains only -gliders to the left of k þ 3pq and only ! -gliders to the right of k. This can happen only if g t ðxÞ½k þ 1; k þ 3pq À 1 does not contain any glider of either type. Then the right bound of g tþ1 ðxÞ is at most k À pq and we are done. Case 2. Assume that the right bound of g t ðxÞ is strictly greater than k for some t 2 N þ and fix the minimal such t. This can happen only if g 1 ðg tÀ1 ðxÞÞ½k À ðp þ qÞ þ 1; k þ ðq þ 1Þp ¼ 010 q 0 and then g 2 ðg 1 ðg tÀ1 ðxÞÞÞ½kÀðpþqÞþ 1;kþðqþ1Þp¼01 pþ1 0: We proceed to Case 2.1 or Case 2.2. Case 2:1. Assume that g 2 ðg 1 ðg tÀ1 ðxÞÞÞ½À1; k þ ðq þ 1Þp does not have suffix 0 qþ1 10 q 1 pþ1 0. Then g 3 ðg 2 ðg 1 ðg tÀ1 ðxÞÞÞÞ½À1; k þ ðq þ 1Þp and g t ðxÞ½À1; k þ ðq þ 1Þp have suffix 01 pþ1 0 ¼ 0 ! 0. This contradicts the choice of t, because the right bound of g t ðxÞ is at most k À ðp þ qÞ.
Proof By inductively applying the previous two lemmas we see that if t 2 N þ tends to 1, then the left bound (resp. the right bound) of g t ðxÞ tends to 1 (resp. to À1). h

Theorem 2
The map g is a diffusive glider automorphism associated to the tuple ð g h i; 0; I ; $ ; spd; 1; GFÞ constructed in this section.
Proof By Lemma 5 we know that for i 2 I, GF i fx 2 X j x is 0-finite and gðxÞ ¼ r spdðiÞ ðxÞg+S i : We prove the other inclusion when i ¼ ', the case i ¼ r being similar. Assume therefore that x 6 2 GF ' is 0-finite and apply the previous lemma for sufficiently large M. By Lemma 5 the set GF i is invariant under the map g, so g t ðxÞ 6 2 GF ' and g t ðxÞ contains an occurrence of ! which is shifted to the right by the map g. Therefore gðg t ðxÞÞ 6 ¼ r pq ðg t ðxÞÞ ¼ r spdð'Þ ðg t ðxÞÞ and g t ðxÞ 6 2 S ' . Since S ' is invariant under the map g, it follows that x 6 2 S ' .
The other conditions necessary for showing that g is a glider automorphism are easy to check. Then the fact that g is a diffusive glider automorphism follows from the previous lemma. h

Finitary Ryan's theorem for transitive SFTs
In this section we prove our finitary version of Ryan's theorem. This is done by applying Lemma 1. As in Example 3, we need a suitable automorphism f to augment the diffusive glider automorphism group of Theorem 2. As earlier, let X be a mixing SFT from the conclusion of Lemma 4 and consider the notation of the previous section. First we define maps f 1 ; f 2 : X ! X as follows. In any x 2 X, • f 1 replaces every occurrence of 0 ! 000 0 by 0 ! 00 00 and vice versa • f 2 replaces every occurrence of 0 ! 00 0 by 00 ! 0 0 and vice versa.
It is easy to see that these maps are well-defined automorphisms of X. The automorphism f : X ! X is then defined as the composition f 2 f 1 . Similarly to Example 3, f has the following properties. First, it replaces any occurrence of 0 ! 000 0 by 00 ! 0 00. Second, if x 2 X is a configuration containing only gliders and ! and every occurrence of is sufficiently far from every occurrence of ! , then f ðxÞ ¼ x.
Proposition 1 Let X A Z and g; f : X ! X be as above.
Then Cð g; f h iÞ ¼ r h i.
Proof Consider the diffusive glider automorphism group ð g h i; 0; I ; $ ; spd; 1; GFÞ from Theorem 2. If we define G ¼ g; f h i, then it directly follows that ðG; 0; I ; $ ; spd; 1; GFÞ is also a diffusive glider automorphism group of X. We want to use Lemma 1 to show that CðGÞ \ Aut ðX; 0Þ ¼ r h i. The bipartite graph B in the statement of the lemma has in this case the set of vertices I ¼ fr; 'g with the partition I 1 ¼ frg and I 2 ¼ f'g, so it suffices to show that there is an edge between r and '.
