Wild Attractors of Polymodal Negative Schwarzian Maps

Abstract:

We study “wild attractors” of polymodal negative Schwarzian interval maps and prove that they are {\it super persistently recurrent} (a polymodal version of persistent recurrence). We also prove that if a map has an attractor which is a cycle of intervals then at almost every point of this cycle the map has properties similar to the Markov property introduced by Martens. Thus, the lack of super persistent recurrence at a critical point $c$ can be considered as a mild topological expanding property, and this expansion prevents ω(c) from being a wild attractor (in the previous paper we have shown that it also prevents the map from being C ²-stable).