 |
 |
|
|
| HAL : hal-00700880, version 1 |
| arXiv : 1205.5226 |
| Fiche détaillée |  Récupérer au format |
|
|
|
|
| Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps |
|
|
| Viviane Baladi 1Stefano Marmi 2 |
|
|
| (23/05/2012) |
|
|
|
| We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f with unique acim mu, a perturbation X, and an observable phi. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Psi(z), associated to precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the nontangential limit of the Psi(z) as z tends to 1 exists and coincides with the derivative of the acim with respect to the map (linear response formula). Applying the Wiener-Wintner theorem, we study the singularity type of nontangential limits as z tends to e^{i\omega}. An additional LIL typicality assumption on the postcritical orbit gives stronger results. |
|
|
|
|
|
|
|
|
|
|
|
| 1 : | Département de Mathématiques et Applications (DMA) |
|
|
|
CNRS : UMR8553 – Ecole normale supérieure de Paris - ENS Paris |
|
|
| 2 : | Gruppo di ricerca di sistemi dinamici (SNS PISA) |
|
|
|
Scuola Normale Superiore |
|
|
| 3 : | Institut de Mécanique Céleste et de Calcul des Ephémérides (IMCCE) |
|
|
|
CNRS : UMR8028 – INSU – Observatoire de Paris – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Lille I - Sciences et technologies |
|
|
|
|
|
|
|
|
|
| Domaine | : | Mathématiques/Systèmes dynamiques Mathématiques/Variables complexes Science non linéaire/Dynamique Chaotique |
|
|
| susceptibility function – transfer operator – natural boundary – linear response – piecewise expanding maps – interval maps |
|
|
|
| Lien vers le texte intégral : |
|
| http://fr.arXiv.org/abs/1205.5226 |
| hal-00700880, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00700880 | |
| oai:hal.archives-ouvertes.fr:hal-00700880 | |
| Contributeur : Viviane Baladi | |
| Soumis le : Jeudi 24 Mai 2012, 11:10:04 | |
| Dernière modification le : Mercredi 10 Octobre 2012, 19:47:28 | |
