Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems. II: The smooth case. (English) Zbl 0665.58024

Blokh, A. M.; Lyubich, M. Yu.

Ergodic Theory Dyn. Syst. 9, No. 4, 751-758 (1989).

This paper is the continuation of part I [see the preceding article]. Let \({\mathcal A}\) be the class of \(C^ 3\)-smooth one dimensional maps f: \(M\to M\) with nondegenerate critical points. Theorem. A map \(f\in {\mathcal A}\) has no wandering intervals.
The theorem is proved by combining the technique of part I with the distortion estimates due to W. de Melo and S. J. van Strien [ibid. 7, 415-462 (1987; Zbl 0609.58023)] (in that paper the absence of wandering intervals for unimodal \(f\in {\mathcal A}\) was established). The main new point is the analysis of intersection multiplicity of intervals in some specific situation.

Reviewer: M.Yu.Lyubich

MSC:

37C70	Attractors and repellers of smooth dynamical systems and their topological structure
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems

Keywords:

topological attractor; solenoid; nondegenerate critical points; wandering intervals

References:

[1]	van Strien, Hyperbolicity and invariant measures for general C pp 87–
[2]	Lyubich, Ergod. Th. & Dyn. Sys. none pp none– (none)
[3]	Melo, A structure theorem in one dimensional dynamics (1986)