Conformal Structures and Rigidity Properties of Anosov and Partially Hyperbolic Systems

ABSTRACT The 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.