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Hyperbolicity, sinks and measure in one dimensional dynamics. (English) Zbl 0583.58016
The author studies the dynamics of \(C^ 2\) maps of the unit circle and the unit interval. The main theorem states that if f is a \(C^ 2\) endomorphism and \(\Lambda\) is a compact invariant subset that does not contain critical points, sinks or nonhyperbolic periodic points, then either \(\Lambda =S^ 1\) and f is topologically equivalent to an irrational rotation or \(\Lambda\) is a hyperbolic set. The author gives another application of the method, proving, for instance, that for every \(C^ 2\) immersion f of the circle, either its Julia set has measure zero or it is the whole circle and then f is ergodic.
Reviewer: Yu.Kifer

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D99 Dynamical systems with hyperbolic behavior
37A99 Ergodic theory
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