Projective convergence of columns for inhomogeneous products of matrices with nonnegative entries (original) (raw)

Let Pn be the n-step right product A1 • • • An, where A1, A2,. .. is a given infinite sequence of d×d matrices with nonnegative entries. In a wide range of situations, the normalized matrix product Pn/ Pn does not converge and we shall be rather interested in the asymptotic behavior of the normalized columns PnUi/ PnUi , where U1,. .. , U d are the canonical d × 1 vectors. Our main result in Theorem A gives a sufficient condition (C) over the sequence A1, A2,. .. ensuring the existence of dominant columns of Pn, having the same projective limit V : more precisely, for any rank n, there exists a partition of {1,. .. , d} made of two subsets Jn = ∅ and J c n such that each one of the sequences of normalized columns, say PnUj n / PnUj n with jn ∈ Jn tends to V as n tends to +∞ and are dominant in the sense that the ratio PnU j n / PnUj n tends to 0, as soon as j n ∈ J c n. The existence of sequences of such dominant columns implies that for any probability vector X with positive entries, the probability vector PnX/ PnX , converges as n tends to +∞. Our main application of Theorem A (and our initial motivation) is related to an Erdős problem concerned with a family of probability measures µ β (for 1 < β < 2 a real parameter) fully supported by a subinterval of the real line, known as Bernoulli convolutions. For some parameters β (actually the so-called PV-numbers) such measures are known to be linearly representable: the µ β-measure of a suitable family of nested generating intervals may be computed by means of matrix products of the form PnX, where An takes only finitely many values, say A(0),. .. , A(a), and X is a probability vector with positive entries. Because, An = A(ξn), where ξ = ξ1ξ2 • • • is a sequence (one-sided infinite word) with ξn ∈ {0,. .. , a}, we shall write Pn = Pn(ξ) the dependence of the n-step product with ξ: when the convergence of Pn(ξ)X/ Pn(ξ)X is uniform w.r.t. ξ, a sharp analysis of the measure µ β (Gibbs structures and multifractal decomposition) becomes possible. However, most of the matrices involved in the decomposition of these Bernoulli convolutions are large, sparse and it is usually not easy to prove the condition (C) of Theorem A. To illustrate the technics, we consider one parameter β for which the matrices are neither too small nor too large and we verify condition (C): this leads to the Gibbs properties of µ β .