Continuous extension of Anosov foliations in 3-manifolds with negatively curved fundamental group

Abstract

We study Anosov flows in 3-manifolds whose stable and unstable foliations in the universal cover have Hausdorff leaf space. We show that the intrinsic ideal boundaries of distinct stable leaves can be canonically identified and similarly for the unstable foliation. This is then applied to the case when the 3manifold has negatively curved fundamental group and leaves of the above foliations extend continuously to the ideal boundaries. We prove that the continuous extension restricted to the ideal boundaries respects the identifications of intrinsic ideal points mentioned above. We also analyse the non injectivity of the extension to the boundaries and show that there are uncountably many almost periodic, non periodic orbits of the flow which lift to flow lines with same ideal point in both directions. Finally we prove that the image of any open set in the domain ideal boundary, contains open sets in the range ideal boundary.