Abstract
We apply a theory of infinite labeled graphs to studying presentations and classifications of symbolic dynamical systems, by introducing a class of infinite labeled graphs, called λ-graph systems. Its matrix presentation is called a symbolic matrix system. The notions of a λ-graph system and symbolic matrix system are generalized notions of a finite labeled graph and symbolic matrix for sofic subshifts to general subshifts. Strong shift equivalence and shift equivalence between symbolic matrix systems are formulated so that two subshifts are topologically conjugate if and only if the associated canonical symbolic matrix systems are strong shift equivalent. We construct several kinds of shift equivalence invariants for symbolic matrix systems. They are the dimension groups, the K-groups, and the Bowen–Franks groups that are generalizations of the corresponding notions for nonnegative matrices. They yield topological conjugacy invariants of subshifts. The entropic quantities called λ-entropy and volume entropy for λ-graph systems are also studied related to the topological entropy of symbolic dynamics.
Keywords
MSC2000: Primary 37B10; Secondary 28D20, 46L80
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References
Bates T, Pask D (2007) C ∗-algebras of labelled graphs. J Oper Theory 57:207–226
Blanchard F, Hansel G (1986) Systems codés. Theor Comput Sci 44:17–49
Bowen R, Franks J (1977) Homology for zero-dimensional nonwandering sets. Ann Math 106:73–92
Boyle M, Krieger W (1988) Almost Markov and shift equivalent sofic systems. In: Proceedings of Maryland special year in dynamics 1986–1987. Lecture Notes in Mathematics, vol 1342. Springer, pp 33–93
Bratteli O (1972) Inductive limits of finite-dimensional C ∗-algebras. Trans Am Math Soc 171:195–234
Brown LG (1983) The universal coefficient theorem for Ext and quasidiagonality. Operator Algebras and Group Representation, vol 17. Pitmann Press, Boston, pp 60–64
Carlsen TM, Eilers S (2004) Matsumoto K-groups associated to certain shift spaces. Doc Math 9:639–671
Carlsen TM, Eilers S (2006) K-groups associated to substitutional dynamics. J Funct Anal 238:99–117
Chomsky N, Schützenberger MP (1963) The algebraic theory of context-free languages. In: Braffort P, Hirschberg D (eds) Computer programing and formal systems. North-Holland, Amsterdam, pp 118–161
Cuntz J, Krieger W (1980) A class of C ∗-algebras and topological Markov chains. Invent Math 56:251–268
Denker M, Grillenberger C, Sigmund K (1976) Ergodic theory on compact spaces. Springer, Berlin, Heidelberg and New York
Effros EG (1981) Dimensions and C ∗-algebras. In: AMS-CBMS Reg Conf Ser Math, vol 46. American Mathematical Society, Providence, RI
Enomoto M, Fujii M, Watatani Y (1984) KMS states for gauge action on \({\mathcal{O}}_{A}\). Math Japon 29:607–619
Fischer R (1975) Sofic systems and graphs. Monats für Math 80:179–186
Franks J (1984) Flow equivalence of subshifts of finite type. Ergod Theory Dyn Syst 4:53–66
Hamachi T, Nasu M (1988) Topological conjugacy for 1-block factor maps of subshifts and sofic covers. In: Proceedings of Maryland special year in dynamics 1986–1987. Lecture Notes in Mathematics, vol 1342. Springer, pp 251–260
Hopcroft JE, Ullman JD (2001) Introduction to automata theory, languages, and computation. Addison-Wesley, Reading, MA
Katayama Y, Matsumoto K, Watatani Y (1998) Simple C ∗-algebras arising from β-expansion of real numbers. Ergod Theory Dyn Syst 18:937–962
Kim KH, Roush FW (1979) Some results on decidability of shift equivalence. J Combin Inf Syst Sci 4:123–146
Kim KH, Roush FW (1999) Williams conjecture is false for irreducible subshifts. Ann Math 149:545–558
Kitchens BP (1998) Symbolic dynamics. Springer, Berlin
Krieger W (1974) On the uniqueness of the equilibrium state. Math Syst Theory 8:97–104
Krieger W (1980) On dimension for a class of homeomorphism groups. Math Ann 252:87–95
Krieger W (1980) On dimension functions and topological Markov chains. Invent Math 56:239–250
Krieger W (1984) On sofic systems I. Isr J Math 48:305–330
Krieger W (1987) On sofic systems II. Isr J Math 60:167–176
Krieger W (2000) On subshifts and topological Markov chains. Numbers, Information and Complexity (Bielefeld 1998). Kluwer, Boston, MA, pp 453–472
Krieger W, Matsumoto K (2002) Shannon graphs, subshifts and lambda-graph systems. J Math Soc Jpn 54:877–900
Krieger W, Matsumoto K (2003) A lambda-graph system for the Dyck shift and its K-groups. Doc Math 8:79–96
Krieger W, Matsumoto K (2004) A class of topological conjugacy of subshifts. Ergod Theory Dyn Syst 24:1155–1172
Krieger W, Matsumoto K (2010) Subshifts and C ∗-algebras from one-counter codes. Contemporary Math AMS (to appear)-10pt]Please update reference [31].
