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Transactions of the American Mathematical Society

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

 

 

The periodic points of Morse-Smale endomorphisms of the circle


Author: Louis Block
Journal: Trans. Amer. Math. Soc. 226 (1977), 77-88
MSC: Primary 58F20
DOI: https://doi.org/10.1090/S0002-9947-1977-0436220-1
MathSciNet review: 0436220
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Abstract: Let $ MS({S^1})$ denote the set of continuously differentiable maps of the circle with finite nonwandering set, which satisfy certain generic properties. For $ f \in MS({S^1})$ let $ P(f)$ denote the set of positive integers which occur as the period of some periodic point of f. It is shown that for $ f \in MS({S^1})$ there are integers $ m \geqslant 1$ and $ n \geqslant 0$ such that $ P(f) = \{ m,2m,4m, \ldots ,{2^n}m\} $. Conversely, if m and n are integers, $ m \geqslant 1,n \geqslant 0$, there is a map $ f \in MS({S^1})$ with $ P(f) = \{ m,2m,4m, \ldots ,{2^n}m\} $.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0436220-1
Keywords: Endomorphism, non wandering set, periodic point
Article copyright: © Copyright 1977 American Mathematical Society