Dynamical behaviour and ergodic theory of quasiperiodically forced maps, with particular attention to the existence and properties of strange non-chaotic attractors

许多具有重要理论和实际意义的动力系统都受到外力的作用。虽然周期强迫的影响已经相当清楚，但关于准周期强迫系统的数学结果仍然很少。因此，拟议的研究项目的主题是一个系统的动力学性质和qpf系统的最重要的类别之一，即qpf循环同胚和同胚的长期行为的研究。因此，主要重点将是两方面的：第一个目标是获得分类结果。在这个方向上的第一步已经表明，这种分类可能与一维（非受迫）动力学的相应结果部分相似，但是有更丰富的可能行为和新的有趣现象出现，这些现象在一维环境中不会发生。最终，这些研究也应该为理解锁模现象和参数化家族（如qpf Arnold圆映射）中Arnold舌的结构等现象提供基础。第二个目标是研究所谓的奇异非混沌吸引子，这似乎经常出现在准周期强迫系统。这样的吸引子表现出一个奇怪的几何和拓扑结构与非混沌动力学的非常不寻常的组合。因此，这些物体在理论物理学中引起了相当大的兴趣，但到目前为止，严格的结果仍然很少，并且只有在少数非常特殊的情况下才能证明它们的存在。然而，最近的两种方法似乎允许更大的普遍性。这将意味着第一次可以严格证明国民核算体系在各种不同模式中的广泛存在。最后，申请人的第三个目标是获得有关二维和三维动力学相关系统的更深入的知识，以使他在这个方向上的研究兴趣多样化。作为一个例子，从理论的一般环面horneomorphisms和无理伪平移的环面，这可能会提供一个出发点，为进一步的研究。

Many dynamical systems both of practical and theoretical importance are subject to external forcing. While the influence of periodic forcing is quite wellunderstood, mathematical results on quasiperiodically forced (qpf) systems are still rather few. The subject of the proposed research project is therefore a systematic study of the dynamical properties and long-time behavior of one of the most important classes of qpf systems, namely qpf circle homeo- and diffeomorphisms. Thereby, the main focus will be two-fold: The first aim is to obtain classification results. First steps in this direction already indicate that such a classification might be in partial analogy to the respective results from one-dimensional (unforced) dynamics, but there is a richer variety of possible behavior and new interesting phenomena show up which do not occur in the one-dimensional setting. Ultimately, such studies should also provide the basis for the understanding of phenomena like mode-locking and the structure of Arnold tongues in parameterized families such as the qpf Arnold circle map. The second objective is the investigation of so-called strange non-chaotic attractors, which seem to occur frequently in quasiperiodically forced systems. Such attractors exhibit the very unusual combination of a strange geometrical and topological structure with non-chaotic dynamics. Consequently these objects have evoked considerable interest in theoretical physics, but so far rigorous results are still few and a proof of their existence is available only in a few very particular situations. However, two recent approaches to this problem seemingly allow for much greater generalization. This would mean that for the first time the widespread existence of SNA in a variety of different models could be proved rigorously. Finally, a third objective of the applicant is to acquire a deeper knowledge about related systems in two and three-dimensional dynamics, in order to diversify his research interests in this direction. As an example, some aspects from the theory of general torus horneomorphisms and irrational pseudo-translations of the torus are included, which might provide a starting point for further research.