Possible heights of Alexandroff square transformation groups
Fatemah Ayatollah Zadeh Shirazi, Fatemeh Ebrahimifar, Reza Yaghmaeian, Hamed Yahyaoghli
TL;DR
The paper investigates the possible heights of transformation groups with phase spaces $\mathbb{A}$ (Alexandroff square), $\mathbb{O}$ (lexicographic unit square), and $\mathbb{U}$ (Euclidean unit square). It develops the orbit-space approach to classify group actions and analyzes topological transitivity and chaos, concluding that Alexandroff square and lexicographic square lack topological transitivity (and Devaney chaos) while the Euclidean unit square action is transitive. It derives explicit height sets: $P_h(\mathbb{A})=\{n:n\geq5\}\cup\{+\infty\}$, $P_h(\mathbb{O})=\{n:n\geq4\}\cup\{+\infty\}$, and $P_h(\mathbb{U})=\{n:n\geq1\}\cup\{+\infty\}$, by constructing subgroup chains with heights $5$, $4$, and $1$ and noting the identity has infinite height. These results provide a complete characterization of possible transformation-group heights on these canonical unit-square spaces, clarifying when chaotic or transitive dynamics can arise in Alexandroff-type topologies.
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 $P_h(\mathbb{A})=\{n:n\geq5\}\cup\{+\infty\}$, $P_h(\mathbb{O})=\{n:n\geq4\}\cup\{+\infty\}$, $P_h(\mathbb{U})=\{n:n\geq1\}\cup\{+\infty\}$ (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.