#### Vol. 4, No. 2, 2011

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On the size of the resonant set for the products of $2\times 2$ matrices

### Jeffrey Allen, Benjamin Seeger and Deborah Unger

Vol. 4 (2011), No. 2, 157–166
##### Abstract

For $\theta \in \left[0,2\pi \right)$ and $\lambda >1$, consider the matrix $h=\left(\genfrac{}{}{0}{}{\lambda }{0}\genfrac{}{}{0}{}{0}{0}\right)$ and the rotation matrix ${R}_{\theta }$. Let ${W}_{n}\left(\theta \right)$ denote some product of $m$ instances of ${R}_{\theta }$ and $n$ of $h$, with the condition $m\le ϵn$ ($0<ϵ<1$). We analyze the measure of the set of $\theta$ for which $\parallel {W}_{n}\left(\theta \right)\parallel \ge {\lambda }^{\delta n}$ ($0<\delta <1$). This can be regarded as a model problem for the Bochi–Fayad conjecture.