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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


On the set of periods for $ \sigma$ maps

Authors: M. Carme Leseduarte and Jaume Llibre
Journal: Trans. Amer. Math. Soc. 347 (1995), 4899-4942
MSC: Primary 58F20; Secondary 54H20
MathSciNet review: 1316856
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Abstract: Let $ \sigma $ be the topological graph shaped like the letter $ \sigma $. We denote by 0 the unique branching point of $ \sigma $, and by $ {\mathbf{O}}$ and $ {\mathbf{I}}$ the closures of the components of $ \sigma \backslash \{ 0\} $ homeomorphics to the circle and the interval, respectively. A continuous map from $ \sigma $ into itself satisfying that $ f$ has a fixed point in $ {\mathbf{O}}$, or $ f$ has a fixed point and $ f(0) \in {\mathbf{I}}$ is called a $ \sigma $ map. These are the continuous self-maps of $ \sigma $ whose sets of periods can be studied without the notion of rotation interval. We characterize the sets of periods of all $ \sigma $ maps.

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PII: S 0002-9947(1995)1316856-7
Keywords: Periodic orbit, set of periods, Sarkovskii theorem
Article copyright: © Copyright 1995 American Mathematical Society