A class of Bernoulli random matrices with continuous singular stationary measures. (English) Zbl 0534.60029

Summary: Assume that we have a measure \(\mu\) on \(Sl_ 2(R)\), the group of 2\(\times 2\) real matrices of determinant 1. We look at measures \(\mu\) on \(Sl_ 2(R)\) supported on two points, the Bernoulli case. Let \(P^ 1\) be real projective one-space. We look at stationary measures for \(\mu\) on \(P^ 1\). 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 \(P^ 1\) for measures on \(Sl_ 2(R)\).


60F15 Strong limit theorems
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
43A05 Measures on groups and semigroups, etc.
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