Affine extensions of a Bernoulli shift

Abstract:

(a) For any automorphism $\phi$ of a compact metric group G, and any $a > 0$, we show the existence of a free finite measure-preserving (m.p.) action of the twisted product $Z{ \times ^\phi } G$ whose restriction to Z is Bernoulli with entropy $a + h(\phi )$, $h(\phi )$ being the entropy of $\phi$ on G with Haar measure. (b) A classification is given of all free finite m.p. actions of $Z \times { ^\phi } G$ such that the action of Z on the $\sigma$-algebra of invariant sets of G is a Bernoulli action. (c) The classification of (b) is extended to “quasifree” actions: those for which the isotropy subgroups are in a single conjugacy class within G. An existence result like that of (a) holds in this case, provided certain necessary and sufficient algebraic conditions are satisfied; similarly, an isomorphism theorem for such actions holds, under certain necessary and sufficient conditions. (d) If G is a Lie group, then all actions of $Z \times { ^\phi } G$ are quasifree; if G is also connected, then the second set of additional algebraic conditions alluded to in (c) is always satisfied, while the first will be satisfied only in an obvious case. (e) Examples are given where the isomorphism theorem fails: by violation of the algebraic conditions in the quasifree case, for other reasons in the non-quasifree case.