## A $K$-theoretic invariant for dynamical systems

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- 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 $โG$ 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 $(X,T)$.## 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, - 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 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 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, 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 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, 385โ407. 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, 191โ204. MR**563206**, DOI 10.1512/iumj.1980.29.29013
G. H. Hardy and E. M. Wright, - 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, 613โ626. 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, 93โ118. MR**587369** - Marc A. Rieffel,
*$C^{\ast }$-algebras associated with irrational rotations*, Pacific J. Math.**93**(1981), no.ย 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.ย 527โ537. MR**1065849**, DOI 10.1216/rmjm/1181073126
I. Putnam, - Marc A. Rieffel,
*$C^{\ast }$-algebras associated with irrational rotations*, Pacific J. Math.**93**(1981), no.ย 2, 415โ429. MR**623572**
C. Sutherland, - 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 10.2307/1970908
โ,

*Graph theory with applications*, North-Holland, New York, 1980.

*An introduction to the theory of numbers*, Oxford Univ. Press, Oxford, 1960.

*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.

*Notes on orbit equivalence*: "

*Kreigerโs Theorem*", Unpublished Lecture Notes, Universitet i Oslo, 1976.

*Strong shift-equivalence of matrices in*$\operatorname {GL} (2,z)$, preprint.

## 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