Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle

TL;DR

This paper establishes quantitative stability and linear response results for circle diffeomorphisms with irrational rotation numbers, showing how invariant measures change under perturbations, especially for Diophantine rotation numbers, using KAM theory and transfer operators.

Contribution

It provides the first quantitative stability estimates for irrational circle diffeomorphisms under small perturbations, including numerical discretizations, and demonstrates linear response for smooth perturbations.

Findings

Invariant measure varies Hölder continuously with perturbations for Diophantine rotation numbers.

Linear response holds for smooth perturbations preserving the rotation number.

Stability results apply to perturbations from spatial discretization and numerical truncation.

Abstract

We prove quantitative statistical stability results for a large class of small $C^{0}$ perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a H\"older way under perturbation of the map and the H\"older exponent depends on the Diophantine type of the rotation number. The set of admissable perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more general ones. This is done {by means of} classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.