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We study the entropy of Sinai-Ruelle-Bowen measure of the geodesic flow onconvex real projective surfaces, and shows that the Hilbert area tends toinfinity if the entropy tends to zero. For the Blaschke metric, we obtain theconverse statement as well.

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