Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces

TL;DR

This paper extends a classical projection theorem to new settings and demonstrates that on certain negatively curved Riemann surfaces, invariant measures can have supports with surprising projection properties, revealing complex geometric structures.

Contribution

It generalizes the Besicovitch-Federer projection theorem to transversal families of mappings and applies this to analyze geodesic flows on specific Riemann surfaces.

Findings

Existence of invariant measures with 2-dimensional supports and negligible projections.

Union of complete geodesics has Hausdorff dimension 2 and is Lebesgue negligible.

Supports of projections are Lebesgue negligible despite having full Hausdorff dimension.

Abstract

We extend the Besicovitch-Federer projection theorem to transversal 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 having 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.