PROJECTIVE CONVERGENCE OF INHOMOGENEOUS 2 × 2 MATRIX PRODUCTS ´ ERIC OLIVIER AND ALAIN THOMAS Abstract. – The 27th of March 2014 – Each digit in a finite alphabet labels an element of a set M of 2 × 2 column-allowable matrices with nonnegative entries; the right inhomogeneous product of these matrices is made up to rank n, according to a given one-sided sequence of digits; then, the n-step matrix is multiplied by a fixed vector with positive entries. Our main result provides a characterization of those M for which the direction of the n-step vector is convergent toward a limit continuous w.r.t. to the digits sequence. The applications are concerned with Bernoulli convolutions and the Gibbs properties of linearly representable measures. 1. Introduction The set M := {M 0 ,...,M s−1 } is made of 2 × 2 column-allowable matrices (i.e. with no null column) having nonnegative entries. We note S N the product space of the one-sided infinite sequences ξ = ξ 0 ξ 1 ··· with digits in S = {0,..., s − 1} and consider the right inhomogeneous matrix product M n (ξ ) := M ξ 0 ··· M ξ n−1 . Given 0 ≤ α ≤ 1 work we study the limit direction map p α : S N → [0 ; 1], where (provided it exists) p α (ξ ) is the limit of the first entry of the probability vector M n (ξ )U α /‖M n (ξ )‖ : here and throughout, U α is the probability vector whose first entry equals α and ‖·‖ stands for the matrix norm obtained by summing the modulus of the matrix entries. When the map ξ → p α (ξ ) is well defined on the whole space S N , we call M = {M 0 ,...,M s−1 } a α-Right Projective Convergent Product (α-RPCP) set; if in addition ξ → p α (ξ ) is continuous on S N (endowed with the product topology), we call M a continuous α-RPCP set. This later notion is to be compared with the Right Convergent Product (RCP) sets of matrices introduced by Daubechies & Lagarias in [DL92, DL01]. Our main result in Theorem A provides a characterization of those M which are continuous RPCP sets (the case of the non-continuous RPCP set is developed in [OT13b, OT13c]). Theorem B shows how existence and continuity of the limit direction map ξ → p α (ξ ) may be related to the Gibbs properties of linearly representable probability measures that we call M-measures. The motivation for studying this question originates in several works concerned with multi- fractal analysis [Oli99][FFW01][FL02][Tes06], the variational principle for Hausdorff dimension [McM84][Bed84][KP96b][KP96a][Yay09][Oli09][Oli10][Fen11] as well as Gibbs structures within different classes of Bernoulli convolutions [SV98][DST99][HL01][FO03][Fen05][OST05][Oli12]. The statements of Theorem A and Theorem B are given in § 1.1. In Section 2, the proof of Theorem B serves as an introductive illustration of the ideas developed throughout the paper, while Theorem A is completely established in Section 3. Special attention is given to applications of the continuous RPCP property. Section 4 shows how the multifractal analysis of the level sets for Birkhoff averages is related (in some cases) to the joint spectral radius of a finite set 1991 Mathematics Subject Classification. 37-XX. Key words and phrases. Inhomogeneous matrix product, joint spectral radius, Gibbs measure, weak Gibbs measure, sofic affine-invariant sets, measure with full dimension, an Erd˝ os problem, Bernoulli convolutions. 1