# Transactions of the American Mathematical Society

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## A \$K\$-theoretic invariant for dynamical systemsHTML articles powered by AMS MathViewer

by Yiu Tung Poon
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 \$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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