Autohomeomorphisms of the finite powers of the double arrow

Let $\mathbb{A}$ and $\mathbb{S}$ denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that any homeomorphism $h{:}^m\mathbb{A}{\to }^m\mathbb{A}$ is locally (outside of a nowhere dense set) a product of monotone embeddings $h_i:J_i\subseteq \mathbb{A}\to \mathbb{A}(i\in m)$ followed by a permutation of the coordinates. We also prove that the symmetric products ${\mathcal{F}}_m(\mathbb{A})$ are not homogeneous for any $m\ge 2$. This partially solves an open question of A. Arhangel’skiǐ. In contrast, we show that symmetric product ${\mathcal{F}}_2(\mathbb{S})$ is homogeneous.