Contraction Groups and Passage to Subgroups and Quotients for   Endomorphisms of Totally Disconnected Locally Compact Groups
  The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup. 
Timothy P. Bywaters, Helge Gl\"ockner, Stephan Tornier
math.GR