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.
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Éric Olivier, Alain Thomas. Projective convergence of inhomogeneous $2 \times 2$ matrix products. Asian Journal of Mathematics, International Press, 2015, 19 (5), pp.811--844. ⟨10.4310/AJM.2015.v19.n5.a2⟩. ⟨hal-01387011⟩

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