Controlling the statistical properties of expanding maps. (26th May 2017)

• Record Type: Journal Article
• Authors: Galatolo, Stefano; Pollicott, Mark
• Is Part Of: Nonlinearity. Volume 30:Number 7(2017:Jul.)
• Page Start: 2737
• Page End: 2751
• Publication Date: 2017-05-26
• Subjects: linear response -- optimal control -- expanding maps; 37C30 -- 37N35 -- 49N05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa714f
• Languages: English
• ISSNs: 0951-7715

Abstract

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.