Infinite products of 2×2 matrices and the Gibbs properties of Bernoulli convolutions
Eric Olivier, Alain Thomas
Abstract
We consider the infinite sequences (A\n)\n∈ of 2×2 matrices with nonnegative entries, where the A\n are taken in a finite set of matrices. Given a vector V=v\1 v\2 with v\1,v\2>0, we give a necessary and sufficient condition for A\1... A\nV|| 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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