4 - Symbolic dynamics and the stable algebra of matrices
Published online by Cambridge University Press: 11 May 2024
Summary
We give an introduction to a topic in the “stable algebra of matrices,” as related to certain problems in symbolic dynamics. We introduce enough symbolic dynamics to explain these connections, but the algebra is of independent interest and can be followed with little attention to the symbolic dynamics. This “stable algebra of matrices” involves the study of properties and relations of square matrices over a semiring S, which are invariant under two fundamental equivalence relations: shift equivalence and strong shift equivalence. When S is a field, these relations are the same, and matrices over S are shift equivalent if and only if the nonnilpotent parts of their canonical forms are similar. We give a detailed account of these relations over other rings and semirings. When S is a ring, this involves module theory and algebraic K theory. We discuss in detail and contrast the problems of characterizing the possible spectra, and the possible nonzero spectra, of nonnegative real matrices.We also review key features of the automorphism group of a shift of finite type; the recently introduced stabilized automorphism group; and the work of Kim, Roush and Wagoner giving counterexamples to Williams’ shift equivalence conjecture.
Keywords
- Type
- Chapter
- Information
- Groups and Graphs, Designs and Dynamics , pp. 266 - 422Publisher: Cambridge University PressPrint publication year: 2024
References
Almkvist, Gert, Erratum: “K-theory of endomorphisms” [J. Algebra 55 (1978), no. 2, 308–340; MR 80i:18018], J. Algebra, 68 (1981), no. 2, 520–521.Google Scholar
Badoian, L. and Wagoner, J. B., Simple connectivity of the Markov partition space, Pacific J. Math., 193 (2000), no. 1. 1–3.Google Scholar
Baker, Kirby A., Strong shift equivalence of 2 × 2 matrices of nonnegative integers, Ergodic Theory Dynam. Systems, 3 (1983), no. 4, 501–508.Google Scholar
Baker, Kirby A., Strong shift equivalence and shear adjacency of nonnegative square integer matrices, Linear Algebra Appl., 93 (1987), 131–147.Google Scholar
Bass, H., Milnor, J., and Serre, J.-P., Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2), Inst. Hautes Études Sci. Publ. Math., 33 (1967), 59–137.Google Scholar
Louis Block, John Guckenheimer, Misiurewicz, Michał, and Sang Young, Lai, Periodic points and topological entropy of one-dimensional maps, In Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), volume 819 of Lecture Notes in Math., pages 18–34. Springer, Berlin, 1980.Google Scholar
Bowen, Rufus and Franks, John, Homology for zero-dimensional nonwandering sets, Ann. Math. (2), 106 (1977), no. 1, 73–92.Google Scholar
Boyle, Mike, Shift equivalence and the Jordan form away from zero. Ergodic Theory Dynam. Systems, 4 (1984), no. 3, 367–379.Google Scholar
Boyle, M., Nasu’s simple automorphisms, In Dynamical systems (College Park, MD, 1986–87), volume 1342 of Lecture Notes in Math., pages 23–32. Springer, Berlin, 1988.Google Scholar
Boyle, Mike, Algebraic aspects of symbolic dynamics, In Topics in symbolic dynamics and applications (Temuco, 1997), volume 279 of London Math. Soc. Lecture Note Ser., pages 57–88, Cambridge Univ. Press, Cambridge, 2000.Google Scholar
Boyle, Mike, Eventual extensions of finite codes. Proc. Amer. Math. Soc., 104 (1988), no. 3, 965–972.Google Scholar
Boyle, Mike, Symbolic dynamics and matrices. In Combinatorial and graph-theoretical problems in linear algebra (Minneapolis, MN, 1991), volume 50 of IMA Vol. Math. Appl., pages 1–38. Springer, New York, 1993.Google Scholar
