Unfolding chaotic quadratic maps --- parameter dependence of natural measures

Abstract

We consider perturbations of quadratic maps $f_a$ admitting an absolutely continuous invariant probability measure, where $a$ is in a certain positive measure set $\mathcal{A}$ of parameters, and show that in any neighborhood of any such an $f_a$, we find a rich fauna of dynamics. There are maps with periodic attractors as well as non-periodic maps whose critical orbit is absorbed by the continuation of any prescribed hyperbolic repeller of $f_a$. In particular, Misiurewicz maps are dense in $\mathcal{A}$. Almost all maps $f_a$ in the quadratic family is known to possess a unique natural measure, that is, an invariant probability measure $\mu_a$ describing the asymptotic distribution of almost all orbits. We discuss weak*-(dis)continuity properties of the map $a\mapsto \mu_a$ near the set $\mathcal{A}$, and prove that almost all maps in $\mathcal{A}$ have the property that $\mu_a$ can be approximated with measures supported on periodic attractors of certain nearby maps. On the other hand, for any $a \in \mathcal{A}$ and any periodic repeller $\Gamma_a$ of $f_a$, the singular measure supported on $\Gamma_a$ can also approximated with measures supported on nearby periodic attractors. It follows that $a\mapsto \mu_a$ is not weak*continuous on any full-measure subset of $(0,2]$. Some of these results extend to unimodal families with critical point of higher order, and even to not-too-flat flat topped families.