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

Each digit in a finite alphabet labels an element of a set $\mathcal{M}$ of $2 \times 2$ 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 $\mathcal{M}$ 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.

Keywords

inhomogeneous matrix product, joint spectral radius, Gibbs measure, weak Gibbs measure, sofic affine-invariant sets, measure with full dimension, Erdős problem, Bernoulli convolutions

2010 Mathematics Subject Classification

28A80, 34D08, 37C45, 37D35, 37F35, 37H15, 37L30, 37-xx

Full Text (PDF format)

Published 20 November 2015