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

Eric Olivier·Alain Thomas
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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.

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Eric Olivier, Alain Thomas. 2006-07-27. Infinite products of $2\times2$ matrices and the Gibbs properties of Bernoulli convolutions. https://arxiv.org/abs/math/0607704

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