I think statistics would be considered math in general. Some professors told me that some of it is pretty much like pure math. The more theoretical part of it definitely is from experience. However I have heard quotes from certain funny as professors, such as this one:
"My brother is a statistician, so I asked him what he does. He told me that people just come up to him and say 'ok this is the data we have, and this is the paper we want to write... can you fix us up' "
As far as statistical dynamics goes... dunno what it is exactly but if it's the same thing as statistical mechanics then I can tell you that there is some really hardcore as well as nice mathematics involved after seeing what one of my mates is working now on. Lot's of combinatorics, generating functions, continued fractions even etc. my mate who pretty much has a bit of a 'pure' taste in math has gotten interested in it. In general I think that in a lot of cases applied math is actually not as bad as you think, gone are the days of G.H Hardy's outdated views that nice mathematics like number theory etc. is pure and has no applications. Nowadays things can get complicated and the fact that we're in the digital age means that things that were mainly pure now find many applications(think cryptography). I think the fact that we learn from old textbooks may give us a bit of an outdated view.