Infinite products of \(2\times2\) matrices and the Gibbs properties of Bernoulli convolutions

Abstract

We consider the infinite sequences \((A\_n)\_{n\in\NN}\) of \(2\times2\) matrices with nonnegative entries, where the \(A\_n\) are taken in a finite set of matrices. Given a vector \(V=\pmatrix{v\_1\cr v\_2}\) with \(v\_1,v\_2>0\), we give a necessary and sufficient condition for \(\displaystyle{A\_1... A\_nV\over|| A\_1... A\_nV||}\) to converge uniformly. In application we prove that the Bernoulli convolutions related to the numeration in Pisot quadratic bases are weak Gibbs.