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Transactions of the American Mathematical Society

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

 

 

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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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