On the dimension of Furstenberg measure for SL2(R) random matrix products

Michael Hochman, Boris Solomyak

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16 Scopus citations

Abstract

Let μ be a measure on SL2(R) generating a non-compact and totally irreducible subgroup, and let ν be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if μ is supported on finitely many matrices with algebraic entries, then (Formula Presented.)where hRW(μ) is the random walk entropy of μ, is the Lyapunov exponent for the random matrix product associated with μ, and dim denotes pointwise dimension. In particular, for every δ> 0 , there is a neighborhood U of the identity in SL2(R) such that if a measure μ∈ P(U) is supported on algebraic matrices with all atoms of size at least δ, and generates a group which is non-compact and totally irreducible, then its stationary measure ν satisfies dim ν= 1.

Original languageEnglish
Pages (from-to)815-875
Number of pages61
JournalInventiones Mathematicae
Volume210
Issue number3
DOIs
StatePublished - 1 Dec 2017

Bibliographical note

Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.

Keywords

  • 37F35

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