Renormalization and Quasi-Periodicity

重整化是动力系统渐近分析中最有力的工具之一。在本研究项目中，我们的目标是将重整化理论的思想和工具应用于与拟周期动力系统渐近行为有关的几个重要问题，即圆微分同态和带奇点的圆映射。我们还计划解决与具有准周期势的Schrödinger算子的谱性质有关的问题，并表明两组问题是紧密相连的。本课题的课题之一是研究具有多个断点的圆映射的重整化行为和刚性理论。最近我们构造了一个具有强对称性的映射的重整化方案。我们可以说，通常应用于线性区间交换变换的rauzy归纳，在Möbius变换的设置中有一个非常重要的对应项。我们打算证明作用于Möbius变换参数的映射具有强双曲性质。当双曲性被建立时，人们可以使用诸如畸变估计等易于理解的工具来发展一个丰富的刚性理论。另一个问题与具有奇点的圆映射族的参数依赖性有关。众所周知，在奇点存在的情况下，映射的旋转数通常是有理数。我们打算证明在“非理性”参数值上可以定义一个具有强普适性的自然条件概率分布。也就是说，这种概率分布的渐近性质仅依赖于奇异点的局部结构，如临界点的阶数或断裂的大小。接下来我们讨论具有拟周期势的Schrödinger算符。众所周知，在一维情况下，当势前的耦合常数增加时，从绝对连续谱向纯点谱转变。我们可以考虑与2D标准型地图的动作最小化轨道相关的一个自然的Schrödinger算子族。这样的轨道是奥布里-马瑟理论的主题。当最小作用轨道属于KAM不变曲线时，我们期望对应的Schrödinger算子具有绝对连续谱的正度量分量。我们推测，在破坏不变曲线后，光谱将是纯点。一个非常困难但有趣的问题是临界不变曲线的情况。很自然地，我们会期望那里有一个奇异的连续光谱。注意，重整化方法可以应用于奥布里-马瑟理论中的不变曲线、临界不变曲线和cantor型不变集的研究，以及相应的薛定谔算子的分析。

Renormalization is one of the most powerful tool in asymptotic analysis of dynamical systems. In the proposed research project our aim is to apply the ideology and the toolsof renormalization theory to several important problems related to the asymptotic behaviour of quasi-periodic dynamical systems, namely circle diffeomorphisms and circle maps withsingularities. We also plan to address problems related to spectral properties of Schrödinger operators with quasi-periodic potentials, and show that two sets of problems are deeply connected. One of the problem in our project is to study renormalization behaviour and rigidity theory for circle maps with multiple break points. Recently we have constructed a renormalizationscheme for such maps which has strong symmetry properties. One can say that Rauzyinduction, which is usually applied in the case of linear interval exchange transformation, has a very non-trivial counterpart in the setting of Möbius transformations. We intend to show that the map acting on the parameters of the Möbius transformations has strong hyperbolicproperties. When hyperbolicity is established one can develop a rich rigidity theory using a wellunderstood tools like distortion estimates etc.Another problem is connected with a parameter dependence for families of circle maps with singularities. It is well known that in the presence of singularities typically the rotation number of a map is rational. We plan to prove that one can define a natural conditional probability distribution on the "irrational" parameter values which exhibits strong universality properties.Namely, the asymptotic properties of such probability distributions depend only on the local structure of singular points, such as the order of the critical points, or the size of a break. We next discuss Schrödinger operators with quasi-periodic potential. It is well-known that in the 1D case there is a transitionfrom absolutely continuous spectrum to the pure point spectrum whenthe coupling constant in front of the potential is increasing. One can consider a natural family of Schrödinger operators related to the action-minimizing orbits for 2D Standard-type maps. Such orbits are a subject of the Aubry-Mather theory. When action-minimizing orbits belong to KAM invariant curves we expect that the corresponding Schrödinger operator will have a positive measure component of the absolutely-continuous spectrum. We conjecture that after the destruction of invariant curves the spectrum will be pure point. A very difficult but interesting questions arise is the case of critical invariant curves. It seems natural to expect a singular continuous spectrum there. Notice that the renormalization approach can be applied to the study of invariant curves, critical invariant curves, and Cantor-type invariant sets in Aubry-Mather theory, as well as in the analysisof the corresponding Schrodinger operators.