Cocycles and stable foliations of Axiom A flows (Q2730710)

Let \(M\) be a compact manifold, \(\{\varphi_t\}\) an Axiom A flow on \(M\), \(\Lambda\) a connected component of its non-wandering set, \(G\) a locally compact Abelian group, \(\pi: \widehat{M}\to M\) a regular covering of \(M\) with covering group \(G\), and \(\{\widehat{\varphi}_t\}\) the natural extension of \(\{\varphi_t\}\) to \(M\). The author proves that the following three conditions (1)--(3) are equivalent: NEWLINENEWLINENEWLINE(1) the strong stable foliation of \(\{\widehat{\varphi}_t\}\) is transitive, NEWLINENEWLINENEWLINE(2) the subgroup generated by the periods and Frobenius elements of the periodic orbits of \(\{\varphi_t\}\) on \(\Lambda\) is dense, NEWLINENEWLINENEWLINE(3) the stable foliation of \(\{\widehat{\varphi}_t\}\) in restriction to \(\pi^{-1}(\Lambda)\) is ergodic with respect to a natural measure.