A 𝐾-theoretic invariant for dynamical systems

Poon, Yiu Tung

doi:10.1090/S0002-9947-1989-0978367-5

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 $‘G$ 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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