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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Concerning periodic points in mappings of continua


Author: W. T. Ingram
Journal: Proc. Amer. Math. Soc. 104 (1988), 643-649
MSC: Primary 54F20; Secondary 54H20, 58F20
DOI: https://doi.org/10.1090/S0002-9939-1988-0962842-8
MathSciNet review: 962842
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Abstract: In this paper we present some conditions which are sufficient for a mapping to have periodic points.

Theorem. If $ f$ is a mapping of the space $ X$ into $ X$ and there exist subcontinua $ H$ and $ K$ of $ X$ such that (1) every subcontinuum of $ K$ has the fixed point property, (2) $ f[K]$ and every subcontinuum of $ f[H]$ are in class $ W$, (3) $ f[K]$ contains $ H$, (4) $ f[H]$ contains $ H \cup K$, and (5) if $ n$ is a positive integer such that $ {(f\vert H)^{ - n}}(K)$ intersects $ K$, then $ n = 2$, then $ K$ contains periodic points of $ f$ of every period greater than 1.

Also included is a fixed point lemma:

Lemma. Suppose $ f$ is a mapping of the space $ X$ into $ X$ and $ K$ is a subcontinuum of $ X$ such that $ f[K]$ contains $ K$. If (1) every subcontinuum of $ K$ has the fixed point property, and (2) every subcontinuum of $ f[K]$ is in class $ W$, then there is a point $ x$ of $ K$ such that $ f(x) = x$.

Further we show that: If $ f$ is a mapping of $ [0,1]$ into $ [0,1]$ and $ f$ has a periodic point which is not a power of 2, then $ \lim \{ [0,1],f\}$ contains an indecomposable continuum. Moreover, for each positive integer $ i$, there is a mapping of $ [0,1]$ into $ [0,1]$ with a periodic point of period $ {2^i}$ and having a hereditarily decomposable inverse limit.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0962842-8
Keywords: Periodic point, fixed point property, class $ W$, indecomposable continuum, inverse limit
Article copyright: © Copyright 1988 American Mathematical Society

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