Infinitely Many Sinks around Nonuniformly Expanding 1d Maps

In this paper we study families of multi-modal 1D maps following the setting of Wang and Young [20]. Under a mild combinatoric assumption, we prove that for generic one parameter families of 1D maps containing a Misiurewicz map, parameters of non-uniformly expanding maps, the measure abundance of which was proved previously in [20], are accumulation points of paramaters admitting super-stable periodic sinks. To motivate the main theorems of this paper, we start with a discussion on the studies of the quadratic family {fa : a ∈ [1, 2]} of 1D maps defined by fa(x) = 1 − ax on I = [−1, 1]. The dynamical properties of fa depend sensitively on the value of the parameter a. For this family, there are two primary dynamical scenarios competing in the space of parameters. The first is the scenario of periodic sinks representing stability. The second is the scenario of positive Lyapunov exponents almost everywhere in I representing chaos. Let ∆1 be the set of all a ∈ [1, 2] such that fa admits a periodic sink, and ∆2 be the set of all a ∈ [1, 2] such that fa has positive Lyapunov exponents almost-everywhere in I. Notice that ∆1 and ∆2 are disjoint by definition. It has been proved that (i) ∆1 is open and dense in [1, 2] ([4], [8]), and (ii) a = 2 is a Lebesgue density point of ∆2 ([6], [1]). Statement (i) implies that the stable scenario of periodic sinks dominates parameter space in the topological sense. On the other hand, (ii) claims that, at least in the vicinity of a = 2, the chaotic scenario dominates in the measure-theoretic sense. We study one parameter families of multi-modal 1D maps following the setting introduced byWang and Young in [20]. This setting includes the quadratic family as a specific example. A correspondence of item (ii) above, in which a = 2 is replaced by the parameter of a Misiurewicz map, was proved in [20]. For a correspondence of item (i), the open part is obvious but the dense part has been a major challenge. The existing proofs for the quadratic family rely heavily on the specifics of the quadratic form therefore can not be modified to apply to other families of 1D maps. As a matter of fact, it is not even clear if this part of the claim remains true for multi-modal 1D families studied in [20]. In this paper we prove that, under a mild combinatoric assumption, parameters of chaotic maps such as those constructed in [20] are accumulation points of the set of parameters admitting periodic sinks. The proofs of items (i) and (ii) above for the quadratic family have a long and celebrated history. That ∆1 is dense in [1, 2] was proved by Graczyk and Świa̧tek [4], Date: November 2006. 2000 Mathematics Subject Classification. Primary: 37D45, 37C40.