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Explicit examples in ergodic optimization Hermes H. Ferreira1 · Artur O. Lopes1   · Elismar R. Oliveira1 Accepted: 18 September 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020

Abstract Denote by T the transformation T(x) = 2 x (mod 1). Given a potential A ∶ S1 → ℝ we are interested in exhibiting in several examples the explicit expression for the calibrated subaction V ∶ S1 → ℝ for A. The action of the 1/2 iterative procedure G , acting on continuous functions f ∶ S1 → ℝ , was analyzed in a companion paper. Given an initial condition f0 , the sequence, Gn (f0 ) 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) = sin2 (2𝜋x) and A(x) = sin(2𝜋x) . 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. Keywords  Subaction · Maximizing probability · Ergodic optimization · Iterative process · Involution kernel · Spectral radius · Eigenfunction of the Ruelle operator

1 Introduction Here we will present several examples in Ergodic Optimization where one can exhibit the maximizing probability and the subaction. The 1/2 iterative procedure is a tool (in some cases) for the corroboration of what is calculated or a helpful instrument to get important information. Comment about this last point: suppose someone in a specific example (not covered by the examples described in the present text) does not know the explicit expression for the maximizing probability and

Communicated by Philip Boyland. * Artur O. Lopes [email protected] 1



Inst. Mat - UFRGS, Porto Alegre, Brazil

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Vol.:(0123456789)



São Paulo Journal of Mathematical Sciences

the subaction. We want to show, through several worked examples, how one can proceed (using the 1/2 iterative procedure) in order to try to get explicit information. Denote by T ∶ S1 → S1 the transformation T(x) = 2 x (mod 1). We also denote by 𝜏1 ∶ [0, 1) → [0, 1∕2) and 𝜏2 ∶ [0, 1) → [1∕2, 1) the two inverse branches of T. Definition 1  For a continuous function A ∶ S1 → ℝ we denote the maximal ergodic value the number

sup

m(A) =

𝜌 is invariant for T

∫

A d𝜌.

Any invariant probability 𝜇 which attains such supremum is called a maximizing probability For general properties of maximizing probabilities see [4, 14, 26, 34, 39]. A recent survey by Jenkinson (see [35]) covers the more recent literature on the topic. We will assume here in most of the cases that A is at least Hölder continuous. The results we consider here can also be applied to the case when A acts on the interval [0, 1] (non periodic setting). Definition 2  The union of the supports o