Given an integer \(k \ge 5\), and a \(C^k\) Anosov flow \(\Phi \) on some compact connected 3-manifold preserving a smooth volume, we show that the measure of maximal entropy is the volume measure if and only if \(\Phi \) is \(C^{k-\varepsilon }\)-conjugate to an algebraic flow, for \(\varepsilon >0\) arbitrarily small. Moreover, in the case of dispersing billiards, we show that if the measure of maximal entropy is the volume measure, then the Birkhoff Normal Form of regular periodic orbits with a homoclinic intersection is linear.

中文翻译：

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3D保守Anosov流动和分散台球的熵刚度

给定一个整数\（k \ ge 5 \）和一个\（C ^ k \） Anosov流\（\ Phi \）在一些保持光滑体积的紧凑连接三歧管上，我们证明了最大熵的度量是当且仅当\（\ Phi \）为\（C ^ {k- \ varepsilon} \）-代数流的共轭时，对于\（\ varepsilon> 0 \）任意小，才可以进行体积测量。此外，在分散台球的情况下，我们表明，如果最大熵的度量是体积度量，那么具有同斜交的规则周期轨道的伯克霍夫正态形式是线性的。