论文标题
较高属表面流动的动力学和“算术”
Dynamics and 'arithmetics' of higher genus surface flows
论文作者
论文摘要
我们调查了表面（区域保护）流的研究（尤其是在平滑面积保护（或本地汉密尔顿）流的典型动力学，千古和光谱特性的研究中，以及最近在更高属中的线性化和刚性问题上的突破。我们特别关注证明这种结果所需的二磷酸条件，这些条件可以认为是托里和圆形差异性上流量的算术条件的概括。我们将解释如何通过控制重新归一化的动力学来实现上较高属流量及其繁殖部分的这些条件及其繁殖部分（即广泛的间隔交换图），但是与One属相比，它们具有更微妙的性质，因为它们通常利用了源自肾术的非均匀超物质的特征。
We survey some recent advances in the study of (area-preserving) flows on surfaces, in particular on the typical dynamical, ergodic and spectral properties of smooth area-preserving (or locally Hamiltonian) flows, as well as recent breakthroughs on linearization and rigidity questions in higher genus. We focus in particular on the Diophantine-like conditions which are required to prove such results, which can be thought of as a generalization of arithmetic conditions for flows on tori and circle diffeomorphisms. We will explain how these conditions on higher genus flows and their Poincare' sections (namely generalized interval exchange maps) can be imposed by controlling a renormalization dynamics, but are of more subtle nature than in genus one since they often exploit features which originate from the non-uniform hyperbolicity of the renormalization.
