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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $โG$ 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 $(X,T)$.
- Roy L. Adler and Brian Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc. 20 (1979), no. 219, iv+84. MR 533691, DOI https://doi.org/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 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. 43โ49. 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), 13โ24. MR 0365157, DOI https://doi.org/10.1016/0022-1236%2875%2990003-8
- Joachim Cuntz, $K$-theory for certain $C^{\ast } $-algebras. II, J. Operator Theory 5 (1981), no. 1, 101โ108. MR 613050
- Joachim Cuntz and Wolfgang Krieger, Topological Markov chains with dicyclic dimension groups, J. Reine Angew. Math. 320 (1980), 44โ51. MR 592141, DOI https://doi.org/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. 2, 385โ407. MR 564479, DOI https://doi.org/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. 2, 191โ204. MR 563206, DOI https://doi.org/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, London Mathematical Society Lecture Note Series, vol. 67, Cambridge University Press, Cambridge-New York, 1982. Statistics: Textbooks and Monographs, 41. 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. 4, 613โ626. MR 753927, DOI https://doi.org/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. 1, 93โ118. MR 587369
- Marc A. Rieffel, $C^{\ast } $-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415โ429. 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. 527โ537. MR 1065849, DOI https://doi.org/10.1216/rmjm/1181073126 I. Putnam, On the non-stable $K$-theory of certain transformation group ${C^{\ast }}$-algebras, preprint. ---, 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. 2, 415โ429. MR 623572 C. Sutherland, Notes on orbit equivalence: "Kreigerโs 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), 120โ153; errata, ibid. (2) 99 (1974), 380โ381. MR 331436, DOI https://doi.org/10.2307/1970908 ---, Strong shift-equivalence of matrices in $\operatorname {GL} (2,z)$, preprint.
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Keywords:
Invariant for dynamical systems,
invariants for crossed products,
ordering in <IMG WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$K$">-groups,
direct limits
Article copyright:
© Copyright 1989
American Mathematical Society