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We consider flows (X,T) with
X compact Hausdorff, and suitable discontinuous functions f : X → W where W is
an arbitrary compact Hausdorff space. A ring extension of the ring of all continuous
complex valued functions on X(C(X)) is formed and equipped with a norm. The
Gelfand-Naimark theorem is applied to the completion of this normed ring to
produce an almost one-to-one extension ρ : (Xf,T) → (X,T).
The question of isomorphism of flows (Xf,T) and (Xg,T) corresponding to
functions f and g is discussed, as well as the lifting of dynamical properties from
(X,T) to (Xf,T). Extension of flows via classes of discontinuous functions is
considered, showing that no new examples arise in this way. A characterization
theorem for extensions is proved when T is locally compact Hausdorff, showing that
every minimal almost one-to-one extension of (X,T) can be obtained using our
construction.
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