Invariant Measures on G/Γ for Split Simple Lie Groups G

Manfred Einsiedler, Anatoly Katok

Research output: Contribution to journalArticle

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)1184-1221
Number of pages38
JournalCommunications on Pure and Applied Mathematics
Volume56
Issue number8
DOIs
StatePublished - Aug 1 2003

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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