Sturmian maximizing measures for the piecewise-linear cosine family

Let T be the angle-doubling map on the circle $\mathbb{T}$ , and consider the 1-parameter family of piecewise-linear cosine functions $f_\theta :\mathbb{T} \to \mathbb{R}$ , defined by $f_\theta (x) = 1 - 4d_\mathbb{T} (x,\theta )$ . We identify the maximizing T-invariant measures for this family: for each ?? the f _?? -maximizing measure is unique and Sturmian (i.e. with support contained in some closed semi-circle). For rational p/q, we give an explicit formula for the set of functions in the family whose maximizing measure is the Sturmian measure of rotation number p/q. This allows us to analyse the variation with ?? of the maximum ergodic average for f _?? .