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# Asian Journal of Mathematics

## Volume 19 (2015)

### Number 5

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

Pages: 811 – 844

DOI: https://dx.doi.org/10.4310/AJM.2015.v19.n5.a2

#### Authors

#### Abstract

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

Published 20 November 2015