Abstract
A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980's that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set.
Highlights
This paper is motivated by the following conjectures in one-dimensional dynamics about the boundary of mappings with positive topological entropy: Given a map f of an interval, I, let Per(f ) = {n ∈ N : f n(p) = p for some p ∈ I, and f j(p) = p for 1 j < n}.We refer to Per(f ) as the set of periods of f .Boundary of Chaos Conjecture I
For C1 mappings, this conjecture implies that the transition to positive entropy for mappings on the interval occurs through successive period doubling bifurcations
There exists an open and dense set of mappings contained in the boundary of positive entropy in Areven,b(I),r > 3, which is a union of disjoint codimension-one submanifolds of Areven,b(I), and each of these submanifolds is contained in the basin of a unimodal, quadraticlike fixed point of renormalization
Summary
∞, ω, with Per(f ) = {2n : n ∈ N ∪ {0}}, are on the boundary of mappings with positive topological entropy and on the boundary of the set of mappings with finitely many periods. Interest in this conjecture is strongly motivated by its implications on the routes to chaos, that is, on the transition from zero to positive entropy, for mappings of the circle or the interval, see [35, 36]. 2010 Mathematics Subject Classification 37E05 (primary), 37B40, 37E20 (secondary) The results in lower regularity are perturbative, and this approach does not seem to work in higher regularity
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