Boyle, Mike, Positive K-theory and symbolic dynamics. In Dynamics and randomness (Santiago, 2000), volume 7 of Nonlinear Phenom. Complex Systems, pages 31–52. Kluwer Acad. Publ., Dordrecht, 2002.Google Scholar
Boyle, Mike, Open problems in symbolic dynamics. In Geometric and probabilistic structures in dynamics, volume 469 of Contemp. Math., pages 69–118. Amer. Math. Soc., Providence, RI, 2008.Google Scholar
Boyle, Mike, Notes on the Perron–Frobenius theory of nonnegative matrices. website www.math.umd.edu/~mboyle/papers/pfnotes.pdf, 2019. 7 pages.Google Scholar
Boyle, Mike, Meier Carlsen, Toke, and Eilers, Søren, Flow equivalence and isotopy for subshifts. Dyn. Syst., 32 (2017), no. 3, 305–325.Google Scholar
Boyle, M., Carlsen, T. M., and Eilers, S., Corrigendum: “Flow equivalence and isotopy for subshifts” [MR3669803]. Dyn. Syst., 32 (2017), no. 3, ii.Google Scholar
Boyle, Mike, Meier Carlsen, Toke, and Eilers, Søren, Flow equivalence of G-SFTs. Trans. Amer. Math. Soc., 373 (2020), no. 4, 2591–2657.Google Scholar
Boyle, Mike and Chuysurichay, Sompong, The mapping class group of a shift of finite type. J. Mod. Dyn., 13 (2018), 115–145.CrossRefGoogle Scholar
Boyle, Mike and Fiebig, Ulf-Rainer, The action of inert finite-order automor-phisms on finite subsystems of the shift. Ergodic Theory Dynam. Systems, 11 (1991), no. 3, 413–425.Google Scholar
Boyle, Mike and Handelman, David, The spectra of nonnegative matrices via symbolic dynamics (including Appendix 4 joint with Kim and Roush). Ann. Math. (2), 133 (1991), no. 2, 249–316.Google Scholar
Boyle, Mike and Handelman, David, Algebraic shift equivalence and primitive matrices. Trans. Amer. Math. Soc., 336 (1993), no. 1, 121–149.Google Scholar
Mike Boyle, K. H. Kim, and F. W. Roush, Path methods for strong shift equivalence of positive matrices. Acta Appl. Math., 126 (2013), 65–115.CrossRefGoogle Scholar
Boyle, Mike and Krieger, Wolfgang, Periodic points and automorphisms of the shift. Trans. Amer. Math. Soc., 302 (1987), no. 1, 125–149.Google Scholar
Boyle, Mike, Lind, Douglas, and Rudolph, Daniel, The automorphism group of a shift of finite type. Trans. Amer. Math. Soc., 306 (1988), no. 1, 71–114.Google Scholar
Boyle, Mike, Marcus, Brian, and Trow, Paul, Resolving maps and the dimension group for shifts of finite type. Mem. Amer. Math. Soc., 70 (1987), no. 377, vi+146pp.CrossRefGoogle Scholar
Boyle, Mike and Schmieding, Scott, Strong shift equivalence and the generalized spectral conjecture for nonnegative matrices. Linear Algebra Appl., 498 (2016), 231–243.CrossRefGoogle Scholar
Boyle, M. and Schmieding, S., Finite group extensions of shifts of finite type: K-theory, Parry and Livšic, Ergodic Theory Dynam. Systems, 37 (2017), no. 4, 1026–1059.Google Scholar
Boyle, Mike and Schmieding, Scott, Strong shift equivalence and algebraic K-theory. J. Reine Angew. Math., 752 (2019), 63–104.CrossRefGoogle Scholar
Mike Boyle and Michael C. Sullivan, Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings. Proc. London Math. Soc. (3), 91 (2005), no. 1, 184–214.Google Scholar
Boyle, M. and Wagoner, J. B., Positive algebraic K-theory and shifts of finite type. In Modern dynamical systems and applications, (ed. Brin, Michael, Hasselblatt, Boris and Pesin, Yakov), pages 45–66. Cambridge Univ. Press, Cambridge, 2004.Google Scholar
Costa, Alfredo and Steinberg, Benjamin, A categorical invariant of flow equivalence of shifts. Ergodic Theory Dynam. Systems, 36 (2016), No. 2. (2):470–513.Google Scholar
