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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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##### References:
 [1] Jacobson, M. V.: On smooth mappings of the circle into itself. Mat. U.S.S.R. Sb.14, 161-185 (1971) [2] Herman, M.: Sur la conjugaison différentiable des difféomorphisms du cercle à des rotations, Publications Mathématiques IHES [3] Lasota, A., Yorke, J.: On the existence of invariant measures for piecewise monotonic transformations, Trans. Am. Math. Soc.186, 481-448 (1973) · Zbl 0298.28015 [4] Malta, I.: On Denjoy theorem for endomorphisms. (preprint, 1983) · Zbl 0657.58018 [5] Nitecki, Z.: Factorization of nonsingular circle endomorphisms, salvador symposium on dynamical systems. Peixoto, M. ed. New York: Academic Press, 1973 · Zbl 0273.58011 [6] Peixoto, M.: Structural stability on two dimensional manifolds. Topology1, 101-120 (1962) · Zbl 0107.07103 [7] Shub, M.: Endomorphisms of compact manifolds. Am. J. Math.91, 175-199 (1969) · Zbl 0201.56305 [8] Young, L. S.: A closing Lemma on the interval. Inv. Math.54, 179-187 (1970) · Zbl 0425.58016
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