Wild Cantor attractors exist

• 成果类型: Article
• 署名作者: Bruin, H; Keller, G; Nowicki, T; vanStrien, S
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2118654
• 发表日期: 1996
• 页码: 97-130
• 关键词: s-unimodal maps; invariant-measures; cubic polynomials; interval maps; DYNAMICS; Iteration

摘要

In this paper we shall show that there exists a polynomial unimodal map f: [0, 1] --> [0, 1] with so-called Fibonacci dynamics which is non-renormalizable and in particular, for each x from a residual set, omega(x) is equal to an interval (here omega(x) is defined to be the set of accumulation points of the sequence x, f(x), f(2)(x), ...); for which the closure of the forward orbit of the critical point c, i.e., omega(c), is a Canter set and for which w(x) = w(c) for Lebesgue-almost all x. So the topological attractor and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor [Mil] in dimension one.