Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications
Debamita Chatterjee, Himanshu Lekharu, Riddhi Shah
TL;DR
This work analyzes distal and expansive dynamics of automorphisms acting on the compact invariant space $Sub^p_G$ of closed one-parameter subgroups, with a focus on the $n$-torus and central tori in connected Lie groups. It establishes a precise distal criterion on $Sub^p_G$ for $G=\mathbb{T}^n$: an automorphism $T$ acts distally on $Sub^p_G$ if and only if some power $T^m$ is the identity, and it connects distal actions on $Sub^p_G$ to distality on $G$ and to compactness properties within $Aut(G)$. The paper further shows that for $n\ge2$, and more generally for groups with a central torus of dimension at least $2$, no automorphism acts expansively on $Sub^p_G$, extending prior results on expansivity. These results generalize and unify earlier work by Shah–Yadav, Chatterjee–Shah, and Prajapati–Shah, linking dynamics on subgroup spaces to the ambient Lie-group dynamics and providing structural insights via the Chabauty topology and Berend-type convergence criteria.
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.