Entropy rigidity for three dimensional volume preserving Anosov flows

Abstract

The original proof has a gap, and need extra hypothesis that the strong stable and strong unstable filiation both to be $C^1$. The argument is like the following: with the regularity, one can show that the weak-stable and weak-unstable foliation both to be $C^{1+Lip}$, and then following the same argument as in the paper one can conclude the proof. But this extra hypothesis seems implying that the flow has a contact structure. Then it will be only a result which improves the Foulon's proof on contract structure for $C^\infty$ regularity to $C^2$, not so interest.