  We consider extensions of non-singular maps which are exact, respectively
K-mixing, or at least have a decomposition into positive-measure exact,
respectively K-mixing, components. The fibers of the extension spaces have
countable (finite or infinite) cardinality and the action on them is assumed
surjective or bijective. We call these systems, respectively, fiber-surjective
and fiber-bijective extensions. Technically, they are skew products, though the
point of view we take here is not the one generally associated with skew
products. Our main results are an Exact and a K-mixing Decomposition Theorem.
The latter can be used to show that a large number of periodic Lorentz gases
(the term denoting here general group extensions of Sinai billiards, including
Lorentz tubes and slabs, in any dimension) are K-mixing.
