  The joint spectral radius of a finite set of real d x d matrices is defined
to be the maximum possible exponential rate of growth of products of matrices
drawn from that set. In previous work with K. G. Hare and J. Theys we showed
that for a certain one-parameter family of pairs of matrices, this maximum
possible rate of growth is attained along Sturmian sequences with a certain
characteristic ratio which depends continuously upon the parameter. In this
paper we answer some open questions from that paper by showing that the
dependence of the ratio function upon the parameter takes the form of a Devil's
staircase. We show in particular that this Devil's staircase attains every
rational value strictly between 0 and 1 on some interval, and attains
irrational values only in a set of Hausdorff dimension zero. This result
generalises to include certain one-parameter families considered by other
authors. We also give explicit formulas for the preimages of both rational and
irrational numbers under the ratio function, thereby establishing a large
family of pairs of matrices for which the joint spectral radius may be
calculated exactly.
