Rigidity of Furstenberg entropy for semisimple Lie group actions

Abstract

We consider the action of a semi-simple Lie group G on a compact manifold (and more generally a Borel space) X , with a measure ν stationary under a probability measure μ on G . We first establish some properties of the fundamental invariant associated with a ( G, μ) -space ( X, ν) , namely the Furstenberg entropy [? ], given by h μ(X,ν)= ∫ G ∫ X − log dg −1ν dν (x) dν(x) dμ(g). We then prove that when ( X, ν) is a P -mixing ( G, μ) -space [? ], and R -rank ( G)= r≥2 , the value of the Furstenberg entropy must coincide with one of the 2 r values h μ ( G/ Q, ν 0) , where Q⊂ G is a parabolic subgroup. We also construct counterexamples to show that this conclusion fails for both non- P -mixing actions and actions of groups with R -rank 1 . We also characterize amenable actions with a stationary measure as the actions having the maximal possible value of the Furstenberg entropy. We give applications to geometric rigidity for actions with low Furstenberg entropy, to orbit equivalence and to the cohomology of actions with stationary measure.