Concerning periodic points in mappings of continua

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 ${(f|H)^{ - n}}(K)$ 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.