Coven, Ethan M., Endomorphisms of substitution minimal sets. Z. Wahrschein-lichkeitstheorie und Verw. Gebiete, 20 (1971/2), 129–133.CrossRefGoogle Scholar
Cuntz, Joachim and Krieger, Wolfgang, Topological Markov chains with dicyclic dimension groups. J. Reine Angew. Math., 320 (1980), 44–51.Google Scholar
Cyr, Van and Kra, Bryna, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma, 3 (2015), e5 (27pp.)CrossRefGoogle Scholar
Cyr, Van and Kra, Bryna, The automorphism group of a minimal shift of stretched exponential growth. J. Mod. Dyn., 10 (2016), 483–495.Google Scholar
Cyr, Van and Kra, Bryna, The automorphism group of a shift of subquadratic growth. Proc. Amer. Math. Soc., 144 (2016), no. 2, 613–621.Google Scholar
Sebastián Donoso, Fabien Durand, Maass, Alejandro, and Petite, Samuel, On au-tomorphism groups of low complexity subshifts. Ergodic Theory Dynam. Systems, 36 (2016), no. 1, 64–95.Google Scholar
Fabien Durand and Dominique Perrin, Dimension groups and dynamical systems—substitutions, Bratteli diagrams and Cantor systems. Cambridge University Press, Cambridge, 2022.Google Scholar
Effros, Edward G., Dimensions and C*-algebras, volume 46 of CBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, D.C., 1981.Google Scholar
Effros, Edward G., Handelman, David E., and Shen, Chao-Liang, Dimension groups and their affine representations. Amer. J. Math., 102 (1980), no. 2, 385–407.Google Scholar
Søren Eilers, Gunnar Restorff, Ruiz, Efren, and Sørenson, Adam P.W., The complete classification of unital graph C*-algebras: geometric and strong. Duke Math. J. 170 (2021), no. 11, 2421–2517.Google Scholar
Farrell, F. T., The nonfiniteness of Nil. Proc. Amer. Math. Soc., 65 (1977), no. 2, 215–216.Google Scholar
Franks, John, Flow equivalence of subshifts of finite type. Ergodic Theory Dynam. Systems, 4 (1984), no. 1, 53–66.Google Scholar
Frisch, Joshua, Schlank, Tomer, and Tamuz, Omer, Normal amenable subgroups of the automorphism group of the full shift. Ergodic Theory Dynam. Systems, 39 (2019), no. 5, 1290–1298.Google Scholar
Gilmer, Patrick M., Topological quantum field theory and strong shift equivalence. Canadian Math. Bulletin, 42 (1999), no. 2, 190–197.Google Scholar
Thierry Giordano, Hiroki Matui, Putnam, Ian F., and Skau, Christian F.. Orbit equivalence for Cantor minimal Zd-systems. Invent. Math., 179 (2010), no. 1, 119–158.Google Scholar
Thierry Giordano, Ian F. Putnam, and Christian F. Skau, Topological orbit equivalence and C*-crossed products. J. Reine Angew. Math., 469 (1995), 51–111.Google Scholar
Goodearl, K. R., Partially ordered abelian groups with interpolation, volume 20 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1986.Google Scholar
Grayson, Daniel R., The K-theory of endomorphisms. J. Algebra, 48 (1977), no. 2, 439–446.Google Scholar
Grunewald, Fritz J., Solution of the conjugacy problem in certain arithmetic groups. In Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), volume 95 of Stud. Logic Foundations Math., pages 101–139. North-Holland, Amsterdam-New York, 1980.Google Scholar
Grunewald, Fritz and Segal, Daniel, Some general algorithms. I. Arithmetic groups. Ann. Math. (2), 112 (1980), no. 3, 531–583.Google Scholar
Handelman, David, Positive matrices and dimension groups affiliated to C*-algebras and topological Markov chains. J. Operator Theory, 6 (1981), no. 1, 55–74.Google Scholar
Handelman, David, Eventually positive matrices with rational eigenvectors. Ergodic Theory Dynam. Systems, 7 (1987), no. 2, 193–196.Google Scholar
