Explicit examples in Ergodic Optimization

TL;DR

This paper explores explicit examples of calibrated subactions in ergodic optimization for a doubling map, using iterative procedures to derive analytical expressions and analyze non-uniqueness and spectral properties.

Contribution

It provides explicit analytical expressions for calibrated subactions in several examples, advancing understanding of ergodic optimization for the doubling map.

Findings

Explicit subactions for A(x) = sin^2(2πx) and A(x) = sin(2πx)

Piecewise analytical expressions for subactions

Analysis of iteration when subaction is not unique

Abstract

Denote by $T$ the transformation $T(x)= 2 \,x $ (mod 1). Given a potential $A:S^1 \to \mathbb{R}$ we are interested in exhibiting in several examples the explicit expression for the calibrated subaction $V: S^1 \to \mathbb{R}$ for $A$. The action of the $1/2$ iterative procedure $\mathcal{G}$, acting on continuous functions $f: S^1 \to \mathbb{R}$, was analyzed in a companion paper. Given an initial condition $f_0$, the sequence, $\mathcal{G}^n(f_0)$ 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 $A(x) = \sin^2 ( 2 \pi x)$ and $A(x) = \sin ( 2 \pi x)$. 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 $\mathcal{G}$ 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.