Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

Abstract

We consider \(C^2\) families \(t \mapsto f_t\) of \(C^4\) nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure \(\mu _t\) 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 of CE parameters \(\Delta \), and, for each \(t_0\in \Delta \), a set \(\Delta _0\subset \Delta \) of polynomially recurrent parameters containing \(t_0\) as a Lebesgue density point, and constants \(C\ge 1\), \(\Gamma >4\), so that, for every \(1/2\)-Holder function \(A\), $$\begin{aligned} \Big |\int A\, d\mu _t -\int A\, d\mu _{t_0}\Big | \le C \Vert A\Vert _{C^{1/2}}|t-t_0|^{1/2}| \log |t-t_0||^\Gamma \!\!, \, \, \forall t \in \Delta _0\, . \end{aligned}$$ In addition, for all \(t\in \Delta _0\), the renormalisation period \(P_t\) of \(f_t\) satisfies \(P_t\le P_{t_0}\), and there are uniform bounds on the rates of mixing of \(f_t^{P_t}\) for all \(t\) with \(P_t=P_{t_0}\). If \(f_t(x)=tx(1-x)\), the set \(\Delta \) contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz–Thurston parameter \(t_0\), we find a set of CE parameters \(\Delta '_{MT}\) accumulating at \(t_0\), a constant \(C\ge 1\), and a \(C^\infty \) function \(A_0\), so that $$\begin{aligned} C |t-t_0|^{1/2}\ge \Big |\int A_0\, d\mu _t -\int A_0\, d\mu _{t_0}\Big | \ge C^{-1} |t-t_0|^{1/2}\!,\, \, \forall t \in \Delta '_{MT}\, . \end{aligned}$$