Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications

TL;DR

This paper investigates how automorphisms of tori and certain Lie groups act on the space of one-parameter subgroups, characterizing when these actions are distal or expansive, and extending previous results in the field.

Contribution

It characterizes automorphisms acting distally or expansively on the space of one-parameter subgroups of tori and Lie groups, generalizing prior work on distal and expansive actions.

Findings

Only finite order automorphisms act distally on Sub$^p_G$ for tori.

Connected Lie groups with large central tori do not admit expansive automorphisms on Sub$^p_G$.

Results extend previous theorems on distal and expansive actions to broader classes of groups.

Abstract

For a connected Lie group $G$, we study the dynamics of actions of automorphisms of $G$ on certain compact invariant subspaces of closed subgroups of $G$ in terms of distality and expansivity. We show that only the finite order automorphisms of $G$ act distally on Sub$^p_G$, the smallest compact space containing all closed one-parameter subgroups of $G$, when $G$ is any $n$-torus, $n\in\mathbb{N}$. This enables us to relate distality of the $T$-action on Sub$^p_G$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of closed subgroups generate by $T$ in the group Aut$(G)$, in case $G$ is not a vector group. We also extend these results to the action of subgroups of automorphisms. We show that any $n$-torus $G$, $n\geq 2$, more generally, any connected Lie group $G$ whose central torus has dimension at least 2, does not admit any automorphism which acts expansively on Sub$^p_G$. Our results generalise some results on distal actions by Shah and Yadav, and by Chatterjee and Shah, and some results on expansive actions by Prajapati and Shah.