Open AccessQuantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle
Author(s) -
Stefano Galatolo,
Alfonso Sorrentino
Publication year - 2022
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2021138
Subject(s) - diophantine equation , rotation number , mathematics , irrational number , rotation (mathematics) , transfer operator , pure mathematics , invariant (physics) , diophantine approximation , mathematical analysis , geometry , mathematical physics
We prove quantitative statistical stability results for a large class of small \begin{document}$ C^{0} $\end{document} perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissible 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.