LENGTH SPECTRUM OF GEODESIC SPHERES IN RANK ONE SYMMETRIC SPACES

2001 ◽  
Author(s):  
TOSHIAKI ADACHI
Keyword(s):  
Symmetric Spaces ◽  
Length Spectrum ◽  
Geodesic Spheres ◽  
Rank One

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

2001 ◽  
Vol 21 (1) ◽  
pp. 93-114 ◽  
Author(s):  
INKANG KIM
Keyword(s):  
Ergodic Theory ◽  
Symmetric Spaces ◽  
Isometry Group ◽  
Length Spectrum ◽  
Compact Type ◽  
Negatively Curved ◽  
Rank One ◽  
Convex Cocompact ◽  
Marked Length Spectrum ◽  
Locally Symmetric Manifolds

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.

scholarly journals Polar actions on rank-one symmetric spaces

1999 ◽  
Vol 53 (1) ◽  
pp. 131-175 ◽  
Author(s):  
Fabio Podestà ◽  
Gudlaugur Thorbergsson
Keyword(s):  
Symmetric Spaces ◽  
Polar Actions ◽  
Rank One
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