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 $alpha$ be an analytic endomorphism of $G$. The contraction group of $alpha$ ist the set of all $xin G$ such that $alpha^n(x)to e$ as $ntoinfty$. Call sequence $(x_{-n})_{ngeq 0}$ in $G$ an $alpha$-regressive trajectory for $xin G$ if $alpha(x_{-n})=x_{-n+1}$ for all $ngeq 1$ and $x_0=x$. The anti-contraction group of $alpha$ is the set of all $xin G$ admitting an $alpha$-regressive trajectory $(x_{-n})_{ngeq 0}$ such that $x_{-n}to e$ as $ntoinfty$. The Levi subgroup is the set of all $xin G$ whose $alpha$-orbit is relatively compact, and such that $x$ admits an $alpha$-regressive trajectory $(x_{-n})_{ngeq 0}$ such that ${x_{-n}colon ngeq 0}$ is relatively compact. The big cell associated to $alpha$ is the set $Omega$ of all all products $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and $z$ in the anti-contraction group. Let $pi$ be the mapping from the cartesian product of the contraction group, Levi subgroup and anti-contraction group to $Omega$ which maps $(x,y,z)$ to $xyz$. We show: $Omega$ is open in $G$ and $pi$ is {e}tale 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.