Possible heights of Alexandroff square transformation groups

Abstract

In the following text we compute possible heights of $\mathbb{A}$ (Alexandroff square), $\mathbb{O}$ (unit square $[0,1]\times [0,1]$ with lexicographic order topology) and $\mathbb{U}$ (unit square $[0,1]\times [0,1]$ with induced topology of Euclidean plane). We prove $P_h(\mathbb{A})=\{n:n\ge 5\}\cup \{+\infty \}$, $P_h(\mathbb{O})=\{n:n\ge 4\}\cup \{+\infty \}$, $P_h(\mathbb{U})=\{n:n\ge 1\}\cup \{+\infty \}$ (where for topological space $X$, by $P_h(X)$ we mean the collection of heights of transformation groups with phase space $X$. In this way we also prove that there is not any topological transitive (resp. Devaney chaotic) Alexandroff square transformation group.

10 pages