Harmon, Dennis R., NK1 of finite groups. Proc. Amer. Math. Soc., 100 (1987), no. 2, 229–232.Google Scholar
Hartman, Yair, Kra, Bryna, and Schmieding, Scott. The stabilized automorphism group of a subshift. Int. Math. Res. Not., 21 (2022), 17112–17186.Google Scholar
Hedlund, G. A., Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3 (1969), :320–375.CrossRefGoogle Scholar
Higman, G., The units of group-rings, Proc. London Math. Soc. 46 (1940), 231–248.CrossRefGoogle Scholar
Host, B. and Parreau, F., Homomorphismes entre systèmes dynamiques dèfinis par substitutions. Ergodic Theory Dynam. Systems, 9 (1989), no. 3, 469–477.Google Scholar
Jeandel, Emmanuel, Strong shift equivalence as a category notion. arXiv:2107.10734, 2021.Google Scholar
Johnson, Charles R., Row stochastic matrices similar to doubly stochastic matrices. Linear and Multilinear Algebra, 10 (1981), no. 2, 113–130.Google Scholar
Johnson, Charles R., Laffey, Thomas J., and Loewy, Raphael, The real and the symmetric nonnegative inverse eigenvalue problems are different. Proc. Amer. Math. Soc., 124 (1996), no. 12, 3647–3651.Google Scholar
Johnson, Charles R., Marijuán, Carlos, Paparella, Pietro, and Pisonero, Miriam. The NIEP. In Operator theory, operator algebras, and matrix theory, volume 267 of Oper. Theory Adv. Appl., pages 199–220. Birkhäuser/Springer, Cham, 2018.Google Scholar
Ki Hang Kim, Nicholas S. Ormes, and Fred W. Roush, The spectra of nonnegative integer matrices via formal power series. J. Amer. Math. Soc., 13 (2000), no. 4, 773–806 (electronic).CrossRefGoogle Scholar
Ki Hang Kim and Fred W. Roush, Decidability of shift equivalence. In Dynamical systems (College Park, MD, 1986–87), volume 1342 of Lecture Notes in Math., pages 374–424. Springer, Berlin, 1988.Google Scholar
Kim, K. H. and Roush, F. W., On the automorphism groups of subshifts. Pure Math. Appl. Ser. B, 1 (1990), no. 4, 203–230.Google Scholar
Kim, K. H. and Roush, F. W., On the structure of inert automorphisms of subshifts. Pure Math. Appl. Ser. B, 2 (1991), no. 1, 3–22.Google Scholar
Kim, K. H. and Roush, F. W., Solution of two conjectures in symbolic dynamics. Proc. Amer. Math. Soc., 112 (1991), no. 4, 1163–1168.Google Scholar
Kim, K. H. and Roush, F. W., Topological classification of reducible subshifts. Pure Math. Appl. Ser. B, 3 (1992), nos. 2–4, 87–102.Google Scholar
Kim, K. H. and Roush, F. W., Williams’s conjecture is false for reducible sub-shifts. J. Amer. Math. Soc., 5 (1992), no. 1, 213–215.Google Scholar
Kim, K. H. and Roush, F. W., The Williams conjecture is false for irreducible subshifts. Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 105–109 (electronic).CrossRefGoogle Scholar
Kim, K. H. and Roush, F. W., The Williams conjecture is false for irreducible subshifts. Ann. Math., (2) 149 (1999), no. 2, 545–558.Google Scholar
Kim, K. H., Roush, F. W., and Wagoner, J. B., Automorphisms of the dimension group and gyration numbers. J. Amer. Math. Soc., 5 (1992), no. 1, 191–212.Google Scholar
Kim, K. H., Roush, F. W., and Wagoner, J. B., Inert actions on periodic points. Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 55–62 (electronic).CrossRefGoogle Scholar
Kim, K. H., Roush, F. W., and Wagoner, J. B., Characterization of inert actions on periodic points. I. Forum Math., 12 (2000), no. 5, 565–602.Google Scholar
Kim, K. H., Roush, F. W., and Wagoner, J. B., Characterization of inert actions on periodic points. II. Forum Math., 12 (2000), no. 6, 671–712.Google Scholar
Kopra, Johan, Glider automorphisms and a finitary Ryan’s theorem for transitive subshifts of finite type. Nat. Comput., 19 (2020), no. 4, 773–786.Google Scholar
