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

Stefano Galatolo; Alfonso Sorrentino

doi:10.3934/dcds.2021138

Article Contents

2022, Volume 42, Issue 2: 815-839. Doi: 10.3934/dcds.2021138

This issue Previous Article Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations Next Article Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

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

Stefano Galatolo^1, and
Alfonso Sorrentino^2,

1.
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
2.
Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della ricerca scientifica 1, 00133, Roma, Italy

Received: February 28, 2021

Revised: June 30, 2021

Early access: September 2021

Published: February 2022

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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 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.

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Mathematics Subject Classification: Primary: 37C40; Secondary: 37A30, 37J40, 37A45, 37C05, 37C75, 37E45.

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