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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.

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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


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 $(X,T)$.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 311 (1989), 515-533
  • MSC: Primary 46L80; Secondary 19K14, 28D20, 46L55
  • DOI:
  • MathSciNet review: 978367