Renormalization and Quasi-Periodicity

重整化和准周期性

• 批准号: RGPIN-2018-04510
• 负责人: Khanin, Konstantin
• 金额: $ 2.55万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712976
• 关键词: Renormalization; Quasi; Periodicity

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 tools of renormalization theory to several important problems related to the asymptotic behaviour of quasi-periodic dynamical systems, namely circle diffeomorphisms and circle maps with singularities. 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 renormalization scheme for such maps which has strong symmetry properties. One can say that Rauzy induction, 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 hyperbolic properties. When hyperbolicity is established one can develop a rich rigidity theory using a well understood 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 when the 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 analysis of the corresponding Schrodinger operators.

重正化是动力学渐近分析的有力工具之一 系统.在拟议的研究项目中，我们的目标是应用思想和工具， 的渐近行为有关的几个重要问题 的拟周期动力系统，即圆同态和圆映射， 奇点我们还计划解决与薛定谔的光谱性质有关的问题 运营商与准周期的潜力，并表明，两套问题是深连接。 我们的课题之一是研究重整化行为和刚性理论 对于具有多个断点的圆贴图。最近我们构造了一个重正化 计划，这样的地图，具有强对称性。可以说劳齐 归纳法通常应用于线性区间交换变换的情况， 在莫比乌斯变换的背景下，一个非常重要的对应物。我们打算证明， 作用于Möbius变换参数的映射具有强双曲性 特性.当双曲性成立时，人们可以使用井来发展丰富的刚性理论。 理解工具，如失真估计等。 另一个问题是与参数依赖的家庭圆地图， 奇点众所周知，在存在奇点的情况下， 是理性的。我们计划证明人们可以定义一个自然条件概率 分布的“无理”参数值，表现出很强的普适性。 也就是说，这种概率分布的渐近性质仅取决于局部 奇异点的结构，如临界点的顺序或断裂的大小。 接下来我们讨论具有拟周期势的薛定谔算子。众所周知，在 1D情况下，存在从绝对连续谱到纯点谱的过渡， 电势前面的耦合常数增加。我们可以认为一个自然的家庭 薛定谔算子与二维标准型映射的作用极小化轨道有关。等 轨道是奥布里-马瑟理论的一个主题当最小作用轨道属于KAM时 不变曲线，我们期望相应的薛定谔算子将有一个正测度 绝对连续光谱的组成部分。 我们推测， 不变曲线的光谱将是纯点。一个非常困难但有趣的问题出现了， 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谱。 注意，重整化方法可以应用于研究不变曲线，临界曲线， 不变曲线，康托型不变集在奥布里-马瑟理论，以及在分析 对应的薛定谔算子。