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Contraction groups and the big cell for endomorphisms of Lie groups over local fields

Let $G$ be a Lie group over a totally disconnected local field and $α$ be an analytic endomorphism of $G$. The contraction group of $α$ ist the set of all $x\in G$ such that $α^n(x)\to e$ as $n\to\infty$. Call sequence $(x_{-n})_{n\geq 0}$ in $G$ an $α$-regressive trajectory for $x\in G$ if $α(x_{-n})=x_{-n+1}$ for all $n\geq 1$ and $x_0=x$. The anti-contraction group of $α$ is the set of all $x\in G$ admitting an $α$-regressive trajectory $(x_{-n})_{n\geq 0}$ such that $x_{-n}\to e$ as $n\to\infty$. The Levi subgroup is the set of all $x\in G$ whose $α$-orbit is relatively compact, and such that $x$ admits an $α$-regressive trajectory $(x_{-n})_{n\geq 0}$ such that $\{x_{-n}\colon n\geq 0\}$ is relatively compact. The big cell associated to $α$ is the set $Ω$ of all all products $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and $z$ in the anti-contraction group. Let $π$ be the mapping from the cartesian product of the contraction group, Levi subgroup and anti-contraction group to $Ω$ which maps $(x,y,z)$ to $xyz$. We show: $Ω$ is open in $G$ and $π$ is étale for suitable immersed Lie subgroup structures on the three subgroups just mentioned. Moreover, we study group-theoretic properties of contraction groups and anti-contraction groups.