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Periodicity and decomposability of basin boundaries with irrational maps on prime ends


Author: Russell B. Walker
Journal: Trans. Amer. Math. Soc. 324 (1991), 303-317
MSC: Primary 54H20; Secondary 58F10, 58F11
DOI: https://doi.org/10.1090/S0002-9947-1991-0992609-0
MathSciNet review: 992609
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Abstract: Planar basin boundaries of iterated homeomorphisms induce homeomorphisms on prime ends. When the basin is connected, simply connected, and has a compact connected boundary, the space of prime ends is a topological circle. If the induced homeomorphism on prime ends has rational rotation number, the basin boundary contains periodic orbits. Several questions as to basin boundary periodics, decomposability, and minimality, when the induced map on prime ends has irrational rotation number, are answered by construction of both homeomorphisms and diffeomorphisms. Examples in the literature of basin boundaries with interesting prime end dynamics have been sparse. Prime end dynamics has drawn recent interest as a natural tool for the study of strange attractors.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-0992609-0
Article copyright: © Copyright 1991 American Mathematical Society

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