R\'eponse lin\'eaire et points p\'eriodiques : cas analytique

TL;DR

This paper clarifies the proof of a formula for linear response involving periodic points for analytic expanding maps and Anosov diffeomorphisms, using dynamical determinants and transfer operators.

Contribution

It provides an explanation of the proof by Pollicott and Vytnova, focusing on the role of dynamical determinants and transfer operators in the analytic case.

Findings

The dynamical determinant can be expressed as a Fredholm determinant.

The proof applies to analytic expanding maps and Anosov diffeomorphisms.

The approach links linear response to periodic points via transfer operators.

Abstract

We explain some points of the demonstration by Pollicott and Vytnova of a formula for linear response in terms of periodic points in the case of analytic expanding maps of the circle or Anosov diffeomorphisms of the torus. The main tool is the dynamical determinant which may be written as the Fredholm determinant of a transfer operator (or a quotient of Fredholm determinants of transfer operators).