  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.
