Possible Heights of Alexandroff Square Transformation Groups

Abstract

In the following text we compute possible heights of A (Alexandroff square), O (unit square [0, 1] × [0, 1] with lexicographic order topology) and U (unit square [0, 1] × [0, 1] with induced topology of Euclidean plane). We prove Ph(A) = {n : n ≥ 5} ∪ {+∞}, Ph(O) = {n : n ≥ 4} ∪ {+∞}, Ph(U) = {n : n ≥ 1} ∪ {+∞} (where for topological space X, by Ph(X) we mean the collection of heights of transformation groups with phase space X. Additionally we show that there is no topological transitive (resp. Devaney chaotic) transformation group (G, A).