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Interactions Between Ergodic Theory, Lie Groups, and Number Theory

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Proceedings of the International Congress of Mathematicians

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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Standford University, Stanford, CA, 94305, USA

    Marina Ratner

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Marina Ratner

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  1. Marina Ratner
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Editors and Affiliations

  1. Département de Mathématiques, EPFL, 1015, Laussane, Switzerland

    S. D. Chatterji

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© 1995 Birkhäser Verlag, Basel, Switzerland

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Ratner, M. (1995). Interactions Between Ergodic Theory, Lie Groups, and Number Theory. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_13

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  • DOI: https://doi.org/10.1007/978-3-0348-9078-6_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9897-3

  • Online ISBN: 978-3-0348-9078-6

  • eBook Packages: Springer Book Archive

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