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On decomposition of one-dimensional dynamical systems into ergodic components. The case of negative Schwarzian. (English. Russian original) Zbl 0718.58024
Leningr. Math. J. 1, No. 1, 137-155 (1990); translation from Algebra Anal. 1, No. 1, 128-145 (1989).
Let f: $$M\to M$$ be a mapping in the negative Schwarzian case: here M is a circle or an interval. One studies measurable dynamics of this kind of mappings: behavior of almost all (Lebesgue measure) trajectories of this mapping. The important role is attached to the notion of metric attractor of J. Milnor [Commun. Math. Phys. 99, 177-195 (1985; Zbl 0595.58028)]. In this paper Milnor has formulated the problem of decomposition of the global attractor of the mapping f into a finite union of indecomposable attractors $$A_ k$$. In the present work one proves ergodicity of the restriction of f on domains of attraction of these attractors. It is therefore the answer on the questions of Milnor.

##### MSC:
 37E99 Low-dimensional dynamical systems 37A99 Ergodic theory 37C70 Attractors and repellers of smooth dynamical systems and their topological structure