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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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