 Interactions Between Ergodic Theory, Lie Groups, and Number TheoryMarina Ratner
A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 157-182 from Springer Abstract: 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. Date: 1995 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-9078-6_13 Ordering information: This item can be ordered from DOI: 10.1007/978-3-0348-9078-6_13 Access Statistics for this chapter More chapters in Springer Books from Springer |
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