Homeomorphisms of universal dendrites (Q1923449)

A dendrite is a locally connected metric continuum containing no simple closed curve. Given a subset \(S\) of \(\{3,4, \dots, \omega\}\), the standard universal dendrite \(D_S \) of orders in \(S\) is any dendrite \(X\) satisfying the following two conditions: the order of each ramification point of \(X\) is in \(S\); given \(m\) in \(S\), each arc in \(X\) contains points of order \(m\). The orbit of a point \(x\) in \(D_S\) is the set \(\{h(x) \mid h\) is an autohomeomorphism of \(D_S\}\). The author proves the following four results about autohomeomorphisms of standard universal dendrites. Given \(x\) and \(y\) in \(D_S\), there is an autohomeomorphism \(h\) of \(D_S\) such that \(h(x) = y\) if and only if \(x\) and \(y\) have the same order. The action on \(D_S\) of the group of autohomeomorphisms of \(D_S\) has exactly \(\text{Card} (S) + 2\) orbits. For each arc \(A\) and orbit \(B\) of \(D_S\), the intersection \(A \cap B\) is dense in \(A\). Each orbit is a dense subset of \(D_S\).