Projective convergence of inhomogeneous $2 \times 2$ matrix products

Abstract

Each digit in a finite alphabet labels an element of a set M of 2 x 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.