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Dynamical systems on one-dimensional branched manifolds. I. (Russian) Zbl 0621.58027

Let K be a compact branched manifold, i.e. a topological space locally homeomorphic to the union of intervals with one common point. Consider a continuous map \(f: K\to K\). In the paper the detailed analysis of the topological dynamics of f is given. A spectral decomposition of three sets \(\overline{Per(f)}\), C(f), \(\omega\) (f) into transitive components is obtained (here Per(f) is the set of periodic points of f, C(f) is the center of f, \(\omega\) (f) is the union of \(\omega\)-limit sets of all orbits). It is shown that there are four kinds of transitive components: periodic orbits, rotations of the circle, so-called solenoidal sets and basic sets. The detailed description of all kinds of components is given. In particular, it follows that \(\omega\) (f)\(\setminus C(f)\) contains at most finitely many points and C(f)\(\setminus \overline{Per(f)}\) is the union of the rotation components.
Reviewer: M.Lyubich

MSC:

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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