Abstract
Let G be a semisimple Lie group acting on a space X, let μ be a symmetric compactly supported measure on G, and let A be a strongly irreducible linear cocycle over the action of G. We then have a random walk on X, and let T be the associated shift map. We show that, under certain assumptions, the cocycle A over the action of T is conjugate to a block conformal cocycle.
This statement is used in the recent paper by Eskin–Mirzakhani on the classification of invariant measures for the SL(2, ℝ) action on moduli space. The ingredients of the proof are essentially contained in the papers of Guivarch and Raugi and also Goldsheid and Margulis.
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Research of the first author is partially supported by NSF grants DMS 0244542, DMS 0604251 and DMS 0905912.
Research of the second author is partially supported by the Balzan project of Jacob Palis and by the French ANR grant “GeoDyM” (ANR-11-BS01-0004).
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Eskin, A., Matheus, C. Semisimplicity of the Lyapunov spectrum for irreducible cocycles. Isr. J. Math. 230, 973–1005 (2019). https://doi.org/10.1007/s11856-019-1841-2
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DOI: https://doi.org/10.1007/s11856-019-1841-2