Controlling the statistical properties of expanding maps

TL;DR

This paper explores how to modify expanding maps of the circle to achieve desired changes in their statistical properties, providing a mathematical framework and solutions, especially for the doubling map example.

Contribution

It introduces a mathematical structure for controlling statistical properties of expanding maps and finds solutions, including optimal ones, with a detailed analysis of the doubling map.

Findings

Many solutions exist for the control problem in expanding maps.

Complete solution provided for the doubling map case.

Existence of optimal solutions in the control problem.

Abstract

How can one change a system, in order to change its statistical properties in a prescribed way? In this note we consider a control problem related to the theory of linear response. Given an expanding map of the unit circle with an associated invariant density we can consider the inverse problem of finding which first order changes in the transformation can achieve a given first order perturbation in the density. We show the general mathematical structure of the problem, the existence of many solutions in the case of expanding maps of the circle and the existence of optimal ones. We investigate in depth the example of the doubling map, where we give a complete solution of the problem.