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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.

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You will find much of that in Neil Hindman and Dona Strauss, Algebra in the Stone-Cech Compactification---Theory and Applications. Series: De Gruyter Textbook.

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