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Attractors with irrational rotation number. (English) Zbl 1267.37040

The authors consider a dissipative orientation-preserving homeomorphism of the plane having a point attractor with an unbounded basin of attraction. The restriction of the map to this basin has a well-defined rotation number, derived from the Carathéodory theory of prime ends. The authors show that if the rotation number is irrational, then such a restriction induces a Denjoy homeomorphism in the space of prime ends. Several examples of maps in this class are presented. This work is a continuation of a previous work by two of the authors on the connections between rotation numbers and stability theory.

MSC:

37E45 Rotation numbers and vectors
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
30D40 Cluster sets, prime ends, boundary behavior
37E10 Dynamical systems involving maps of the circle
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[1] Kerékjártó, C.R. Acad. Sci. 198 pp 317– (1934)
[2] Hale, Asymptotic behavior of dissipative systems (1988) · Zbl 0642.58013
[3] DOI: 10.2307/1969308 · doi:10.2307/1969308
[4] DOI: 10.4064/fm182-1-1 · Zbl 1099.37030 · doi:10.4064/fm182-1-1
[5] DOI: 10.1016/S0166-8641(97)00179-X · Zbl 0929.37013 · doi:10.1016/S0166-8641(97)00179-X
[6] Birkhoff, Bull. Soc. Math. France 60 pp 1– (1932) · Zbl 0005.22002 · doi:10.24033/bsmf.1182
[7] DOI: 10.2307/1969299 · Zbl 0061.18910 · doi:10.2307/1969299
[8] Pommerenke, Boundary behaviour of conformal maps. (1991)
[9] DOI: 10.1017/S0143385700006842 · Zbl 0767.58023 · doi:10.1017/S0143385700006842
[10] DOI: 10.4171/JEMS/288 · Zbl 1253.37048 · doi:10.4171/JEMS/288
[11] Arrowsmith, An Introduction to Dynamical Systems (1992) · Zbl 0702.58002
[12] Mather, Selected Studies pp 225– (1982)
[13] DOI: 10.1112/plms/s3-20.4.688 · Zbl 0194.54904 · doi:10.1112/plms/s3-20.4.688
[14] Barge, Continuum Theory and Dynamical Systems pp 15– (1993)
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