On the ergodic properties of Cartan flows in ergodic actions of SL₂(R) and SO(n, 1)

Let G = SL₂(R) (or G = SO(n, 1)) act ergodically on a probability space (X, m). We consider the ergodic properties of the flow (X, m, {g,}), where {g_t} is a Cartan subgroup of G. The geodesic flow on a compact Riemann surface is an example of such a flow; here X = G/ Γ is a transitive G-space, G = SL₂(R) and Γ ⊂ G is a lattice. In this case the flow is Bernoullian. For the general ergodic G-action, the flow (X, m, {g_t}) is always a K-flow, however there are examples in which it is not Bernoullian.