Conformal Structures and Rigidity Properties of Anosov and Partially Hyperbolic Systems

动力系统的研究是数学的一个现代分支，它起源于物理学、力学和微分方程。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膨胀产生了具有复杂而稳定的轨道结构的混沌行为。这导致了丰富的理论与应用在自然科学和数学的各个领域。 Anosov和部分双曲系统的收缩和膨胀表现出一定的保形性，即扭曲的形状只有适度。在高维中，这个条件是必不可少的研究规律性的不变叶理和光滑的共轭小扰动或代数模型。 它也可以产生显著的刚性不存在于低维的情况下。PI计划进一步研究各种类型的保形性在正则性中的作用。另一个目标是在较弱或替代的假设下研究刚性，例如叶理的光滑性和几何结构的保留。

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.