Kopra, Johan, Glider automata on all transitive sofic shifts. Ergodic Theory Dynam. Systems, 42 (2022), no. 12, 3716–3744.Google Scholar
Krieger, Wolfgang, On dimension functions and topological Markov chains. Invent. Math., 56 (1980), no. 3, 239–250.Google Scholar
Laffey, Thomas J., A constructive version of the Boyle-Handelman theorem on the spectra of nonnegative matrices. Linear Algebra Appl., 436 (2012), no. 6, 1701–1709.Google Scholar
Laffey, Thomas J., Loewy, Raphael, and Šmigoc, Helena, Power series with positive coefficients arising from the characteristic polynomials of positive matrices. Math. Ann., 364 (2016), nos. 1–2, 687–707.Google Scholar
Laffey, Thomas J. and Meehan, Eleanor, A characterization of trace zero nonnegative 5 × 5 matrices, 1999. Special issue dedicated to Hans Schneider (Madison, WI, 1998).Google Scholar
Latimer, Claiborne G. and MacDuffee, C. C., A correspondence between classes of ideals and classes of matrices. Ann. Math., 34 (1933), no. 2, 313–316.Google Scholar
Lin, Chao-Hui and Rudolph, Daniel, Sections for semiflows and Kakutani shift equivalence. In Modern dynamical systems and applications, pages 145-161, Cambridge Univ. Press, Cambridge, 2004.Google Scholar
Lind, D. A., The entropies of topological Markov shifts and a related class of algebraic integers. Ergodic Theory Dynam. Systems, 4 (1984), no. 2, 283–300.Google Scholar
Douglas Lind and Brian Marcus, An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.Google Scholar
Douglas Lind and Brian Marcus, An Introduction to Symbolic Dynamics and Coding, second edition. Cambridge University Press, Cambridge, 2021.Google Scholar
Loewy, Raphael and London, David, A note on an inverse problem for nonnega-tive matrices. Linear and Multilinear Algebra, 6 (1978/9), no. 1, 83–90.CrossRefGoogle Scholar
Long, Nicholas, Mixing shifts of finite type with non-elementary surjective dimension representations. Acta Appl. Math., 126 (2013), 277–295.CrossRefGoogle Scholar
Maller, M. and Shub, M., The integral homology of Smale diffeomorphisms. Topology, 24 (1985), 153–164.CrossRefGoogle Scholar
Marcus, Brian and Tuncel, Selim, The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains. Ergodic Theory Dynam. Systems, 11 (1991), no. 1, 129–180.Google Scholar
Marcus, Daniel A., Number fields. Universitext, Springer-Verlag, New York– Heidelberg, 1977.CrossRefGoogle Scholar
Milnor, John, Introduction to algebraic K-theory. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72.Google Scholar
Mischaikow, Konstantin and Weibel, Charles, Computing the Conley Index: a Cautionary Tale. arXiv:2303.06492, 2023.Google Scholar
Marston Morse and Gustav A. Hedlund, Symbolic Dynamics. Amer. J. Math., 60(4):815–866, 1938.CrossRefGoogle Scholar
Nasu, Masakazu, Topological conjugacy for sofic systems and extensions of au-tomorphisms of finite subsystems of topological Markov shifts. In Dynamical systems (College Park, MD, 1986–87), volume 1342 of Lecture Notes in Math., pages 564–607. Springer, Berlin, 1988.Google Scholar
Nasu, Masakazu, Textile systems for endomorphisms and automorphisms of the shift. Mem. Amer. Math. Soc., 114 (1995), no. 546, viii+215pp.CrossRefGoogle Scholar
Nasu, Masakazu, The dynamics of expansive invertible onesided cellular automata. Trans. Amer. Math. Soc., 354 (2002), no. 10, 4067–4084.Google Scholar
Nasu, Masakazu, Nondegenerate q-biresolving textile systems and expansive au-tomorphisms of onesided full shifts. Trans. Amer. Math. Soc., 358 (2006), no. 2, 871–891.Google Scholar