Kumjian A, Pask D, Raeburn I, Renault J (1997) Graphs, groupoids and Cuntz–Krieger algebras. J Funct Anal 144:505–541
Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge
Matsumoto K (1999) A simple C ∗-algebra arising from certain subshift. J Oper Theory 42:351–370
Matsumoto K (1999) Dimension groups for subshifts and simplicity of the associated C ∗-algebras. J Math Soc Jpn 51:679–698
Matsumoto K (1999) Presentations of subshifts and their topological conjugacy invariants. Doc Math 4:285–340
Matsumoto K (2000) Stabilized C ∗-algebras constructed from symbolic dynamical systems. Ergod Theor Dyn Syst 20:821–841
Matsumoto K (2001) Bowen–Franks groups for subshifts and Ext-groups for C ∗-algebras. K Theor 23:67–104
Matsumoto K (2001) Bowen–Franks groups as an invariant for flow equivalence of subshifts. Ergod Theory Dyn Syst 21:1831–1842
Matsumoto K (2002) C ∗-algebras associated with presentations of subshifts. Doc Math 7:1–30
Matsumoto K (2003) On strong shift equivalence of symbolic matrix systems. Ergod Theory Dyn Syst 23:1551–1574
Matsumoto K (2005) Topological entropy in C ∗-algebras associated with λ-graph systems. Ergod Theor Dyn Syst 25:1935–1951
Matsumoto K (2005) K-theoretic invariants and conformal measures on the Dyck shifts. Int J Math 16:213–248
Matsumoto K (2007) Actions of symbolic dynamical systems on C ∗-algebras. J Reine Angew Math 605:23–49
Matsumoto K A class of simple C ∗-algebras arising from certain nonsofic subshifts. Preprint, arxiv:0805.2767
Matsumoto K, Watatani Y, Yoshida M (1998) KMS-states for gauge actions on C ∗-algebras associated with subshifts. Math Z 228:489–509
Nasu M (1986) Topological conjugacy for sofic shifts. Ergod Theory Dyn Syst 6:265–280
Nasu M (1995) Textile systems for endomorphisms and automorphisms of the shift. Mem Am Math Soc 114:546
Parry W, Sullivan D (1975) A topological invariant for flows on one-dimensional spaces. Topology 14:297–299
Tuncel S (1983) A dimension, dimension modules, and Markov chains. Proc Lond Math Soc 46:100–116
Weiss B (1973) Subshifts of finite type and sofic systems. Monatsh Math 77:462–474
Williams RF (1973) Classification of subshifts of finite type. Ann Math 98:120–153. Erratum (1974) Ann Math 99:380–381
Acknowledgements
The author would like to deeply thank Matthias Dehmer and Jun Ichi Fujii for their invitation to the author to write this chapter and for their helpful suggestions in the presentation of this paper.
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Matsumoto, K. (2011). Application of Infinite Labeled Graphs to Symbolic Dynamical Systems. In: Dehmer, M. (eds) Structural Analysis of Complex Networks. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4789-6_6
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DOI: https://doi.org/10.1007/978-0-8176-4789-6_6
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