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A $ K$-theoretic invariant for dynamical systems


Author: Yiu Tung Poon
Journal: Trans. Amer. Math. Soc. 311 (1989), 515-533
MSC: Primary 46L80; Secondary 19K14, 28D20, 46L55
DOI: https://doi.org/10.1090/S0002-9947-1989-0978367-5
MathSciNet review: 978367
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Abstract: Let $ (X,T)$ be a zero-dimensional dynamical system. We consider the quotient group $ G = C(X,Z)/B(X,T)$, where $ C(X,Z)$ is the group of continuous integer-valued functions on $ X$ and $ B(X,T)$ is the subgroup of functions of the form $ f - f \circ T$. We show that if $ (X,T)$ is topologically transitive, then there is a natural order on $ G$ which makes $ G$ an ordered group. This order structure gives a new invariant for the classification of dynamical systems. We prove that for each $ n$, the number of fixed points of $ {T^n}$ is an invariant of the ordered group $ G$. Then we show how $ G$ can be computed as a direct limit of finite rank ordered groups. This is used to study the conditions under which $ \lq G$ is a dimension group. Finally we discuss the relation between $ G$ and the $ {K_0}$-group of the crossed product $ {C^{\ast}}$-algebra associated to the system $ (X,T)$.


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  • [1] R. L. Adler and B. Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc., no. 219, 1979. MR 533691 (83h:28027)
  • [2] L. Asimow and A. Ellis, Convexity theory and its applications in functional analysis, Academic Press, New York, 1980. MR 623459 (82m:46009)
  • [3] B. Blackadar, $ K$-theory for operator theory, MSRI publication 5, Springer-Verlag, New York, 1986. MR 859867 (88g:46082)
  • [4] J. A. Bondy and U. S. R. Murty, Graph theory with applications, North-Holland, New York, 1980.
  • [5] R. Bowen and O. E. Lanford III, Zeta functions of restrictions of the shift transformation, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 43-50. MR 0271401 (42:6284)
  • [6] J. Bunce and J. Deddens, A family of simple $ {C^{\ast}}$-algebras related to weighted shift operators, J. Funct. Anal. 19 (1975), 13-24. MR 0365157 (51:1410)
  • [7] J. Cuntz, $ K$-theory for certain $ {C^{\ast}}$-algebras. II, J. Operator Theory 5 (1981), 101-108. MR 613050 (84k:46053)
  • [8] J. Cuntz and W. Krieger, Topological Markov chains with dycyclic dimension groups, J. Reine Angew. Math. 320 (1980), 44-51. MR 592141 (81m:54074)
  • [9] M. Denker, C. Grillenberger and K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Math., vol. 527, Springer-Verlag, Berlin and New York, 1976. MR 0457675 (56:15879)
  • [10] E. G. Effros, Dimensions and $ {C^{\ast}}$-algebras, CBMS Regional Conf. Ser. in Math., no. 46, Amer. Math. Soc., Providence, R.I., 1981. MR 623762 (84k:46042)
  • [11] E. G. Effros and F. Hahn, Locally compact transformation groups and $ {C^{\ast}}$-algebras, Mem. Amer. Math. Soc., no. 75, 1967. MR 0227310 (37:2895)
  • [12] E. G. Effros, D. Handelman and C. L. Shen, Dimension groups and their affine representation, Amer. J. Math. 102 (198), 385-407. MR 564479 (83g:46061)
  • [13] E. G. Effros and C. L. Shen, Approximately finite $ {C^{\ast}}$-algebras and continued fractions, Indiana J. Math. 29 (1980), 191-204. MR 563206 (81g:46076)
  • [14] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford Univ. Press, Oxford, 1960.
  • [15] W. Parry and S. Tuncel, Classification problems in ergodic theory, London Math. Soc. Lecture Notes Series 67, Cambridge Univ. Press, 1982. MR 666871 (84g:28024)
  • [16] G. K. Pedersen, $ {C^{\ast}}$-algebras and their automorphism groups, Academic Press, London, 1979. MR 548006 (81e:46037)
  • [17] M. Pimsner, Embedding some transformation group $ {C^{\ast}}$-algebra into $ AF$-algebras, Ergodic Theory Dynamical Systems 3 (1983), 613-626. MR 753927 (86d:46054)
  • [18] M. Pimsner and D. Voiculescu, Exact sequences for $ K$-groups and Ext groups of certain crossed product $ {C^{\ast}}$-algebras, J. Operator Theory 4 (1980), 93-118. MR 587369 (82c:46074)
  • [19] -, Imbedding the irrational rotations, Pacific J. Math. 93 (1981), 415-429. MR 623572 (83b:46087)
  • [20] Y. T. Poon, $ AF$ subalgebras of certain crossed products, Rocky Mountain J. Math. (to appear). MR 1065849 (91k:46078)
  • [21] I. Putnam, On the non-stable $ K$-theory of certain transformation group $ {C^{\ast}}$-algebras, preprint.
  • [22] -, The $ {C^{\ast}}$-algebras associated with minimal homeomorphisms of the Cantor set, preprint.
  • [23] M. Rieffel, $ {C^{\ast}}$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429. MR 623572 (83b:46087)
  • [24] C. Sutherland, Notes on orbit equivalence: "Kreiger's Theorem", Unpublished Lecture Notes, Universitet i Oslo, 1976.
  • [25] P. Walters, An introduction to ergodic theory, Graduate Texts in Math., vol. 79, Springer-Verlag, Berlin and New York, 1982. MR 648108 (84e:28017)
  • [26] R. F. Williams, Classifications of subshifts of finite type, Ann. of Math. (2) 98 (1973), 120-153; Errata, ibid., 99 (1974), 380-381. MR 0331436 (48:9769)
  • [27] -, Strong shift-equivalence of matrices in $ \operatorname{GL} (2,z)$, preprint.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0978367-5
Keywords: Invariant for dynamical systems, invariants for crossed products, ordering in $ K$-groups, direct limits
Article copyright: © Copyright 1989 American Mathematical Society

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