Nasu, Masakazu, Textile systems and one-sided resolving automorphisms and endomorphisms of the shift. Ergodic Theory Dynam. Systems, 28 (2008), no. 1, 167-209.Google Scholar
Neeman, Amnon and Ranicki, Andrew, Noncommutative localisation in algebraic K-theory. I. Geom. Topol., 8 (2004), :1385–1425.CrossRefGoogle Scholar
Nenashev, A., K1 by generators and relations. J. Pure Appl. Algebra, 131 (1998), no. 2, 195–212.Google Scholar
Newman, Morris, Integral matrices. Pure and Applied Mathematics, Vol. 45. Academic Press, New York–London, 1972.Google Scholar
Oliver, Robert, Whitehead groups of finite groups, volume 132 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1988.CrossRefGoogle Scholar
Olli, Jeanette, Endomorphisms of Sturmian systems and the discrete chair substitution tiling system. Discrete Contin. Dyn. Syst., 33 (2013), no. 9, 4173–4186.Google Scholar
William Parry, Intrinsic Markov chains. Trans. Amer. Math. Soc., 112 (1964), 55–66.CrossRefGoogle Scholar
Parry, Bill and Sullivan, Dennis, A topological invariant of flows on 1-dimensional spaces. Topology, 14 (1975), no. 4, 297–299.Google Scholar
William Parry and Selim Tuncel, Classification problems in ergodic theory, volume 67 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1982.Google Scholar
Parry, William and Tuncel, Selim, On the stochastic and topological structure of Markov chains. Bull. London Math. Soc., 14 (1982), no. 1, 16–27.Google Scholar
Parry, William and Williams, R. F., Block coding and a zeta function for finite Markov chains. Proc. London Math. Soc. (3), 35 (1997), no. 3, 483–495.Google Scholar
Perrin, Dominique, On positive matrices. Theoret. Comput. Sci., 94 (1992), no. 2, 357–366.Google Scholar
Reiner, I., Maximal orders, volume 28 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original, With a foreword by M. J. Taylor.CrossRefGoogle Scholar
Restorff, Gunnar, Classification of Cuntz-Krieger algebras up to stable isomorphism. J. Reine Angew. Math., 598 (2006), 185–210.Google Scholar
Riedel, Norbert, An example on strong shift equivalence of positive integral matrices. Monatsh. Math., 95 (1983), no. 1, 45–55.Google Scholar
Roberts, Leslie G., K2 of some truncated polynomial rings. In Ring theory (Proc. Conf., Univ. Waterloo, Waterloo, 1978), volume 734 of Lecture Notes in Math., pages 249–278. Springer, Berlin, 1979. With a section written jointly with S. Geller.Google Scholar
Rørdam, Mikael, Classification of Cuntz-Krieger algebras. K-Theory, 9 (1995), no. 1, 31–58.Google Scholar
Rosenberg, Jonathan, Algebraic K-theory and its applications, volume 147 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
Patrick Ryan, J., The shift and commutativity. Math. Systems Theory, 6 (1972), 82–85.CrossRefGoogle Scholar
Patrick Ryan, J., The shift and commutativity. II. Math. Systems Theory, 8 (1974/5), no. 3, 249–250.CrossRefGoogle Scholar
Salo, Ville, Transitive action on finite points of a full shift and a finitary Ryan’s theorem. Ergodic Theory Dynam. Systems, 39 (2019), no. 6, 1637–1667.Google Scholar
Salo, Ville, Gate lattices and the stabilized automorphism group. J. Mod. Dyn., 19 (2023), 717–749.CrossRefGoogle Scholar
Ville Salo and Ilkka Törmä, Block maps between primitive uniform and Pisot substitutions. Ergodic Theory Dynam. Systems, 35 (2015), no. 7, 2292–2310.Google Scholar
Schmieding, Scott and Yang, Kitty, The mapping class group of a minimal sub-shift. Colloq. Math., 163 (2021), no. 2, 233–265.Google Scholar
Schmieding, Scott, Local P entropy and stabilized automorphism groups of sub-shifts Invent. Math., 227 (2022), no. 3, 963-995.Google Scholar
