Unique ergodicity of the horocycle flow of a higher genus compact surface with no conjugate points and continuous Green bundles

TL;DR

This paper proves the unique ergodicity of the horocycle flow on certain higher genus surfaces without conjugate points, extending classical results to broader geometric contexts.

Contribution

It generalizes classical ergodic results to higher genus surfaces with no conjugate points and continuous Green bundles, including nonpositively curved surfaces.

Findings

Horocycle flow is uniquely ergodic on specified surfaces.

The proof introduces a uniformly expanding parametrization technique.

Results apply to nonflat, nonpositively curved surfaces.

Abstract

We show that the horocyclic flow of an orientable compact higher genus surface without conjugate points and with continuous Green bundles is uniquely ergodic. The result applies to nonflat nonpositively curved surfaces and generalizes a classical result of Furstenberg and Marcus in negative curvature. The proof relies on the definition of a uniformly expanding parametrization on the quotient by the strips of the surface.