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Eigenvalues, K-theory and Minimal Flows

Let $(Y,T)$ be a minimal suspension flow built over a dynamical system $(X,S)$ and with (strictly positive, continuous) ceiling function $f\colon X\to\R$. We show that the eigenvalues of $(Y,T)$ are contained in the range of a trace on the $K_0$-group of $(X,S)$. Moreover, a trace gives an order isomorphism of a subgroup of $K_0(C(X)\rtimes_S\mathbb{Z})$ with the group of eigenvalues of $(Y,S)$. Using this result, we relate the values of $t$ for which the time-$t$ map on minimal suspension flow is minimal, with the $K$-theory of the base of this suspension.