| In cohomological terms we characterize the ergodicity of a skew product {dollar}Tsb{lcub}f{rcub}{dollar}: X x Y {dollar}longrightarrow{dollar} X x Y, {dollar}(x,y)mapsto(Tx,f(x) + y),{dollar} where T is an ergodic transformation on a probability space X and f is a cocycle on X into a compact Hausdorff topological group Y. This has already been done by Zimmer for the special case where X is a standard Borel space and Y is second countable.; Skew products satisfy certain algebraic properties which can be exploited using the fact that separability automatically occurs in situations where one would not necessarily expect it. For example, the following is a simple fact which possesses more practicality than the Fourier method, at least in some circumstances. If X and Y are measure spaces such that either X is complete and {dollar}sigma{dollar}-finite or else X and Y are {dollar}sigma{dollar}-finite, and if {dollar}fin Lsp1(Xtimes Y),{dollar} then there is a measurable subset {dollar}Xsb0{dollar} of X whose complement has measure zero such that {dollar}{lcub}f(x,{lcub}cdot{rcub}):xin Xsb0{rcub}{dollar} is a separable subset of {dollar}Lsp1(Y).{dollar} This is a consequence of the main sectional topology theorem 2.3. Theorem 2.11 provides a nonintegrable version of this result.; The sectional topology provides a powerful means for analyzing the geometry of the invariant sets of the skew product. Specifically an invariant set can not be oddly shaped, and can only be formed from a measurable rectangle of the form X x E, by shifting each of its sections "up" or "down" to obtain a skewed rectangle.; Zimmer defines minimal cocycle and we show that this is equivalent to cohomological irreducibility, the property that the field of invariant sets for the corresponding skew product consists solely of measurable rectangles of the form X x E.; If Y is second countable or there is a {dollar}{lcub}cal P{rcub}{dollar}-system for {dollar}{lcub}cal A{rcub}(Tsb{lcub}f{rcub}){dollar} with the countable intersection property (Definition 4.10), then {dollar}{lcub}cal A{rcub}(Tsb{lcub}f{rcub}) = Isbphi(Tsb{lcub}g{rcub}),{dollar} where g is a cohomologically irreducible cocycle. Here {dollar}{lcub}cal A{rcub}(Tsb{lcub}f{rcub}){dollar} is the field of invariant sets for the skew product {dollar}Tsb{lcub}f{rcub},{dollar} and I is the identity transformation on X. Thus {dollar}{lcub}cal A{rcub}(Tsb{lcub}f{rcub}){dollar} is a skew field. We generalize to skew products on X x {dollar}Omega{dollar} where {dollar}Omega{dollar} is a probability space on which Y acts. |
