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

• 成果类型: Article
• 署名作者: Baladi, Viviane; Benedicks, Michael; Schnellmann, Daniel
• 署名单位: University of Copenhagen; Royal Institute of Technology; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0554-8
• 发表日期: 2015
• 页码: 773-844
• 关键词: linear-response; susceptibility function; STATISTICAL PROPERTIES; stochastic stability; invariant-measures; DYNAMICAL-SYSTEMS; quadratic family; deformations; analyticity; dependence

摘要

We consider families of nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure of , as a function of on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters , and, for each , a set of polynomially recurrent parameters containing as a Lebesgue density point, and constants , , so that, for every -Holder function , In addition, for all , the renormalisation period of satisfies , and there are uniform bounds on the rates of mixing of for all with . If , the set contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter , we find a set of CE parameters accumulating at , a constant , and a function , so that C vertical bar t - t(0)vertical bar(1/2) >= vertical bar integral A(0)d mu(t) - integral A(0)d mu(t0) vertical bar >= C-1 vertical bar t - t(0)vertical bar(1/2), for all t is an element of Delta(MT)'.