Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces, Geom. Dedicata 161

Abstract

ABSTRACT. We extend the Besicovitch-Federer projection theorem to transver-sal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow hav-ing 2-dimensional supports such that their projections to the base manifold are 2-dimensional but the supports of the projections are Lebesgue negligible. In particular, the union of complete geodesics has Hausdorff dimension 2 and is Lebesgue negligible. 1

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