Category Theory and Ergodic Theory I am very much interested in finding out about any category theoretical work on dynamical systems and on ergodic theory. On the face of it, it seems that a categorical language can go a long way, at least this is my impression by reading the first few pages of the great book by Furstenberg, ``Recurrence in Ergodic Theory and Combinatorial Number Theory.'' I have also seen some categorical language used in Terry Tao's lectures on ergodic theory (MATH 254A : Topics in Ergodic Theory.) Does anyone know of any other work? Especially, are there non-trivial results in ergodic theory that are proven using categorical constructions and theorems?
Thanks,
Esfan Haghverdi 
 A: You could look at the paper
Mackey, George W.
Ergodic theory and virtual groups.
Math. Ann. 166 1966 187–207. 
which ends up by discussing the notion of ergodic groupoid, andfollow this up with the citations of this paper. The intuitive idea is that while a transitive action of a group corresponds to a subgroup, then what does an ergodic action, correspond to?  His theory went through various stages, and ended up with the notion of ergodic groupoid. This introduction of groupoids into analysis is part of the historical background to Noncommutative geometry! 
Mackey told me of this work in 1967, and made me realise that there was more in groupoids than I had then thought; the idea did not come just from algebraic topology. 
Of course groupoid theory is not the same as category theory, but is in that direction. At least, people who liked category theory found it easy to be happy with groupoids. 
A: I don't know of any "non-trivial results in ergodic theory that are proven using categorical constructions and theorems", and more generally I'm not aware that categorical thinking has penetrated ergodic theory to any significant extent.  
I can think of one more reference for you: a (1987?) preprint by Lawvere, Functorial remarks on the general concept of chaos.  (I never knew a remark could be functorial, but apparently so.)  Without having read it, I guess that most people would judge it to be much more categorical than dynamical.  Lawvere tends to mount very long-range attacks.  
A: There is this post, and its continuation 
A: You might want to have a look on the recent paper by Gromov ,
In a Search for a Structure, Part 1: On Entropy. June 19, 2012, 
http://www.ihes.fr/~gromov/PDF/structres-entropy-june-2012.pdf ;
it mentions words such as Functorial Bernulli [Entropy], 
v-Categories and Measure Spaces...
A: Well, because this thread resurrected, I think I should mention the recent work by Bader and Furman generalizing Margulis' superrigidity theorem (and one might say simplifying it).
In their work, their prime construction is of something they call generalized Weyl group, which should appear in every nice LCSC group (there are some open questions about this concepet, especially when one leaves the classes of nice groups, such as linear algebraic groups).
Anyways, their construction is functorial in its nature, and involves categorial language (initial objects in a category and so on), but as far as I know, they didn't prove anything using explicit category theory. If you're looking for Yoneda's lemma, you should probably look in anoter place.
The advantage in their proof over Margulis' is in its generality, and probably in its simplicity, as opposing Margulis' explicit proof which is considered quite hard (for example, I think they can dodge the Danni-Margulis lemma, aka non-divergence of unipotents).
