Explicit examples in ergodic optimization

Abstract

Denote by T the transformation T(x) = 2 x (mod 1). Given a potential A ∶ S 1 → ℝ we are interested in exhibiting in several examples the explicit expression for the calibrated subaction V ∶ S 1 → ℝ for A. The action of the 1/2 iterative procedure G , acting on continuous functions f ∶ S 1 → ℝ , was analyzed in a companion paper. Given an initial condition f 0 , the sequence, G n (f 0) will converge to a subaction. The sharp numerical evidence obtained from this iteration allow us to guess explicit expressions for the subaction in several worked examples: among them for A(x) = sin 2 (2 x) and A(x) = sin(2 x). Here, among other things, we present piecewise analytical expressions for several calibrated subactions. The iterative procedure can also be applied to the estimation of the joint spectral radius of matrices. We also analyze the iteration of G when the subaction is not unique. Moreover, we briefly present the version of the 1/2 iterative procedure for the estimation of the main eigenfunction of the Ruelle operator.

An iterative process for approximating subactions

“…We call algorithm this iterative process. In a companion paper [13] we will consider several examples. The sharp numerical evidence obtained from the algorithm permits to guess explicit expressions for the subaction.…”

“…One interesting question is the dependence of the calibrated subaction u on the potential A (we will address this question on [13]).…”

“…In [13] we will present several examples where the calibrated subaction can be explicitly expressed. When the potential A is analytic the calibrated subaction will be piecewise analytic.…”

“…One can also consider a similar kind of algorithm for approximating eigenfunctions of the Ruelle operator (see [13]). We will leave for a future paper a more careful analysis of this case.…”

“…The algorithm we use here can be helpful for estimating the joint spectral radius (see [13]). This question is related to the recent work [28].…”