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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=685921
• 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.

重正化是动力系统渐近分析中最有力的工具之一。在拟议的研究项目中，我们的目标是应用重整化理论的思想和工具来解决与准周期动力系统的渐近行为有关的几个重要问题，即圆同态和具有 * 奇点的圆映射。我们还计划解决薛定谔 * 运营商的准周期电位的频谱特性相关的问题，并表明，两组问题有很深的联系。** 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，我们构造了一个重正化 * 计划，这类映射具有强对称性。可以说，通常应用于线性区间交换变换的Rauzy* 归纳法在莫比乌斯变换的情况下有一个非常非平凡的对应物。我们打算证明 *** 映射作用于Möbius变换的参数具有强双曲 *** 性质。当双曲性建立时，可以使用像失真估计等很好理解的工具来开发丰富的刚性理论。另一个问题是与一个参数依赖家庭的圆地图与 * 奇点。众所周知，在存在奇点的情况下，映射的旋转数 * 是有理数。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，它具有很强的普适性。也就是说，这种概率分布的渐近性质只取决于奇点的局部 * 结构，如临界点的阶数或突变点的大小。** 我们接下来讨论具有准周期势的薛定谔算子。我们知道，在一维情况下，当势前耦合常数增大时，会出现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，它与二维标准型映射的作用最小化轨道有关。这种轨道是奥布里-马瑟理论的一个主题。当作用极小化轨道属于KAM * 不变曲线时，我们期望相应的薛定谔算子将具有绝对连续谱的正测度 * 分量。 我们猜想，当 * 不变曲线破坏后，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谱。* 注意，重整化方法可以应用于研究Aubry-Mather理论中的不变曲线，临界 * 不变曲线和Cantor型不变集，以及相应的薛定谔算子的分析。