Entropy degeneration of convex projective surfaces

TL;DR

This paper demonstrates that the volume entropy of the Hilbert metric on convex projective surfaces diminishes to zero as the associated Pick differential grows without bound, revealing a deep link between geometric structures and differential growth.

Contribution

It establishes a new asymptotic relationship between Pick differentials and volume entropy on convex projective surfaces, leveraging comparability of Hilbert and Blaschke metrics.

Findings

Volume entropy tends to zero as Pick differential tends to infinity

Hilbert and Blaschke metrics are comparable, enabling the main result

Provides insight into geometric degeneration of convex projective surfaces

Abstract

We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the theorem, due to Benoist and Hulin, that the Hilbert metric and Blaschke metric are comparable.