On the Hilbert geometry of simplicial Tits sets

TL;DR

This paper investigates the behavior of the Hilbert metric's entropy on the moduli space of convex projective structures on simplicial hyperbolic Coxeter orbifolds, showing it tends to zero at infinity in certain cases.

Contribution

It proves that the entropy of the Hilbert metric approaches zero at infinity in the moduli space for non-trivial cases, answering a question posed by M. Crampon.

Findings

Entropy tends to zero at infinity in the moduli space.

The moduli space is either a point or the real line.

Confirmed the behavior in the case where the space is the real line.

Abstract

The moduli space of convex projective structures on a simplicial hyperbolic Coxeter orbifold is either a point or the real line. Answering a question of M. Crampon, we prove that in the latter case, when one goes to infinity in the moduli space, the entropy of the Hilbert metric tends to 0.