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

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.