Sinai-Ruelle-Bowen measure Entropy of geodesic flow on Convex Projective Surfaces

TL;DR

This paper investigates the relationship between entropy of Sinai-Ruelle-Bowen measures and geometric properties of convex real projective surfaces, revealing that certain areas tend to infinity as entropy approaches zero.

Contribution

It establishes a connection between entropy and area growth for geodesic flows on convex projective surfaces, especially under the Blaschke metric.

Findings

Entropy tends to zero as Hilbert area tends to infinity.

For the Blaschke metric, area tends to infinity if and only if entropy tends to zero.

Abstract

We study the entropy of Sinai-Ruelle-Bowen measure of the geodesic flow on convex real projective surfaces, and shows that the Hilbert area tends to infinity if the entropy tends to zero. For the Blaschke metric, the area tends to infinity if and only if the entropy tends to zero.