The beta-representation formula for the stabilisability radius

Let $\beta\in\mathbb{R}_{>1}$, let $c$ be the contraction factor associated with $A_2$, and let ${\cal M}(\theta,c,\beta)$ denote the corresponding switched-system matrix family. If $\bar{d}(\beta)$ is the limiting average digit in the beta representation associated with the switching sequence, then the beta-representation formula.

$$\lim_{\theta\to 0}\tilde{\rho}({\cal M}(\theta,c,\beta))=c^{1/(\bar{d}(\beta)+1)}$$

This formula is obtained under the assumption that the simplifications in the switched-system analysis are valid, by relating the average number of $A_1$ transformations to the average beta digit. The supplied text does not state whether the formula has been proved or identify the precise definition of $\bar{d}(\beta)$, so its status remains open.