论文标题
3D保守的Anosov流动和分散台球的熵刚度
Entropy rigidity for 3D conservative Anosov flows and dispersing billiards
论文作者
论文摘要
给定一个整数$ k \ geq 5 $，以及在某些紧凑的连接$ 3 $ 3 $ manifold上保留平稳体积的$ c^k $ anosov flow $φ$，我们表明，最大熵（mme）的量度是当$ c $ c^$ c^{k- \ varepsilon} $ conge and and and conjauga的量度（mme）是一个体积度量。 $ \ varepsilon> 0 $任意小。除刚度外，我们还研究了熵的柔韧性，并表明，相对于体积度量和悬架流的拓扑熵，悬架流的拓扑熵在$ 2 $ -TORUS上的Anosov diffemoririsps of Anosov diffemoritism of $ 2 $ -torus实现了所有可能的值。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，那么带有同质层次交点的常规周期轨道的Birkhoff正常形式是线性的。
Given an integer $k \geq 5$, and a $C^k$ Anosov flow $Φ$ on some compact connected $3$-manifold preserving a smooth volume, we show that the measure of maximal entropy (MME) is the volume measure if and only if $Φ$ is $C^{k-\varepsilon}$-conjugate to an algebraic flow, for $\varepsilon>0$ arbitrarily small. Besides the rigidity, we also study the entropy flexibility, and show that the metric entropy with respect to the volume measure and the topological entropy of suspension flows over Anosov diffeomorphisms on the $2$-torus achieve all possible values subject to natural normalizations. 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.
