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Metric attractors for smooth unimodal maps. (English) Zbl 1055.37041
The classification of measure theoretic attractors of general \(C^3\) unimodal maps with quadratic critical points is studied. In particular, the work solves in the affirmative Milnor’s problem (whether the metric and topological attractors coincide for a given smooth unimodal map) for smooth unimodal maps with quadratic critical point. Certain of the results obtained are the following theorem and two corollaries:
Let \(I\) be a compact interval and \(f:I\to I\) be a \(C^3\) unimodal map with \(C^3\) nonflat critical point of order 2. Then the \(\omega\)-limit set of a Lebesgue almost every point of \(I\) is either
1. a nonrepelling periodic orbit, or
2. a transitive cycle of intervals, or
3. a Cantor set of solenoid type.
Corollary 1: Every metric attractor of \(f\) is either
1. a topologically attracting periodic orbit, or
2. a transitive cycle of intervals, or
3. a Cantor set of solenoid type.
Corollary 2. The metric and topological attractors of \(f\) coincide.
The main ingredient is the decay of geometry.

MSC:
37E05 Dynamical systems involving maps of the interval
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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