Denis Gaidashev: Renormalization and wild attractors for Fibonacci maps

Abstract: A Fibonacci map is a piecewise defined map of a subset of an interval I onto I with a unique critical point of order d whose orbit undergoes nearest returns at Fibonacci times. It has been shown by Bruin, Keller, Nowicki and van Strien that such maps exhibit "wild" attractors: Cantor sets of zero Lebesgue measure whose basin of attraction is meager but has positive Lebesgue measure. We will discuss real renormalization, and a trichotomy for Fibonacci maps, similar to the Avila-Lyubich trichotomy for Feigenbaum Julia sets, which, in particular, allows us to show that Fibonacci maps admit wild attractors for d=5.1, and do not for d=3.9 (and, conjecturally, for 2<d<=3.9).
This is a joint work with Artem Dudko of IMPAN, Warsaw