{"publication":"arXiv:2101.02981","_id":"34806","external_id":{"arxiv":["2101.02981"]},"language":[{"iso":"eng"}],"date_updated":"2022-12-22T07:48:29Z","user_id":"178","date_created":"2022-12-22T07:47:35Z","citation":{"mla":"Glöckner, Helge. “Contraction Groups and the Big Cell for Endomorphisms of Lie Groups over Local Fields.” *ArXiv:2101.02981*, 2021.","bibtex":"@article{Glöckner_2021, title={Contraction groups and the big cell for endomorphisms of Lie groups over local fields}, journal={arXiv:2101.02981}, author={Glöckner, Helge}, year={2021} }","chicago":"Glöckner, Helge. “Contraction Groups and the Big Cell for Endomorphisms of Lie Groups over Local Fields.” *ArXiv:2101.02981*, 2021.","apa":"Glöckner, H. (2021). Contraction groups and the big cell for endomorphisms of Lie groups over local fields. In *arXiv:2101.02981*.","ieee":"H. Glöckner, “Contraction groups and the big cell for endomorphisms of Lie groups over local fields,” *arXiv:2101.02981*. 2021.","short":"H. Glöckner, ArXiv:2101.02981 (2021).","ama":"Glöckner H. Contraction groups and the big cell for endomorphisms of Lie groups over local fields. *arXiv:210102981*. Published online 2021."},"type":"preprint","year":"2021","abstract":[{"lang":"eng","text":"Let $G$ be a Lie group over a totally disconnected local field and $\\alpha$\r\nbe an analytic endomorphism of $G$. The contraction group of $\\alpha$ ist the\r\nset of all $x\\in G$ such that $\\alpha^n(x)\\to e$ as $n\\to\\infty$. Call sequence\r\n$(x_{-n})_{n\\geq 0}$ in $G$ an $\\alpha$-regressive trajectory for $x\\in G$ if\r\n$\\alpha(x_{-n})=x_{-n+1}$ for all $n\\geq 1$ and $x_0=x$. The anti-contraction\r\ngroup of $\\alpha$ is the set of all $x\\in G$ admitting an $\\alpha$-regressive\r\ntrajectory $(x_{-n})_{n\\geq 0}$ such that $x_{-n}\\to e$ as $n\\to\\infty$. The\r\nLevi subgroup is the set of all $x\\in G$ whose $\\alpha$-orbit is relatively\r\ncompact, and such that $x$ admits an $\\alpha$-regressive trajectory\r\n$(x_{-n})_{n\\geq 0}$ such that $\\{x_{-n}\\colon n\\geq 0\\}$ is relatively\r\ncompact. The big cell associated to $\\alpha$ is the set $\\Omega$ of all all\r\nproducts $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and\r\n$z$ in the anti-contraction group. Let $\\pi$ be the mapping from the cartesian\r\nproduct of the contraction group, Levi subgroup and anti-contraction group to\r\n$\\Omega$ which maps $(x,y,z)$ to $xyz$. We show: $\\Omega$ is open in $G$ and\r\n$\\pi$ is \\'{e}tale for suitable immersed Lie subgroup structures on the three\r\nsubgroups just mentioned. Moreover, we study group-theoretic properties of\r\ncontraction groups and anti-contraction groups."}],"title":"Contraction groups and the big cell for endomorphisms of Lie groups over local fields","author":[{"full_name":"Glöckner, Helge","id":"178","last_name":"Glöckner","first_name":"Helge"}],"status":"public","department":[{"_id":"10"},{"_id":"87"},{"_id":"93"}]}