Concerning Periodic Points in Mappings of Continua

• Published in: Proceedings of the American Mathematical Society 1988-10, Vol.104 (2), p.643-649
• Main Author: Ingram, W. T.
• Format: Article
• Language: English
• Subjects: Diagonal lemma; Fixed point property; Homeomorphism; Integers; Mathematical theorems; Sine waves
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-1988-0962842-8

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) $f\lbrack K \rbrack$ and every subcontinuum of $f\lbrack H \rbrack$ are in class $W$, (3) $f\lbrack H \rbrack$ contains $H$, (4) $f\lbrack H \rbrack$ contains $H \cup K$, and (5) if $n$ is a positive integer such that $(f\mid H)^{-n}(K)$ 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 $f\lbrack K \rbrack$ contains $K$. If (1) every subcontinuum of $K$ has the fixed point property, and (2) every subcontinuum of $f\lbrack K \rbrack$ 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 $\lbrack 0, 1 \rbrack$ into $\lbrack 0, 1 \rbrack$ and $f$ has a periodic point which is not a power of 2, then $\lim\{\lbrack 0, 1 \rbrack, f \}$ contains an indecomposable continuum. Moreover, for each positive integer $i$, there is a mapping of $\lbrack 0, 1 \rbrack$ into $\lbrack 0, 1 \rbrack$ with a periodic point of period $2^i$ and having a hereditarily decomposable inverse limit.