On the Size of the Resonant Set for the Products of 2x2 Matrices

Abstract

For {\theta} \in [0, 2{\pi}), consider the rotation matrix R? and h = ({\lambda}, 0; 0, 0), {\lambda} > 1. Let W_n({\theta}) denote the product of m R?'s and n h's with the condition m \leq [\epsilon\astn], (0 < \epsilon < 1). We analyze the measure of the set of {\theta} for which ||W_n({\theta})|| \geq {\lambda}?^(n*{\delta}), (0 < {\delta} < 1). This can be regarded as a model problem for the so-called Bochi-Fayad conjecture.