Measure Rigidity and Smooth Dynamics

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This project deals with a long-standing problem related to randomness. Consider a ball on a frictionless table. If the ball is set in motion, it will travel forever, making perfectly elastic collisions with the walls. If the table is a square or an equilateral triangle, there are only two possible behaviors: either the ball repeats the same periodic path forever, or it travels completely randomly in the entire polygon, eventually visiting every part of the table. This project is directed toward the basic mathematical problem of understanding the behavior of a ball when the table is a more general polygon. This is a basic problem arising in physics and statistical mechanics. The PI and his collaborators will also study other systems with similar behaviour. The project concerns the study of measure rigidity for actions of non-abelian groups, in the general context of smooth ergodic theory. Measure rigidity is traditionally studied in the context of homogeneous dynamics. Some spectacular results, such as Ratner's theorem and other advances in this direction are at the heart of the subject and its many applications. In recent work with M. Mirzakhani and in part with A. Mohammadi, the PI was able to prove dynamical rigidity results similar to Ratner's theorem in the context of Teichmueller dynamics, which is a non-homogeneous setting (and is not conjecturally conjugate to a homogenous action). In joint work with E. Lindenstrauss, these ideas were used in the homogeneous dynamics setting to prove an improved version of the celebrated Benoist-Quint theorem. In this project, the PI intends to go beyond Teichmueller and homogeneous dynamics and prove similar rigidity theorems in the much more general setting of smooth dynamics. It is expected that the results will be applied to some explicit examples, without the need for perturbation. This should have some applications outside the field, for examples, to geometry and number theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.