Conformal Structures and Rigidity Properties of Anosov and Partially Hyperbolic Systems

动力系统研究是现代数学的一个分支，起源于物理学、力学和微分方程式。双曲和部分双曲系统一直是光滑动力学领域的主要研究对象之一。这些系统的指数收缩和指数膨胀产生了具有复杂和稳定轨道结构的混沌行为。PI考虑了Anosov系统和部分双曲系统，这些系统的收缩和膨胀表现出一定的一致性，即只适度地扭曲形状。在高维中，这个条件对于研究不变叶的正则性和对小扰动或对代数模型的共轭的光滑性是必不可少的。它还可以产生低维情况下不存在的显著刚性。PI计划进一步调查各种类型的一致性在正则性属性中的作用。另一个目标是研究在较弱或替代假设下的刚性，例如叶面的光滑性和几何结构的保留性。

ABSTRACTThe study of dynamical systems is a modern branch of mathematics which originated from physics, mechanics, and differential equations. Hyperbolic and partially hyperbolic systems have been one of the main objects of study in the area of smooth dynamics. The exponential contraction and expansion in these systems produces a chaotic behavior with complex and stable orbit structure. This results in a rich theory with applications in various areas of natural sciences and mathematics.The PI considers Anosov and partially hyperbolic systems whose contraction and expansion exhibit some conformality, i.e. distort shapes only moderately. In higher dimensions, this condition is essential for the study of regularity of the invariant foliations and smoothness of the conjugacy to a small perturbation or to an algebraic model. It may also yield remarkable rigidity not present in the low-dimensional case. The PI plans to investigate further the role of various types of conformality in the regularity properties. Another goal is to study rigidity under weaker or alternative assumptions such as smoothness of foliations and preservation of geometric structures.