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 \(\Lambda\).