arXiv:1006.4050v1 [math.RA] 21 Jun 2010 INFINITE PRODUCTS OF NONNEGATIVE 2 × 2 MATRICES BY NONNEGATIVE VECTORS ALAIN THOMAS Abstract. Given a finite set {M 0 ,...,M d-1 } of nonnegative 2 × 2 matrices and a non- negative column-vector V , we associate to each (ω n ) ∈{0,...,d − 1} N the sequence of the column-vectors M ω1 ...M ωn V ‖M ω1 ...M ωn V ‖ . We give the necessary and sufficient condition on the matrices M k and the vector V for this sequence to converge for all (ω n ) ∈{0,...,d − 1} N such that ∀n, M ω1 ...M ωn V =  0 0  . 2000 Mathematics Subject Classification: 15A48. Introduction Let M = {M 0 ,...,M d−1 } be a finite set of nonnegative 2×2 matrices and V =  v 1 v 2  a non- negative column-vector. We use the notation Y n = Y ω n := M ω 1 ...M ωn and give the neces- sary and sufficient condition for the pointwise convergence of Y n V ‖Y n V ‖ ,(ω n ) ∈{0,...,d − 1} N such that Y n V =  0 0  for any n, where ‖·‖ is the norm-sum. The idea of the proof is that, if the conditions are satisfied, either both columns of Y n tends to the same limit, or they tend to different limits with different orders of growth, so in case V is positive the limit points of Y n V ‖Y n V ‖ only depend on the limit of the dominant column. This problem is obviously very different from the one of the convergence of Y n ‖Y n ‖ , or the convegence of the Y n itselves, see the intoduction of [5] for some counterexamples and [8, Proposition 1.2] for the infinite products of 2 × 2 stochastic matrices. The conditions for the pointwise convergence of Y n V ‖Y n V ‖ also differ from the conditions for its uniform convergence, see [2]. The uniform convergence can be used for the multifractal analysis of some continuous singular measures called Bernoulli convolutions (see [6] for the Key words and phrases. Infinite products of nonnegative matrices. 1