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 $\hat{I}¼$ defined on the completion of the Borel $\ddot{I}ƒ$-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 $\hat{I}¼$ along $\mathcal{F}$, namely: - they are atomic for almost every leaf, or - for almost every leaf they are equivalent to the measure $\hat{I}{»}_x$ induced by the invariant arc-length system over $\mathcal{F}$, or - 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.