Classification of conditional measures along certain invariant one-dimensional foliations

Let f:M→ M be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure μ defined on the completion of the Borel σ -algebra, and F a f -invariant one-dimensional continuous foliation of M by C¹ -leaves. Then, if f preserves a continuous F -arc length system, then we only have three possibilities for the conditional measures of μ along F , namely: (i) they are atomic for almost every leaf; or (ii) for almost every leaf, they are equivalent to the measure x3bb ₓ induced by the invariant arc-length system over F ; or (iii) for almost every leaf, their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.