An iterative process for approximating subactions

Abstract

We describe a procedure of iterating an initial function by an appropriated operator, acting on functions, in order to derive a fixed point. This fixed point will be a calibrated subaction for the doubling map on the circle and a fixed Hölder potential. We study analytical and generic properties of this process and provide some computational evaluations of subactions using a discretization of the circle. We proceed a careful analysis of the dynamics of this operator close by the fixed point in order to explain the difficulty in understanding its asymptotic behavior. The fixed point is unique if the maximizing probability is unique. Even in this case we will show that the convergence rate can be in some moments like 1/2 and sometimes arbitrarily close to 1.With the help of the obtained calibrated subaction we also show how to approximate measures maximizing with respect to that potential, which is the main interest in Ergodic Optimization.