The linear request problem

TL;DR

This paper introduces a novel method for perturbing dynamical systems to control their invariant measures, using infinitesimal conjugacies instead of transfer operators, applicable in any dimension without hyperbolicity assumptions.

Contribution

It presents a simple, general approach to the linear request problem that works in any dimension and does not require hyperbolic conditions, expanding the scope of previous methods.

Findings

Works in any dimension without hyperbolicity assumptions

Provides an infinite-dimensional space of solutions under expansion

Avoids the use of transfer operators for perturbation analysis

Abstract

We propose a simple approach to a problem introduced by Galatolo and Pollicott, consisting in perturbing a dynamical system in order for its absolutely continuous invariant measure to change in a prescribed way. Instead of using transfer operators, we observe that restricting to an infinitesimal conjugacy already yields a solution. This allows us to work in any dimension and dispense from any dynamical hypothesis. In particular, we don't need to assume hyperbolicity to obtain a solution, although expansion moreover ensures the existence of an infinite-dimensional space of solutions.