Sturmian maximizing measures for the piecewise-linear cosine family V. Anagnostopoulou, K. D´ ıaz-Ordaz, O. Jenkinson & C. Richard Abstract. Let T be the angle-doubling map on the circle T, and consider the 1- parameter family of piecewise-linear cosine functions f θ : T → R, defined by f θ (x)= 1 − 4d T (x, θ). We identify the maximizing T -invariant measures for this family: for each θ the f θ -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 θ . 1. Introduction For T : X → X a continuous map on a compact metric space X , let M denote the (weak-∗ compact) set of T -invariant Borel probability measures on X . Given a continuous function f : X → R, a measure µ ∈M is said to be maximizing for f if  f dµ = max m∈M  f dm , i.e. if µ attains the largest possible ergodic average for f . This ergodic optimization problem, concerning the maximizing measure(s) (and corresponding maximum ergodic average) for a given triple (X,T,f ), has been the focus of considerable recent attention (see e.g. [29] for an overview). Most of this work has concerned theoretical aspects of the subject, including abstract information on the nature of maximizing measures [8, 10, 11, 13, 19, 30, 36, 37, 39, 40, 41, 42, 44, 47, 48, 49, 51], approximation of maximizing measures [9, 15, 18], connections with thermodynamic formalism [12, 17, 27, 28, 33, 35, 38, 46], and connections with partial orders on M [2, 31, 32, 34]. Comparatively little work has focused on the concrete problem of precisely iden- tifying the maximizing measure(s) for specific triples (X,T,f ). One reason for the relative lack of progress in this area stems from the intrinsic difficulty of the problem: for interesting T (e.g. hyperbolic maps) the set M is large (an infinite-dimensional simplex), and the bona fide maximizing measure(s) can be approximated arbitrarily well by ‘almost maximizing’ measures, thus complicating the task of identifying (even conjecturally) the maximizing measure(s). Nevertheless, much of the initial impetus behind the development of ergodic opti- mization revolved around specific choices of (X, T ), in particular the archetypal hyper- bolic map T (x)=2x (mod 1) on the circle X = T = R/Z. For this choice of (X, T ), 1