I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets.

I mean cases where adding a topology to the sets or enlarging them to topological spaces enables one to use topological methods to either prove a new result about the behavior of the maps or give a new proof of an old one.

No-go theorems (that a topological method won't be of any help for a given problem) would be interesting as well.