A characterization of monotonely homogeneous dendrites (Q658392)

The author proves that the class of monotonely homogeneous dendrites is precisely the class of dendrites which contain a copy of the Omiljanowski dendrite \(L_0\). A continuum is a nonempty connected compact metric space. A dendrite is a locally connected continuum that contains no simple closed curve. Let \(X\) be a dendrite and \(Y\) a subcontinuum of \(X\). The order of \(Y\) in \(X\) is the number of components of \(Y\setminus X\). The order of a point \(p\in X\) is the order of \(\{p\}\) in \(X\). The points of order \(3\) or more in \(X\) are called ramification points of \(X\). A map is a continuous function. A map \(f:X\rightarrow Y\) between continua is monotone if the preimage of each point is connected. Given a class \(\mathcal M\) of maps, a space \(X\) is \(\mathcal M\)-homogeneous if for every \(x,y\in X\), there is a map \(f\in \mathcal M\) from \(X\) onto itself such that \(f(x)=y\). The following main result of the paper is used to prove that the class of monotonely homogeneous dendrites is precisely the class of dendrites which contain a copy of the Omiljanowski dendrite \(L_0\): Let \(X\) be a nondegenerate monotonely homogeneous dendrite. Then there exists a monotone map from \(X\) onto a dendrite with a dense set of ramification points.