## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 4 (1983), 931-938.

### A Class of Bernoulli Random Matrices with Continuous Singular Stationary Measures

#### Abstract

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 $Sl_2(\mathbf{R})$.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 4 (1983), 931-938.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993442

**Digital Object Identifier**

doi:10.1214/aop/1176993442

**Mathematical Reviews number (MathSciNet)**

MR714956

**Zentralblatt MATH identifier**

0534.60029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 28A70 43A05: Measures on groups and semigroups, etc.

**Keywords**

Random matrices Bernoulli random matrices stationary measures singular measures

#### Citation

Pincus, Steve. A Class of Bernoulli Random Matrices with Continuous Singular Stationary Measures. Ann. Probab. 11 (1983), no. 4, 931--938. doi:10.1214/aop/1176993442. https://projecteuclid.org/euclid.aop/1176993442