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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A canonical partition of the periodic orbits of chaotic maps


Author: Kathleen T. Alligood
Journal: Trans. Amer. Math. Soc. 292 (1985), 713-719
MSC: Primary 58F12; Secondary 34C35, 58F08, 58F13, 58F22
MathSciNet review: 808749
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Abstract: We show that the periodic orbits of an area-contracting horseshoe map can be partitioned into subsets of orbits of minimum period $ k,\;2k,\;4k,\;8k \ldots $, for some positive integer $ k$. This partition is natural in the following sense: for any parametrized area-contracting map which forms a horseshoe, the orbits in one subset of the partition are contained in a single component of orbits in the full parameter space. Furthermore, prior to the formation of the horseshoe, this component contains attracting orbits of minimum period $ {2^m}k$, for each nonnegative integer $ m$.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0808749-X
Article copyright: © Copyright 1985 American Mathematical Society