Classification of conditional measures along certain invariant one-dimensional foliations

Abstract Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu $ defined on the completion of the Borel $\sigma $ -algebra, and $\mathcal F$ a f -invariant one-dimensional continuous foliation of M by $C^1$ -leaves. Then, if f preserves a continuous $\mathcal {F}$ -arc length system, then we only have three possibilities for the conditional measures of $\mu $ along $\mathcal F$ , namely: (i) they are atomic for almost every leaf; or (ii) for almost every leaf, they are equivalent to the measure $\unicode{x3bb} _x$ induced by the invariant arc-length system over $\mathcal F$ ; 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.