Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications
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| Format: | Preprint |
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2025
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| _version_ | 1866915601961713664 |
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| author | Chatterjee, Debamita Lekharu, Himanshu Shah, Riddhi |
| author_facet | Chatterjee, Debamita Lekharu, Himanshu Shah, Riddhi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_04111 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications Chatterjee, Debamita Lekharu, Himanshu Shah, Riddhi Dynamical Systems Group Theory 37B05, 22E15, 22D45 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. |
| title | Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications |
| topic | Dynamical Systems Group Theory 37B05, 22E15, 22D45 |
| url | https://arxiv.org/abs/2511.04111 |