On the connection between a skew product IFS and the ergodic optimization for a finite family of potentials

TL;DR

This paper explores the relationship between skew product iterated function systems (IFS) on the cylinder and ergodic optimization for a finite set of potentials, revealing invariant sets, SRB measures, and solutions to Bellman equations.

Contribution

It establishes a connection between ergodic optimization and the geometric structure of invariant sets in skew product IFSs, including characterization of maximizing measures.

Findings

Existence of a compact invariant set with attractive behavior

Identification of a random SRB measure supported on that set

Connection between maximizing measures and solutions to the Bellman equation

Abstract

We study a skew product IFS on the cylinder defined by Baker-like maps associated to a finite family of potential functions and the doubling map. We show that there exist a compact invariant set with attractive behavior and a random SRB measure whose support is in that set. We also study the IFS ergodic optimization problem for that finite family of potential functions and characterize the maximizing measures and the critical value through a discounting limit. This shows the connection between this maximization problem and the superior boundary of the compact invariant set, which is described as a graph of the solution of the Bellman equation.