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

TL;DR

This paper investigates the Whitney-Holder continuity of the SRB measure for families of unimodal maps, establishing bounds on how the measure varies with parameters under certain conditions.

Contribution

It provides new bounds on the regularity of SRB measures in parameter families of unimodal maps, including both upper and lower bounds under different dynamical assumptions.

Findings

Upper bounds on measure variation with Hölder continuity

Lower bounds demonstrating non-smooth dependence

Results apply to almost all Collet-Eckmann parameters

Abstract

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 >4, 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