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。在所提出的研究项目中，我们的目的是将重正化理论的思想和工具应用于与拟周期动力系统的渐近行为有关的几个重要问题，即具有奇异性的圆微分同胚和圆映射。我们还计划解决与具有拟周期势的薛定谔算子的谱性质有关的问题，并表明这两组问题是紧密相连的。我们项目中的一个问题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，我们为这类映射构造了一个具有强对称性的重整化方案。可以说，通常应用于线性区间交换变换的劳兹归纳法在Möbius变换的设置中具有非常不平凡的对应物。我们要证明作用在Möbius变换参数上的映射具有强双曲性质。当双曲性被建立时，人们可以使用众所周知的工具如失真估计等来发展丰富的刚性理论。另一个问题与具有奇点的圆映射族的参数相关性有关。众所周知，在奇点存在的情况下，映射的旋转数通常是有理的。我们计划证明人们可以在“无理”参数值上定义一个自然的条件概率分布，它表现出很强的普适性。同样，这种概率分布的渐近性质只依赖于奇点的局部结构，如临界点的顺序或断裂点的大小。接下来，我们讨论具有拟周期势的薛定谔算子。众所周知，在一维情况下，当势前的耦合常数增大时，存在从绝对连续谱到纯点谱的转变。我们可以考虑一个自然的薛定谔算子族，它与2D标准型映射的极小化轨道有关。这样的轨道是奥布里-马瑟理论的一个主题。当作用量最小的轨道属于KAM不变曲线时，我们期望相应的薛定谔算子具有绝对连续谱的正的测度分量。我们推测，在不变曲线被破坏后，谱将是纯点的。出现了一个非常困难但有趣的问题是临界不变曲线的情况。在那里，似乎很自然地会有一个奇异的连续光谱。注意，重整化方法可以应用于Aubry-Mather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.