Sturmian maximizing measures for the piecewise-linear cosine family

Abstract

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 .

Explicit examples in ergodic optimization

“…We point out the interesting papers [1,9] where an estimation of the support of the maximizing probability is obtained for a certain class of potentials [but not using the approach described by (3)].…”

Ergodic optimization in dynamical systems

“…Yet another approach to non-convergence in the zero temperature limit has been introduced by Coronel & Rivera-Letelier [53], partially based on the methods of [154], establishing a certain persistence of the non-convergence phenomenon: 3 The paper [37] uses ideas from analytic geometry (semi-algebraic and sub-analytic maps) which are outside the standard toolkit of most ergodic theorists, and despite its elegant brevity, the approach of [37] has not subsequently been pursued. 4 It was noted in [46] that van Enter & Ruszel [154] had already given an example of non-convergence in the zero temperature limit, albeit in a somewhat different context: a nearest neighbour potential model with the shift map acting on a subset of (R/Z) Z , the significant difference being that the state space R/Z is non-discrete.…”

Optimizing Properties of Balanced Words

“…For example, let, as above, T be the doubling map, and let g θ (x) = cos 2π(x − θ ) or f θ (x) = 1 − 4dist T (x, θ ), where dist T is the distance on the circle R/Z. In both cases maximizing measures are Sturmian -see [3,1] and references therein.…”

“…Consequently, ρ(A ) = (ρ(A 0 A 1 )) 1/2 = (1 + √ 5)/2. 1]. Is it true that for any fixed α any maximizing sequence is "essentially periodic" like in the case α = 1?…”