Entropy rigidity for three dimensional volume preserving Anosov flows

TL;DR

This paper discusses the gap in the original proof of entropy rigidity for three-dimensional volume-preserving Anosov flows, highlighting the need for additional regularity assumptions that imply contact structure, thus limiting the result's novelty.

Contribution

The paper clarifies the limitations of existing proofs and the implications of regularity assumptions on the structure of Anosov flows.

Findings

The original proof has a gap.

Extra regularity assumptions imply contact structure.

Result mainly extends Foulon's work under higher regularity.

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.