Interval-expressed tree-like continua with the fixed point property
Keywords:
Fixed point property, inverse limit, tree-like, k-tail sequence, k-tail sequence with interval-valued inverse functions.Abstract
Let $\mathcal T$ be the class of tree-like continua that admit representations as inverse limits on $[0,1]$ with surjective upper semicontinuous set-valued functions. We show (1) if $X\in\mathcal T$ with interval-valued bonding functions $f_i$, where there exists $m\geq 1$ such that, for each $i\geq m$, the graph of $f_i^{-1}$ contains the graph of an interval-valued function, then $X$ has the fixed point property, and (2) if $X\in\mathcal T$ with set-valued bonding functions $f_i$, where for each $i\geq 1$, $f_i^{-1}$ is an interval-valued function, then $X$ is a $\lambda$-dendroid. We also provide an example of an indecomposable, non-arclike continuum in $\mathcal T$ that has the fixed point property.
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