Singularity of projections of 2-dimensional measures invariant under the geodesic flow

TL;DR

This paper constructs a special measure on a negatively curved Riemann surface that remains invariant under geodesic flow but projects to a singular measure on the surface, revealing complex measure-theoretic properties.

Contribution

It demonstrates the existence of invariant ergodic measures with 2-dimensional singular projections on negatively curved surfaces, extending understanding of measure invariance and singularity.

Findings

Existence of invariant ergodic measures with singular projections

Construction applicable to any compact Riemann surface with negative curvature

Highlights complex measure behavior under geodesic flow

Abstract

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.