Let G=SL(2,R), or SO(n,1) or SU(n,1) act ergodically on a probability space (X,m). Consider the ergodic properties of the flow (X,m,{g_t}), where {g_t} is a Cartan subgroup of G. The geodesic flow on the (frame bundle) of a compact (real or complex) hyperbolic manifold is an example of such flow, here X=G/L is a transitive G-space, G=SO(n,1) or SU(n,1) and L is a lattice in G. In this case the flow is known to be Bernoullian.
For the general ergodic G-action, the flow (X,m,{g_t}) is always a K-flow. We construct examples where such flows are not Bernoullian. (In the paper we discuss the groups SO(n,1), however the constructions and the proofs apply to SU(n,1) as well).