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Interactions Between Ergodic Theory, Lie Groups, and Number Theory

In: Proceedings of the International Congress of Mathematicians

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  • Marina Ratner

    (Standford University, Department of Mathematics
    University of California, Department of Mathematics)

Abstract

In this paper we discuss the use of dynamical and ergodic-theoretic ideas and methods to solve some long-standing problems originating from Lie groups and number theory. These problems arise from looking at actions of Lie groups on their homogeneous spaces. Such actions, viewed as dynamical systems, have long been interesting and rich objects of ergodic theory and geometry. Since the 1930s ergodic-theoretic methods have been applied to the study of geodesic and horocycle flows on unit tangent bundlesof compact surfaces of negative curvature. From the algebraic point of view the latter flows are examples of semisimple and unipotent actions on finite-volume homogeneous spaces of real Lie groups. It was established in the 1960s through the fundamental work of D. Ornstein that typical semisimple actions are all statistically the same due to their extremal randomness caused by exponential instability of orbits. Their algebraic nature has little to do with the isomorphism problem for such actions: they are measure-theoretically isomorphic as long as their entropies coincide.

Suggested Citation

  • Marina Ratner, 1995. "Interactions Between Ergodic Theory, Lie Groups, and Number Theory," Springer Books, in: S. D. Chatterji (ed.), Proceedings of the International Congress of Mathematicians, pages 157-182, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-9078-6_13
    DOI: 10.1007/978-3-0348-9078-6_13
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