Seneta, Eugene, Non-negative matrices and Markov chains, Revised reprint of the second (1981) edition, Springer-Verlag, New York, 2006.CrossRefGoogle Scholar
Claude, E. Shannon and Warren Weaver, The Mathematical Theory of Communication. The University of Illinois Press, Urbana, Ill., 1949.Google Scholar
Sheiham, Desmond, Whitehead groups of localizations and the endomorphism class group. J. Algebra, 270 (2003), no. 1, 261–280.Google Scholar
Silver, Daniel S. and Williams, Susan G., Knot invariants from symbolic dynamical systems. Trans. Amer. Math. Soc., 351 (1999), no. 8, 3243-3265.Google Scholar
Silvester, John R., Introduction to algebraic K-theory. Chapman & Hall, London-New York, 1981.Google Scholar
H. R. Sule˘ımanova, Stochastic matrices with real characteristic numbers. Doklady Akad. Nauk SSSR (N.S.), 66 (1949), :343-345.Google Scholar
Taussky, Olga, On a theorem of Latimer and MacDuffee. Canad. J. Math., 1 (1949), 300–302.CrossRefGoogle Scholar
Taussky, Olga, On matrix classes corresponding to an ideal and its inverse. Illinois J. Math., 1 (1957), 108–113.CrossRefGoogle Scholar
Wilberd van der Kallen, Le K2 des nombres duaux. C. R. Acad. Sci. Paris Sèr. A-B, 273 (1971), A1204–A1207.Google Scholar
Wagoner, J. B., Markov partitions and K2. Inst. Hautes Études Sci. Publ. Math., 65 (1987), :91–129.CrossRefGoogle Scholar
Wagoner, J. B., Eventual finite order generation for the kernel of the dimension group representation. Trans. Amer. Math. Soc., 317 (1990), no. 1, 331–350.Google Scholar
Wagoner, J. B., Higher-dimensional shift equivalence and strong shift equivalence are the same over the integers. Proc. Amer. Math. Soc., 109 (1990), no. 2, 527–536.Google Scholar
Wagoner, J. B., Triangle identities and symmetries of a subshift of finite type. Pacific J. Math., 144 (1990), no. 1, 181–205.Google Scholar
Wagoner, J. B., Strong shift equivalence theory and the shift equivalence problem. Bull. Amer. Math. Soc. (N.S.), 36 (1999), no. 3, 271–296.Google Scholar
Wagoner, J. B., Strong shift equivalence and K2 of the dual numbers. J. Reine Angew. Math., 521 (2000), 119–160. With an appendix by K. H. Kim and F. W. Roush.Google Scholar
Weibel, Charles, NK0 and NK1 of the groups C4 and D4. Addendum to “Lower algebraic K-theory of hyperbolic 3-simplex reflection groups” by J.-F. Lafont and I. J. Ortiz [mr2495796]. Comment. Math. Helv., 84 (2009), no. 2, 339–349.Google Scholar
Weibel, Charles A., An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.CrossRefGoogle Scholar
Weibel, Charles A., The K-book: an Introduction to Algebraic K-theory, volume 145 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2013.Google Scholar
Williams, R. F., Classification of one dimensional attractors. In Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pages 341–361. Amer. Math. Soc., Providence, R.I., 1970.Google Scholar
Williams, R. F., Classification of subshifts of finite type. Ann. Math. (2), 98 (1973), 120–153; erratum, ibid. 99 (1974), 380–381.Google Scholar
Williams, R. F., Strong shift equivalence of matrices in GL(2,Z). In Symbolic dynamics and its applications (New Haven, CT, 1991), volume 135 of Contemp. Math., pages 445–451. Amer. Math. Soc., Providence, RI, 1992.Google Scholar
Yang, Kitty, Normal amenable subgroups of the automorphism group of shifts of finite type. Ergodic Theory Dynam. Systems, 41 (2021), no. 4, 1250-1263.Google Scholar
Zakharevich, Inna, Attitudes of K-theory: topological, algebraic, combinatorial. Notices Amer. Math. Soc, 66 (2019), 1034–1044.Google Scholar