Unfolding of chaotic unimodal maps and the parameter dependence of natural measures (Q2712983)

Let \(f\) be a continuous interval map. An invariant measure \(\mu \) of \(f\) is said to be ``natural'' if \((1/n)\sum_{k=0}^{n-1}\delta _{f^k(x)}\) converges to \(\mu \) in the weak\(^{*}\) topology for all \(x\) in a positive measure set. Examples of natural measures are measures supported on periodic attractors and absolutely continuous invariant probability measures (ACIP) of S-unimodal maps with non-flat critical point [\textit{A. M. Blokh} and \textit{M. Yu. Lyubich}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 24, 545-573 (1991; Zbl 0790.58024)]. The author studies the (dis)continuity properties of the map \(a\to\mu_a:=\)the natural measure of \(f_a\), where \(f_a\) are some one-parameter families of unimodal maps: post-critically finite Misiurewicz maps and Benedicks-Carleson maps. The author shows that this map is severely discontinuous at values of the parameter for which \(f_a\) is chaotic.