Recall that we denote p ¼ j0j, q ¼ j1j. Using the same notation as in the statement of Lemma 1, let d ¼ e ¼ 1, ðN m Þ m2N with N m ¼ 2mq þ 3 and ðg 0 m Þ m2N with g 0 m ¼ g Àðmþ1Þ f g m . Let x ! 2 1 0L r , y 2 L ' 0 1 be arbitrary. Fix some m 2 N. Since X is an edge shift, it is clear that x ! :0 N m y 2 X and it is easy to verify that It follows that there is an edge between r and ', so CðGÞ \ Aut ðX; 0Þ ¼ r h i. Now let h 2 CðGÞ be arbitrary. Let us show that h 2 Aut ðX; 0Þ. Namely, assume to the contrary that hð0 Z Þ ¼ w Z 6 2 Oð0 Z Þ for some w ¼ w 1 Á Á Á w p (w i 2 A). The maps g k in the definition of g have been defined so that g k ðxÞ½i ¼ x½i whenever x contains occurrences of 0 only at positions strictly greater than i, so in particular gðw Z Þ ¼ w Z . Consider x ¼ 1 0: 0 1 2 GF ' with the glider at the origin. Note that hðxÞ½ði À 1Þp; ip À 1 6 ¼ w for some i 2 Z (otherwise hðxÞ ¼ w Z ¼ hð0 Z Þ, contradicting the injectivity of h) and hðxÞ½À1; ip À ðjqÞp À 1 ¼ Á Á Á www for some j 2 N þ . By the earlier observation on the maps g k it follows that g t ðhðxÞÞ½À1; ip À ðjqÞp À 1 ¼ Á Á Á www for every t 2 Z but hðg j ðxÞÞ½ip À ðj þ 1Þqp; ip À ðjqÞp À 1 ¼ hðr ðpqÞj ðxÞÞ½ip À ðj þ 1Þqp; ip À ðjqÞp À 1 ¼ hðxÞ½ip À qp; ip À 1 6 ¼ w q , contradicting the commutativity of h and g. Thus h 2 Aut ðX; 0Þ: We have shown that h 2 CðGÞ \ Aut ðX; 0Þ ¼ r h i, so we are done. h Theorem 3 kðXÞ ¼ 2 for every nontrivial mixing SFT X.
Proof Every nontrivial mixing SFT is conjugate to a subshift X of the form given in the conclusion of Lemma 4, so kðXÞ 2 follows from the previous proposition. Clearly Aut ðXÞ 6 ¼ r h i, so by Theorem 1 it is not possible that kðXÞ\2 and therefore kðXÞ ¼ 2.
h Corollary 1 (Finitary Ryan's theorem) kðXÞ ¼ 2 for every transitive SFT X which is not the orbit of a single point.
Proof Let X be a transitive SFT given as the edge subshift of a graph G ¼ ðV; EÞ containing more than a single cycle. By Sect. 4.5 in Lind and Marcus (1995) there is a partition E ¼ S n i¼1 E n with the following properties. First, the ending states of E i can be starting states only for edges of E iþ1 (where i þ 1 is considered modulo n) and this induces a partition X ¼ S n i¼1 X i such that X i ¼ fx 2 X j x½0 2 E i g and rðX i Þ ¼ X iþ1 . Second, the edge shift X 0 of the graph G 0 ¼ ðV 0 ; E 0 Þ is a nontrivial mixing SFT where V 0 V contains the starting states of edges in E 1 and E 0 contains all paths w ¼ w 1 Á Á Á w n of length n in G with w 1 2 E 1 and we let iðwÞ ¼ iðw 1 Þ, sðwÞ ¼ sðw n Þ. There is a natural homeomorphism / : X 0 ! X 1 such that / r ¼ r n /. By the previous theorem there are f 0 1 ; f 0 2 2 Aut ðX 0 Þ which commute with only r X 0 h i and there are unique f 1 ; f 2 2 Aut ðXÞ such that f iX 1 ¼ / f 0 i / À1 . By Theorem 1 it remains to show that Cðff 1 ; f 2 gÞ ¼ r h i. Assume therefore that h commutes with f 1 and f 2 and without loss of generality (by composing h with some power of r X if necessary) that hðX 1 Þ ¼ X 1 . There is h 0 2 Aut ðX 0 Þ commuting with f 0 i such that / h 0 ¼ h /. It follows that h 0 ¼ r k X 0 and h ¼ r nk X . h

A nontrue finitary version of Ryan's theorem
Finitary Ryan's theorem can be interpreted as a compactness result saying that, for nontrivial mixing SFT X, the group Aut ðXÞ has a finite subset S such that CðSÞ ¼ r h i. One may wonder whether this compactness phenomenon is more general: if G Aut ðXÞ is an arbitrary infinite set such that CðGÞ ¼ r h i, does there exist a finite F G such that CðFÞ ¼ r h i? We will show by an example that this is not true for general G even if X is the binary full shift.
