Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Inventiones Mathematicae Année : 2014

Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps

Résumé

We consider C^2 families t->f_t of C^4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure m_t of f_t, as a function of t on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set D of CE parameters, and, for each s in D, a subset D0 of D of polynomially recurrent parameters containing s as a Lebesgue density point, and constants C>1, G >3, so that, for every 1/2-Holder function A (of 1/2-Holder norm |A|) and all t in D0, |\int A dm_t -\int A dm_s| < C |A| |t-s|^{1/2} |log|t-s||^G (If f_t(x)=tx(1-x), the set D contains almost all CE parameters.) Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter s, we find a set of CE parameters D' accumulating at s, a constant C >1, and an infinitely differentiable function B, so that for all t in D' C |t-s|^{1/2} > |\int B dm_t -\int B dm_s| > |t-s|^{1/2}/C

Dates et versions

hal-00788636 , version 1 (14-02-2013)

Identifiants

Citer

Viviane Baladi, Michael Benedicks, Daniel Schnellmann. Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps. Inventiones Mathematicae, 2014, 201 (3), pp.773-844. ⟨10.1007/s00222-014-0554-8⟩. ⟨hal-00788636⟩
126 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More