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

• 成果类型: Article
• 署名作者: Hochman, Michael; Solomyak, Boris
• 署名单位: Hebrew University of Jerusalem
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0740-6
• 发表日期: 2017
• 页码: 815-875
• 关键词: overlaps

摘要

Let mu be a measure on SL2(R) generating a non-compact and totally irreducible subgroup, and let nu be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if mu is supported on finitely many matrices with algebraic entries, then dim nu = min {1, h(RW)(mu)/2 chi(mu)} where h(RW)(mu) is the random walk entropy of mu, chi(mu) is the Lyapunov exponent for the random matrix product associated with mu, and dim denotes pointwise dimension. In particular, for every delta > 0, there is a neighborhood U of the identity in SL2(R) such that if a measure mu is an element of P(U) is supported on algebraic matrices with all atoms of size at least delta, and generates a group which is non-compact and totally irreducible, then its stationary measure nu satisfies dim nu = 1.