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November, 1983 A Class of Bernoulli Random Matrices with Continuous Singular Stationary Measures
Steve Pincus
Ann. Probab. 11(4): 931-938 (November, 1983). DOI: 10.1214/aop/1176993442


Assume that we have a measure $\mu$ on $Sl_2(\mathbf{R})$, the group of $2 \times 2$ real matrices of determinant 1. We look at measures $\mu$ on $Sl_2(\mathbf{R})$ supported on two points, the Bernoulli case. Let $\mathbf{P}^1$ be real projective one-space. We look at stationary measures for $\mu$ on $\mathbf{P}^1$. The major theorem that we prove here gives a general sufficiency condition in the Bernoulli case for the stationary measures to be singular with respect to Haar measure and nowhere atomic. Furthermore, this condition gives the first general examples we know about of continuous singular invariant (stationary) measures of $\mathbf{P}^1$ for measures on $Sl_2(\mathbf{R})$.


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Steve Pincus. "A Class of Bernoulli Random Matrices with Continuous Singular Stationary Measures." Ann. Probab. 11 (4) 931 - 938, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0534.60029
MathSciNet: MR714956
Digital Object Identifier: 10.1214/aop/1176993442

Primary: 60F15
Secondary: 28A70 , 43A05

Keywords: Bernoulli random matrices , random matrices , singular measures , Stationary measures

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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