Invariant measures on  G /Γ for split simple Lie groups  G | Zendy

Einsiedler Manfred | Zendy; Katok Anatole | Zendy

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Invariant measures on G /Γ for split simple Lie groups G

Author(s) -

Einsiedler Manfred,

Katok Anatole

Publication year - 2003

Publication title -

communications on pure and applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 3.12

H-Index - 115

eISSN - 1097-0312

pISSN - 0010-3640

DOI - 10.1002/cpa.10092

Subject(s) - mathematics , lie group , simple lie group , pure mathematics , haar measure , invariant (physics) , entropy (arrow of time) , combinatorics , mathematical physics , physics , quantum mechanics

We study the left action α of a Cartan subgroup on the space X = G /Γ, where Γ is a lattice in a simple split connected Lie group G of rank n > 1. Let μ be an α‐invariant measure on X . We give several conditions using entropy and conditional measures, each of which characterizes the Haar measure on X . Furthermore, we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure. The main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures. © 2003 Wiley Periodicals, Inc.

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