In this section let X ¼ f0; 1g Z . For every n 2 N þ we define two automorphisms g n;1 ; g n;2 : X ! X as follows. In any x 2 X, • g n;1 replaces every occurrence of 001 2nÀ1 0 by 011 2nÀ1 0 and vice versa. • g n;2 replaces every occurrence of 01 2nÀ1 10 by 01 2nÀ1 00 and vice versa.
Lemma 9 The tuple ðG 0 ; 0; I ; $ ; spd; 1; GFÞ defined above is a glider automorphism group of X, i.e. for n 2 N þ • GF n;' is the set of 0-finite configurations x for which g n ðxÞ ¼ rðxÞ • GF n;r is the set of 0-finite configurations x for which g n ðxÞ ¼ r À1 ðxÞ.
Proof By the previous lemma G 0 is a glider automorphism group. For the diffusion property it is sufficient to prove for all 0-finite x 2 X and N 2 N the existence of a g 2 G 0 such that gðxÞ 2 1 0ðð[ i2S L i Þ0 N Þ Ã 0 1 . To do this we define for every 0-finite x 2 X the quantity jocc ' ðx; 01 n 0Þj; i.e. the total number of consecutive runs of ones in x. We remark that N x ¼ N gðxÞ for g 2 G 0 , because this clearly holds for g 2 fg n;1 ; g n;2 j n 2 N þ g and these generate a group containing G 0 . We prove the diffusion property by induction on N x . As the base case we choose x 2 GF s (s 2 I), for which the claim is trivial. Assume therefore that x 6 2 GF s for all s 2 I and fix N 2 N. If the leftmost occurrence of 1 in x is at position i 2 Z, then x½i À 1; 1 has the prefix 01 2nÀ1 0 or 01 2n 0 for some n 2 N þ . We assume without loss of generality that the prefix is of the form 01 2nÀ1 0 (otherwise in the following we replace the map g n by its inverse g À1 n ). Note that by definition g n treats words of the form 01 2nÀ1 0 and 01 2n 0 in all 0-finite y 2 X as gliders which rebound from words of the form 01 2n 0 À1 0 and 01 2n 0 0 (n 0 6 ¼ n) that remain stationary under the action of g n . For

Conclusions
We have constructed diffusive glider automorphisms g for nontrivial mixing SFTs X (with some fixed periodic point 0 Z ) that decompose all 0-finite configurations into two fleets of gliders traveling into opposing directions. This construction was somewhat complicated and finding a simpler construction (and/or a simpler proof) would be desirable. One might also want to construct diffusive glider automorphisms on general mixing SFTs with several different types of gliders that travel at different speeds and that would satisfy some carefully specified collision rules (this is simpler on full shifts when the cardinality of the alphabet is not a prime, see e.g. Salo and Törmä (2017) for the case of the full shift A Z with jAj ¼ 16).
We have applied these glider maps to prove for any nontrivial mixing SFT X that kðXÞ ¼ 2. As a simple corollary we have also shown that kðXÞ ¼ 2 for any transitive SFT X that consists of more than a single orbit. It would be interesting to find more sensitive isomorphism invariants of Aut ðXÞ. As one possible invariant related to k(X) we suggest k 2 ðXÞ ¼ minfjSj j S Aut ðXÞ contains only involutions and CðSÞ ¼ r h ig: It is previously known by Theorem 7.17 of Salo (2017) that k 2 ðA Z Þ 2 N when jAj ¼ 4. Some upper bounds for this quantity for general transitive SFTs can be given by noting that the automorphisms in Proposition 1 can be represented as compositions of involutions. However, it might be difficult to recognize an optimal upper bound when it has been found. For example, we do not know the answer to the following.
Problem 1 Does there exist a mixing SFT X such that k 2 ðXÞ ¼ 2? Do all mixing SFTs have this property?
We have also given an example of a finitary variant of Ryan's theorem which is not true, i.e. there exists G Aut ðf0; 1g Z Þ such that CðGÞ ¼ r h i but CðFÞ) r h i for every finite F G.