# Endomorphisms of Lie groups over local fields

## TL;DR

This paper explores the structure and properties of endomorphisms in Lie groups over local fields, focusing on scale, tidy subgroups, and contraction subgroups, contributing to the understanding of totally disconnected, locally compact groups.

## Contribution

It provides a detailed analysis of endomorphisms in Lie groups over local fields, extending previous work on automorphisms and emphasizing new subgroup structures.

## Key findings

- Analysis of scale and tidy subgroups for endomorphisms
- Characterization of contraction subgroups in these groups
- Connections to totally disconnected, locally compact groups

## Abstract

Lie groups over local fields furnish prime examples of totally disconnected, locally compact groups. We discuss the scale, tidy subgroups and further subgroups (like contraction subgroups) for analytic endomorphisms of such groups. The text is both a research article and a worked out set of lecture notes for a mini-course held June 27-July 1, 2016 at the MATRIX research center in Creswick (Australia) as part of the "Winter of Disconnectedness". The text can be read in parallel to the earlier lecture notes arXiv:0804.2234 which are devoted to automorphisms, with sketches of proof. Complementary aspects are emphasized.