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 θ 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 θ of the maximum ergodic average for f. [ABSTRACT FROM AUTHOR]