Header

UZH-Logo

Maintenance Infos

On mixing diffeomorphisms of the disc


Avila, Artur; Fayad, Bassam; Le Calvez, Patrice; Xu, Disheng; Zhang, Zhiyuan (2019). On mixing diffeomorphisms of the disc. Inventiones Mathematicae:Epub ahead of print.

Abstract

We prove that a Ck, k≥2 pseudo-rotation f of the disc with non-Brjuno rotation number is Ck−1-rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its C1 norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f, that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Rüssmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.

Abstract

We prove that a Ck, k≥2 pseudo-rotation f of the disc with non-Brjuno rotation number is Ck−1-rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its C1 norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f, that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Rüssmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.

Statistics

Citations

Dimensions.ai Metrics

Altmetrics

Downloads

10 downloads since deposited on 08 Jan 2020
10 downloads since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Uncontrolled Keywords:General Mathematics
Language:English
Date:11 December 2019
Deposited On:08 Jan 2020 12:38
Last Modified:15 Jan 2020 14:20
Publisher:Springer
ISSN:1432-1297
OA Status:Green
Publisher DOI:https://doi.org/10.1007/s00222-019-00937-7

Download

Green Open Access

Download PDF  'On mixing diffeomorphisms of the disc'.
Preview
Content: Submitted Version
Language: English
Filetype: PDF
Size: 431kB
View at publisher
Content: Published Version
Language: English
Filetype: PDF - Registered users only
Size: 748kB