On linear response for discontinuous perturbations of smooth endomorphisms

TL;DR

This paper investigates how physical measures of smooth endomorphisms respond Lipschitz continuously to discontinuous perturbations, establishing linear response under certain mixing conditions and applying it to a specific example.

Contribution

It introduces conditions under which linear response holds for discontinuous perturbations of smooth endomorphisms and demonstrates their applicability to a concrete case.

Findings

Physical measure varies Lipschitz continuously with parameter under perturbations.

Linear response theory is established for a class of discontinuous perturbations.

Applicability of assumptions is demonstrated through a concrete example.

Abstract

We consider discontinuous perturbations of smooth endomorphisms and show that if the perturbed family satisfies uniform mixing assumptions on standard pairs the physical measure is Lipschitz in the parameter defying the perturbation. We also study the problem of linear response for this class of perturbations. Finally we discuss the applicability of the abstract assumptions proving linear response for a concrete example.