A -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 be a zero-dimensional dynamical system. We consider the quotient group , where is the group of continuous integer-valued functions on and is the subgroup of functions of the form . We show that if is topologically transitive, then there is a natural order on which makes an ordered group. This order structure gives a new invariant for the classification of dynamical systems. We prove that for each , the number of fixed points of is an invariant of the ordered group . Then we show how can be computed as a direct limit of finite rank ordered groups. This is used to study the conditions under which is a dimension group. Finally we discuss the relation between and the -group of the crossed product -algebra associated to the system .

**[1]**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**, https://doi.org/10.1090/memo/0219**[2]**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****[3]**Bruce Blackadar,*𝐾-theory for operator algebras*, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR**859867****[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*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 43–49. MR**0271401****[6]**John W. Bunce and James A. Deddens,*A family of simple 𝐶*-algebras related to weighted shift operators*, J. Functional Analysis**19**(1975), 13–24. MR**0365157**, https://doi.org/10.1016/0022-1236(75)90003-8**[7]**Joachim Cuntz,*𝐾-theory for certain 𝐶*-algebras. II*, J. Operator Theory**5**(1981), no. 1, 101–108. MR**613050****[8]**Joachim Cuntz and Wolfgang Krieger,*Topological Markov chains with dicyclic dimension groups*, J. Reine Angew. Math.**320**(1980), 44–51. MR**592141**, https://doi.org/10.1515/crll.1980.320.44**[9]**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****[10]**Edward G. Effros,*Dimensions and 𝐶*-algebras*, CBMS Regional Conference Series in Mathematics, vol. 46, Conference Board of the Mathematical Sciences, Washington, D.C., 1981. MR**623762****[11]**Edward G. Effros and Frank Hahn,*Locally compact transformation groups and 𝐶*- algebras*, Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence, R.I., 1967. MR**0227310****[12]**Edward G. Effros, David E. Handelman, and Chao Liang Shen,*Dimension groups and their affine representations*, Amer. J. Math.**102**(1980), no. 2, 385–407. MR**564479**, https://doi.org/10.2307/2374244**[13]**Edward G. Effros and Chao Liang Shen,*Approximately finite 𝐶*-algebras and continued fractions*, Indiana Univ. Math. J.**29**(1980), no. 2, 191–204. MR**563206**, https://doi.org/10.1512/iumj.1980.29.29013**[14]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, Oxford Univ. Press, Oxford, 1960.**[15]**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****[16]**Gert K. Pedersen,*𝐶*-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006****[17]**Mihai V. Pimsner,*Embedding some transformation group 𝐶*-algebras into AF-algebras*, Ergodic Theory Dynam. Systems**3**(1983), no. 4, 613–626. MR**753927**, https://doi.org/10.1017/S0143385700002182**[18]**M. Pimsner and D. Voiculescu,*Exact sequences for 𝐾-groups and Ext-groups of certain cross-product 𝐶*-algebras*, J. Operator Theory**4**(1980), no. 1, 93–118. MR**587369****[19]**Marc A. Rieffel,*𝐶*-algebras associated with irrational rotations*, Pacific J. Math.**93**(1981), no. 2, 415–429. MR**623572****[20]**Yiu Tung Poon,*AF subalgebras of certain crossed products*, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 527–537. MR**1065849**, https://doi.org/10.1216/rmjm/1181073126**[21]**I. Putnam,*On the non-stable*-*theory of certain transformation group*-*algebras*, preprint.**[22]**-,*The*-*algebras associated with minimal homeomorphisms of the Cantor set*, preprint.**[23]**Marc A. Rieffel,*𝐶*-algebras associated with irrational rotations*, Pacific J. Math.**93**(1981), no. 2, 415–429. MR**623572****[24]**C. Sutherland,*Notes on orbit equivalence*: "*Kreiger's Theorem*", Unpublished Lecture Notes, Universitet i Oslo, 1976.**[25]**Peter Walters,*An introduction to ergodic theory*, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR**648108****[26]**R. F. Williams,*Classification of subshifts of finite type*, Ann. of Math. (2)**98**(1973), 120–153; errata, ibid. (2) 99 (1974), 380–381. MR**331436**, https://doi.org/10.2307/1970908**[27]**-,*Strong shift-equivalence of matrices in*, preprint.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46L80,
19K14,
28D20,
46L55

Retrieve articles in all journals with MSC: 46L80, 19K14, 28D20, 46L55

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0978367-5

Keywords:
Invariant for dynamical systems,
invariants for crossed products,
ordering in -groups,
direct limits

Article copyright:
© Copyright 1989
American Mathematical Society