s'authentifier
version française rss feed
HAL : hal-00700880, version 1

Fiche détaillée  Récupérer au format
Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps
Viviane Baladi 1, Stefano Marmi 2, David Sauzin 2, 3
(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 – École normale supérieure [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
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