A Devil's staircase associated to the joint spectral radii of a family pair of matrices

The joint spectral radius of a finite set of square matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. In joint work with Nikita Sidorov, Kevin Hare and Jacques Theys, we examine a certain one-parameter family of pairs of matrices in detail, showing that the matrix products which realise this optimal growth rate correspond to Sturmian sequences with a particular characteristic ratio. We investigate the dependence of this characteristic ratio on the parameter, and show that it takes the form of a Devil's staircase. We establish some fine properties of this Devil's staircase, answering a question posed by T. Bousch.