A global shadow lemma and logarithm law for geometrically finite Hilbert geometries

TL;DR

This paper establishes a global shadow lemma, a Dirichlet-type theorem, and a logarithm law for geodesic excursions in geometrically finite Hilbert geometries, extending hyperbolic geometry results to convex projective settings.

Contribution

It introduces a global shadow lemma and logarithm law for Hilbert geometries, broadening the scope of geometric finiteness and measure theory in convex projective spaces.

Findings

Proved a global shadow lemma for Patterson-Sullivan measures.

Established a Dirichlet-type theorem for geodesic excursions.

Derived a logarithm law for geodesic behavior in convex Hilbert geometries.

Abstract

For geometrically finite group actions on hyperbolic metric spaces and under certain assumptions on the growth of parabolic subgroups, we prove a global shadow lemma for Patterson-Sullivan measures, as well as a Dirichlet-type theorem and a logarithm law for excursion of geodesics into cusps. We then apply these results to geometrically finite quotients of strictly convex Hilbert geometries with C^1 boundary.