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 .
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 .
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Petru Mironescu 1
@article{CRMATH_2015__353_12_1087_0, author = {Petru Mironescu}, title = {Profile decomposition and phase control for circle-valued maps in one dimension}, journal = {Comptes Rendus. Math\'ematique}, pages = {1087--1092}, publisher = {Elsevier}, volume = {353}, number = {12}, year = {2015}, doi = {10.1016/j.crma.2015.09.030}, language = {en}, }
Petru Mironescu. Profile decomposition and phase control for circle-valued maps in one dimension. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1087-1092. doi : 10.1016/j.crma.2015.09.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.030/
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