We present methods to construct flows with varying set of KMS states on a given simple unital AF-algebra. It follows, for example, that for any pair D+ and D of non-empty compact metric spaces there is a flow σ=(σt)tR on the CAR algebra whose set of KMS states is homeomorphic to D+ while the set of KMS states for the inverted flow (σt)tR is homeomorphic to D. Remarkably the flows that realize all such pairs D± can be chosen to have isomorphic KMS bundles.

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38.

In general, when the flow is not a simple flow, some intersections SβσSβσ with ββ′ may be non-empty in which case the map πσ is not defined.

39.

Some parts of the construction are in fact not really necessary for this; only for the arguments in the next section.

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