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Comptes Rendus Mathématique
Volume 353, n° 12
pages 1087-1092 (décembre 2015)
Doi : 10.1016/j.crma.2015.09.030
Received : 18 September 2015 ;  accepted : 30 September 2015
Profile decomposition and phase control for circle-valued maps in one dimension
Décomposition en profils et contrôle des phases des applications unimodulaires en dimension un
 

Petru Mironescu
 Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43 bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France 

Abstract

When  , maps f in   have   phases φ , but the  -seminorm of φ is not controlled by the one of f . Lack of control is illustrated by “the kink”:  , where the phase φ moves quickly from 0 to 2π . A similar situation occurs for maps  , with Moebius maps playing the role of kinks. We prove that this is the only loss of control mechanism: each map   satisfying   can be written as  , where   is a Moebius map vanishing at  , while the integer   and the phase ψ are controlled by M . In particular, we have   for some  . When  , we obtain the sharp value of  , which is  . As an application, we obtain the existence of minimal maps of degree one in   with  .

The full text of this article is available in PDF format.
Résumé

Si  , les applications f appartenant à   ont des phases φ dans  , mais la seminorme   de φ n'est pas contrôlée par celle de f . L'absence de contrôle est illustrée par « le pli » :  , où la phase φ augmente rapidement de 0 à 2π . Pour des applications  , le même phénomène apparaît, avec les transformations de Moebius jouant le rôle des plis. Nous prouvons que cet exemple est essentiellement le seul : toute application   telle que   s'écrit  , où   est une transformation de Moebius s'annulant en  , tandis que l'entier   et la phase ψ sont contrôlés par M . En particulier, nous avons   pour une constante  . Pour  , nous obtenons la valeur optimale de  , qui est  . Comme application, nous obtenons l'existence d'une application minimale de degré un dans   avec  .

The full text of this article is available in PDF format.


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