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On typical behaviour of trajectories of transformations of a segment. (Russian) Zbl 0666.58031
Let f: [0,1]\(\to [0,1]\) be a map of the interval with negative Schwarzian derivative. In the paper the decomposition of the global attractor of f into the finite union of indecomposable attractors is described. This provides an answer to certain of Milnor’s questions. It is proved that each indecomposable attractor is of one of the following types: A1. \(A=\{f^ ka\}^{p-1}_{k=0}\) is a limit cycle. A2. \(A=\cup^{p- 1}_{k=0}f^ kI\) is a cycle of periodic interval I, such that \(f| A\) is topologically transitive. A3. \(A=\cap^{\infty}_{m=1}\cup^{p_ m-1}_{k=0}f^ kI_ m\) is a solenoid. Here \(I_ m\) is a periodic interval of period \(p_ m\to \infty\), \(I_ 1\supset I_ 2\supset..\). and \(f| A\) is topologically conjugate to a group shift. A4. A is a non-solenoidal Cantor set coinciding with an \(\omega\)-limit set \(\omega\) (c) of some critical point \(c\in A\). Moreover, for almost every \(x\in [0,1]\) (with respect to the Lebesgue measure \(\lambda)\), \(\omega\) (x) coincides with one of the indecomposable attractors. At the present time the authors have established many new properties of the decomposition which are published elsewhere. They enable us to prove the following result: if \(\mu\) is a finite invariant measure absolutely continuous with respect to \(\lambda\), then \(h_{\mu}(f)>0\).
Reviewer: M.Yu.Lyubich

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A99 Ergodic theory