Genericity of weak-mixing measures on geometrically finite manifolds

TL;DR

This paper proves that on geometrically finite negatively curved manifolds, the set of mixing and weak-mixing invariant measures is dense in the space of all invariant probability measures, extending to manifolds with cusps or constant curvature.

Contribution

It establishes the density of mixing and weak-mixing measures in the space of invariant measures for a broad class of negatively curved manifolds, including those with cusps or constant curvature.

Findings

Mixing measures are dense in the space of invariant measures.

Weak-mixing measures form a dense $G_{\delta}$ subset.

Results extend to manifolds with cusps or constant negative curvature.

Abstract

Let $M$ be a manifold with pinched negative sectional curvature. We show that when $M$ is geometrically finite and the geodesic flow on $T^1 M$ is topologically mixing then the set of mixing invariant measures is dense in the set $\mathscr{M}^1(T^1M)$ of invariant probability measures. This implies that the set of weak-mixing measures which are invariant by the geodesic flow is a dense $G_{\delta}$ subset of $\mathscr{M}^1(T^1 M)$. We also show how to extend these results to manifolds with cusps or with constant negative curvature.