Unfolding of Chaotic Quadratic Maps — Parameter Dependence of Natural Measures

Abstract. We consider perturbations of quadratic maps fa admitting an absolutely continuous invariant probability measure, where a is in a certain positive measure set A of parameters, and show that in any neighborhood of any such an fa, 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 fa. In particular, Misiurewicz maps are dense in A. Almost all maps fa in the quadratic family is known to possess a unique natural measure, that is, an invariant probability measure μa describing the asymptotic distribution of almost all orbits. We discuss weak*-(dis)continuity properties of the map a 7→ μa near the set A, and prove that almost all maps in A have the property that μa can be approximated with measures supported on periodic attractors of certain nearby maps. On the other hand, for any a ∈ A and any periodic repeller Γa of fa, the singular measure supported on Γa can also approximated with measures supported on nearby periodic attractors. It follows that a 7→ μa 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.