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 $Sl_2(\mathbf{R})$.