An iterative process for approximating subactions

TL;DR

This paper introduces an iterative method to approximate calibrated subactions for the doubling map on the circle, analyzing its properties, convergence behavior, and computational aspects.

Contribution

It presents a novel iterative procedure for approximating subactions and provides analytical and numerical insights into its convergence and dynamics.

Findings

Fixed point is unique when the maximizing probability is unique.

Convergence rate varies, sometimes like 1/2, other times close to 1.

The method effectively computes subactions using discretization.

Abstract

We describe a procedure based on the iteration of an initial function by an appropriated operator, acting on continuous functions, in order to get a fixed point. This fixed point will be a calibrated subaction for the doubling map on the circle and a fixed Lipschitz potential. We study analytical and generic properties of this process and we provide some computational evaluations of subactions using a discretization of the circle. The fixed point is unique if the maximizing probability is unique. We proceed a careful analysis of the dynamics of this operator close by the fixed point in order to explain the difficulty in estimating its asymptotic behavior. We will show that the convergence rate can be in some moments like $1/2$ and sometimes arbitrarily close to $1$.