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

Eric Olivier (1) , Alain Thomas (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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 $\displaystyle{A_1\dots A_nV\over\vert\vert A_1\dots A_nV\vert\vert}$ to converge uniformly. In application we prove that the Bernoulli convolutions related to the numeration in Pisot quadratic bases are weak Gibbs.

Mots clés: Infinite products of matrices; weak Gibbs measures; Bernoulli convolutions; Pisot numbers; $\beta$-numeration

Domaines: Théorie des nombres [math.NT]

Eric Olivier, Alain Thomas. Infinite products of $2\times2$ matrices and the Gibbs properties of Bernoulli convolutions. 2006. ⟨hal-00087679⟩