Lyubich, Mikhail Combinatorics, geometry and attractors of quasi-quadratic maps. (English) Zbl 0821.58014 Ann. Math. (2) 140, No. 2, 347-404 (1994). This paper answers a question which was raised by John Milnor in a 1985 paper. The result characterizes measure theoretic attractors, and needs to be contrasted with the topological counterpart which has been known since the late 1970’s. The measure theoretic version is as follows: Suppose that \(f : [0,1] \to [0,1]\) is an \(S\)-unimodal map with non- degenerate critical point \(c\) (a so-called quasi-quadratic map), normalized by the requirement that \(f\) takes \(c\) to 1, and 1 to 0. Then there is a unique set \(A\) (a measure theoretic attractor) such that \(A\) is the \(\omega\)-limit set of \(\chi\) for (Lebesgue) almost all \(\chi \in [0,1]\), and only one of the following three possibilities can occur: (1) \(A\) is a limit cycle; (2) \(A\) is a cycle of intervals; or (3) \(A\) is a Feigenbaum-like attractor.The topological version of this theorem has the same hypotheses, and the same conclusions for a set \(\Lambda\) (a topological attractor) which is the \(\omega\)-limit set for a generic \(\chi \in [0,1]\). Thus, by the measure theoretic attractor theorem, the map \(f\) has a unique measure theoretic attractor \(A\) coinciding with the topological attractor \(\Lambda\). Reviewer: W.J.Satzer jun.(St.Paul) Cited in 2 ReviewsCited in 34 Documents MSC: 37E99 Low-dimensional dynamical systems 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 54H20 Topological dynamics (MSC2010) Keywords:quasi-quadratic maps; maps of the unit interval; measure theoretic attractors; topological attractor PDFBibTeX XMLCite \textit{M. Lyubich}, Ann. Math. (2) 140, No. 2, 347--404 (1994; Zbl 0821.58014) Full Text: DOI arXiv