FLOW EQUIVALENCE OF G-SFTS MIKE BOYLE, TOKE MEIER CARLSEN, AND SØREN EILERS Abstract. In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G = Z2, we compute explicit complete invariants. We relate our matrix struc- tures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata. Contents 1. Introduction 1 2. Applications of G-SFTs 3 3. Background 7 4. (G; P) coset structures 14 5. The main results 21 6. From G-flow equivalence to positive ElP (H) equivalence 23 7. The Factorization Theorem: setting 35 8. The Factorization Theorem: proof 42 9. Conclusion of proofs 50 Appendix A. Cohomology as positive equivalence 52 Appendix B. Permutation similarity as positive equivalence 55 Appendix C. Resolving extensions 56 Appendix D. G-SFTs following Adler-Kitchens-Marcus 59 Appendix E. A special case 67 References 79 1. Introduction The shifts of finite type (SFTs) are the fundamental building blocks of symbolic dynamics. Elaborations of these include the class of G- SFTs: SFTs equipped with a continuous action by a group G which Date: August 16, 2019. 1 FLOW EQUIVALENCE OF G-SFTS 2 commutes with the shift. Apart from a few remarks, in this paper G- SFT means G-SFT with G finite and acting freely (gx = x only when g = e). We will give an algebraic classification of these G-SFTs up to equi- variant flow equivalence (G-flow equivalence). This generalizes the G- flow equivalence classification for irreducible G-SFTs in [12] and the Huang flow equivalence classification for general SFTs without group action [4, 9]. Square matrices over Z+G (the positive cone in the integral group ring of G) present G-SFTs. When such matrices A and B present nontrivial1 mixing G-SFTs, these G-SFTs are G-flow equivalent if and only if there exist n in N and identity matrices Ij, Ik such that there are matrices U; V in the elementary group El(n; ZG) such that U((I −A)⊕ Ij)V = (I − B) ⊕ Ik. This reduces the dynamical classification of G- SFTs up to G-flow equivalence to the algebraic classification of square matrices over ZG up to the stabilized elementary equivalence described above. The algebraic classification is in general highly nontrivial, but far more manageable than the dynamical problem. For nonmixing G-SFTs, this stabilized elementary ZG equivalence no longer implies G-flow equivalence. To get an analogous result (Theorem 5.1) for general G-SFTs, we consider G-SFTs presented by matrices in a special block triangular form, with entries in an ij block lying in ZHij for some union Hij of double cosets of G, and their equivalence by el- ementary matrices from a restricted class subordinate to this blocked coset structure. Using this, we classify G-SFTs up to G-flow equiva- lence (Theorems 5.1 and 5.3). The paper is organized as follows. In Section 2, we discuss uses of G-SFTs, and application of the results of this paper. This includes a new use of the G-SFT structure, for cellular automata. Especially, the main theorem of the current paper is a key result for the classification up to flow equivalence of a large class of irreducible sofic shifts in [7]. In Section 3, we give a bare- bones review of the necessary background. In Section 4, we introduce the notion of coset structure, crucial for defining our restricted class of elementary matrices, and define various classes of related matrices. We prove Proposition 4.23 which tells us that, in order to classify G-SFTs up to G-flow equivalence, it is enough to work with square matrices over Z+G having a special block form. In Section 5, we present our classification, with comments. The full classification statement is Theorem 5.3; the essence is the simpler statement of Theorem 5.1 for the case the G-flow equivalence respects the ordering of irreducible components. In each statement, for matrices in a suitable class chosen to present the G-SFTs, a matrix condition 1By definition, an irreducible SFT is trivial iff it contains only one orbit. The only trivial mixing SFT is a single point; so, a trivial mixing G-SFT has G = feg. FLOW EQUIVALENCE OF G-SFTS 3 is necessary and sufficient for the G-flow equivalence. In Section 6, we prove a strengthened version of necessity of the matrix condition, Theorem 6.2. In Section 7, we present the functorial Factorization Theorem 7.2, which we later use to prove sufficiency of the matrix condition in Theorem 5.3, and develop the setting for its proof. In