Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A $K$-theoretic invariant for dynamical systems
HTML articles powered by AMS MathViewer

by Yiu Tung Poon PDF
Trans. Amer. Math. Soc. 311 (1989), 515-533 Request permission

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 $โ€˜G$ 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 $(X,T)$.
References
  • Roy L. Adler and Brian Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc. 20 (1979), no.ย 219, iv+84. MR 533691, DOI 10.1090/memo/0219
  • L. Asimow and A. J. Ellis, Convexity theory and its applications in functional analysis, London Mathematical Society Monographs, vol. 16, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 623459
  • Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
  • J. A. Bondy and U. S. R. Murty, Graph theory with applications, North-Holland, New York, 1980.
  • R. Bowen and O. E. Lanford III, Zeta functions of restrictions of the shift transformation, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp.ย 43โ€“49. MR 0271401
  • John W. Bunce and James A. Deddens, A family of simple $C^{\ast }$-algebras related to weighted shift operators, J. Functional Analysis 19 (1975), 13โ€“24. MR 0365157, DOI 10.1016/0022-1236(75)90003-8
  • Joachim Cuntz, $K$-theory for certain $C^{\ast }$-algebras. II, J. Operator Theory 5 (1981), no.ย 1, 101โ€“108. MR 613050
  • Joachim Cuntz and Wolfgang Krieger, Topological Markov chains with dicyclic dimension groups, J. Reine Angew. Math. 320 (1980), 44โ€“51. MR 592141, DOI 10.1515/crll.1980.320.44
  • Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
  • Edward G. Effros, Dimensions and $C^{\ast }$-algebras, CBMS Regional Conference Series in Mathematics, vol. 46, Conference Board of the Mathematical Sciences, Washington, D.C., 1981. MR 623762
  • Edward G. Effros and Frank Hahn, Locally compact transformation groups and $C^{\ast }$- algebras, Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence, R.I., 1967. MR 0227310
  • Edward G. Effros, David E. Handelman, and Chao Liang Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), no.ย 2, 385โ€“407. MR 564479, DOI 10.2307/2374244
  • Edward G. Effros and Chao Liang Shen, Approximately finite $C^{\ast }$-algebras and continued fractions, Indiana Univ. Math. J. 29 (1980), no.ย 2, 191โ€“204. MR 563206, DOI 10.1512/iumj.1980.29.29013
  • G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford Univ. Press, Oxford, 1960.
  • William Parry and Selim Tuncel, Classification problems in ergodic theory, Statistics: Textbooks and Monographs, vol. 41, Cambridge University Press, Cambridge-New York, 1982. MR 666871
  • Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
  • Mihai V. Pimsner, Embedding some transformation group $C^{\ast }$-algebras into AF-algebras, Ergodic Theory Dynam. Systems 3 (1983), no.ย 4, 613โ€“626. MR 753927, DOI 10.1017/S0143385700002182
  • M. Pimsner and D. Voiculescu, Exact sequences for $K$-groups and Ext-groups of certain cross-product $C^{\ast }$-algebras, J. Operator Theory 4 (1980), no.ย 1, 93โ€“118. MR 587369
  • Marc A. Rieffel, $C^{\ast }$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no.ย 2, 415โ€“429. MR 623572
  • Yiu Tung Poon, AF subalgebras of certain crossed products, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp.ย 527โ€“537. MR 1065849, DOI 10.1216/rmjm/1181073126
  • I. Putnam, On the non-stable $K$-theory of certain transformation group ${C^{\ast }}$-algebras, preprint. โ€”, The ${C^{\ast }}$-algebras associated with minimal homeomorphisms of the Cantor set, preprint.
  • Marc A. Rieffel, $C^{\ast }$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no.ย 2, 415โ€“429. MR 623572
  • C. Sutherland, Notes on orbit equivalence: "Kreigerโ€™s Theorem", Unpublished Lecture Notes, Universitet i Oslo, 1976.
  • Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
  • R. F. Williams, Classification of subshifts of finite type, Ann. of Math. (2) 98 (1973), 120โ€“153; errata, ibid. (2) 99 (1974), 380โ€“381. MR 331436, DOI 10.2307/1970908
  • โ€”, Strong shift-equivalence of matrices in $\operatorname {GL} (2,z)$, preprint.
Similar Articles
Additional Information
  • © Copyright 1989 American Mathematical Society
  • 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