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Ergodic theory and rigidity on the symmetric space of non-compact type

Published online by Cambridge University Press:  26 March 2001

INKANG KIM
Affiliation:
Department of Mathematics, Seoul National University, Seoul, Korea (e-mail: inkang@math.snu.ac.kr)

Abstract

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

Type
Research Article
Copyright
© 2001 Cambridge University Press

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