Section 8, we give that proof. Section 9 contains the proof of Theorem 5.3 (a short argument appealing to Theorems 6.2 and 7.2), a result on range of invariants and a finiteness result. In Appendices A, B, and C, we establish three types of positive ElP (H)-equivalences which we use in the paper. In Appendix D, we relate the group actions viewpoint on G-SFTs developed in the Adler-Kitchens-Marcus paper [2] to our matrix-based setup. (This is analogous to the relation of matrices and linear trans- formations in elementary linear algebra.) Adler, Kitchens and Marcus were concerned only with nonwandering SFTs. We explain in Remark 4.25 how our coset structures give a kind of algebraic calculus to de- scribe the transitions among irreducible components of a general G- SFT. In Appendix E, we work out algebraic invariants of G-flow equiva- lence for a special class of systems with exactly two irreducible com- ponents. For the subclass with G = Z2, we give a complete algebraic classification, with algorithms to answer all questions (at least, the natural ones we thought of). This supplies a tractable collection of ex- amples and points to some of the issues involved in a general algebraic classification. 2. Applications of G-SFTs G-SFTs arise in several contexts, including the following. (Below, we occasionally assume background reviewed in Section 3.) (i) In [2], Adler, Kitchens and Marcus introduced invariants of non- wandering G-SFTs (and a more general class), and used these in [1] to classify factor maps between irreducible SFTs up to almost topological conjugacy. Here, a construction replaces a given factor map with a map ': S0 ! S, equivalent up to almost conjugacy, which is constant n-to-1. The map ' gives rises to a continuous function τ : S ! Sn (with Sn the group of permutations on n symbols). This τ is used as a skewing function to generate a G- SFT (with G = Sn) extension T of S, with maps π : T ! S and 0 α : T ! S such that π = '◦α. Group invariants of the Sn-action on T are then related to ' and used for the classification, which requires further constructions. (ii) Field, Golubitsky and Nicol used G-SFTs in studies of \symmetry in chaos" and related equivariance [15, 16, 17]. FLOW EQUIVALENCE OF G-SFTS 4 (iii) The Livˇsictheorem, restricted to dimension zero, states that two real-valued H¨older functions on an irreducible SFT are cohomolo- gous if and only if on each periodic orbit the sum of their output values is the same. G-SFTs arise as a case of the study of anal- ogous rigidity possibilities for skew products in terms of group- valued skewing functions (see [24, 28, 10]). (iv) In [31, 30, 32], Michael Sullivan introduced \twistwise flow equiva- lence". Here G = Z2. For an SFT basic set of the return map to a cross-section under a flow on a 3-manifold, the return map to the cross-section induces a map on the local stable set which is orien- tation preserving or reversing. This additional data is given by a function X ! Z2, which can be encoded as a matrix A over ZG. Sullivan found G-FE invariants of A to produce new invariants of the template (branched two-manifold fitted with an expansive semiflow) for the flow. The complete algebraic G-FE classification (for G = Z2) for the mixing case gave further invariants (see [12, Sec. 7] and [33] for more). (v) The mapping class group of a nontrivial irreducible SFT (see [8]) is the countable group of homeomorphisms of the mapping torus of the SFT which respect orientation of the suspension flow, up to isotopy. This is a challenging group to understand. The classifica- tion of irreducible G-SFTs up to G-flow equivalence has provided at least a little information: for example (see [8, Theorem 8.6]), if the irreducible SFT is defined by a matrix A such that det(I − A) is odd, then the free orientation preserving involutions of the map- ping torus are contained in a finite set of conjugacy classes of this mapping class group. (For a speculative application related to mapping class groups, see Remark 7.3.) (vi) The main result of the current paper has already been applied to the classification of sofic shifts up to flow equivalence in [7]. We will give more detail on this below. The automorphism group Aut(S) of an SFT S is the group of home- omorphisms commuting with S. The G-SFTs offer a different tool set to the study of finite subgroups of Aut(S).