arXiv subjects

Debamita Chatterjee

Publications and source records attributed to Debamita Chatterjee.

2 recordsLinked to original sources

arXiv2025

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

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.

math.DS↗

arXiv2024

Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups

For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the action of $T$ on Sub$_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. We study the action of $T$ on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of $G$. We relate the 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 the closed subgroup generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central torus and $G$ is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$, and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group $G$ acts distally on Sub$^p_G$ by conjugation if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. Some of our results extend those of Shah and Yadav.

math.DS↗