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Let $\nu$ be a measure on the set $\{0,1,\dots,b-1\}^\mathbb N$, linearlyrepresentable by means of a finite set $\mathcal M$ of square nonnegativematrices. To prove that $\nu$ has the weak Gibbs property, one generally usesthe uniform convergence on $\mathcal M^\mathbb N$ of the sequence of vectors$n\mapsto\frac{A_1\cdots A_nc}{\Vert A_1\cdots A_nc\Vert}$, where $c$ is apositive column-vector. We give a sufficient condition for this convergence tohold, that we apply to some measures defined by Bernoulli convolution.Secondarily we prove that the sequence of matrices $n\mapsto\frac{A_1\cdotsA_n}{\Vert A_1\cdots A_n\Vert}$ in general diverges. Both examples we give arerelated with the fundamental curves studied for intance by Micchelli,Prautzsch, Daubechies, Lagarias, Berger, Wang.

Normalized image of a vector by an infinite product of nonnegative matrices