\(\frac{1}{3}\)-homogeneous dendrites
For a positive integer \(n\), a topological space \(X\) is \textit{\(\frac{1}{n}\)-homogeneous} if the autohomeomorphism group action on \(X\) has exactly \(n\) orbits. The study of \(\frac{1}{n}\)-homogeneity for \(n>1\) seems to have had its beginning in [\textit{J. Krasinkiewicz}, Pr. Mat. 12, 255--257 (1969; Zbl 0235.54039)], where it is proved that Sierpiński's universal curve is \(\frac{1}{2}\)-homogeneous. In [\textit{S. B. Nadler jun.} et al., Topology Appl. 154, No. 10, 2154--2166 (2007; Zbl 1144.54022)], it is shown that the only \(\frac{1}{2}\)-homogeneous dendrite is an arc. In the present paper the much more difficult classification of \(\frac{1}{3}\)-homogeneous dendrites is given. Along the way to this result, the authors show: (1) all \(\frac{1}{3}\)-homogeneous dendrites have either just one ramification point or infinitely many; (2) all dendrites with just one ramification point are \(\frac{1}{3}\)-homogeneous; and (3) the \(\frac{1}{3}\)-homogeneous dendrites with no free arcs comprise the universal dendrites \(D_n\), \(3\leq n\leq\omega\). They then classify all \(\frac{1}{3}\)-homogeneous dendrites with free arcs and infinitely many ramification points. This task is divided into two cases, dictated by whether or not the set of end points is closed. The seven sections of the paper culminate in the classification (Theorem 7.16) of \(\frac{1}{3}\)-homogeneous dendrites, and the (apparently) formidable problem of classifying the \(\frac{1}{n}\)-homogeneous dendrites for \(n>2\).
scientific article; zbMATH DE number 6684521
scientific article