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.