双曲动力系统的仿微分方法及其应用
A paradifferential approach for hyperbolic dynamical systems and applications
摘要：
本文从微局部的观点发展了一种研究非光滑双曲动力系统及相关非线性偏微分方程的仿微分方法。作为应用，我们描述了Anosov流不稳定丛$E_u$的微局部正则性，即所有$s$的$H^s$波前集.我们还恢复了Hurder-Katok和Hasselblatt在Sobolev类而不是Hölder类中的刚性结果：存在$s_0>0$，使得如果$E_u$对$s>s_0$具有$H^s$正则性，则它是光滑的（对于保体积的$3$维Anosov流具有$s_0=2$）。在Guedes Bonthonneau的附录中，也表明它可以应用于处理非光滑流和势。这项工作可以作为其他应用程序的工具箱。
英文摘要：
We develop a paradifferential approach for studying non-smooth hyperbolic dynamics and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the $H^s$ wave-front set for all $s$, of the unstable bundle $E_u$ for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than Hölder: there is $s_0>0$ such that if $E_u$ has $H^s$ regularity for $s>s_0$ then it is smooth (with $s_0=2$ for volume preserving $3$-dimensional Anosov flows). In the appendix by Guedes Bonthonneau, it is also shown that it can be applied to deal with non-smooth flows and potentials. This work could serve as a toolbox for other applications.
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