# 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

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Look at the work of Ellis on topological dynamics using enveloping semigroups. It is not quite categorical, but in the right direction. –  Benjamin Steinberg Jan 20 '12 at 18:51
It's probably worth mentioning that similar questions have been asked before, but this one has not been answered in any thread I'm aware of: mathoverflow.net/questions/83363, mathoverflow.net/questions/83437, mathoverflow.net/questions/38752 –  David White Jan 20 '12 at 19:45
One thing that I have noticed is that a solenoid may be defined as an inverse limit, which is an object that is studied in dynamics. As far as results, well... –  Lunasaurus Rex Jan 20 '12 at 20:53
Well, some basic constructions in ergodic theory involve products, inverse limits, etc., so the categorical language is definitely there, but I'm not aware of any theorems (in ergodic theory) whose proof uses some abstract, deep facts in category theory. –  Mark Jan 21 '12 at 0:10
Try this post golem.ph.utexas.edu/category/2008/12/bridge_building.html for some category theoretic discussion of Tao's description of cohomology in dynamic systems. –  David Corfield Jan 21 '12 at 12:13

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.

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Thanks for the Lawvere paper Tom! And I do agree with you on the range of his attacks :-) –  Esfan Haghverdi Jan 21 '12 at 16:10

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.

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Many thanks for the reference. I also found a nice review of Mackey's ideas written by Calvin C. Moore called Virtual Groups 45 Years Later''. I am perfectly happy working with groupoids, ultimately I wish to bridge the algebra/category theory side where I am comfortable working to additive/arithmetic combinatorics where I wish to work. I think, given the strong Ergodic Theory/Additive Combinatorics link (e.g Furstenberg's work or more recently the work of Bryna Kra, Ergodic methods in additive combinatorics) this might be a most efficient bridge construction. –  Esfan Haghverdi Jan 23 '12 at 16:07
Glad that was helpful, and I look forward to reading the review by Calvin Moore. My impression is still that the full algebra of groupoids is not yet fully utilised: for example my book "Topology and groupoids" has a whole chapter on orbit space and orbit groupoids, which has been little utilised. Also having been brought up in homotopy theory I worked really hard to realise applications of double and higher groupoids, see my web page "Higher dimensional group theory". One gasps at the thought of "double ergodic theory"!!! –  Ronnie Brown Jan 24 '12 at 17:50

There is this post, and its continuation

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Thanks for the pointer. Even though after a cursory look I think the goal of the post is to understand a categorical phenomenon, namely the fact that an epi or mono endomorphism in certain categories (eg. locally finite ones) but also in the category of finite dimensional vector spaces is an iso, nevertheless it is a good starting point to reformulate more dynamical systems notions in categorical language and use them to solve a category theoretic problem. –  Esfan Haghverdi Jan 21 '12 at 14:40
As far as I was concerned, the idea was more to understand infinite iteration in a general setting, or to put my finger on what a "dynamical concept" is. I certainly wouldn't claim to have done so, but that's what was motivating me. –  Tom Leinster Jan 21 '12 at 14:44
Thanks, that is very helpful. In the past I did some work generalizing the partially additive categories (PAC) of Manes and Arbib to what I called Unique Decomposition Categories. The motivatuion for Manes and Arbib was of course to offer an algebraic denotational semantics as opposed to domain theoretic one of Scott. Each PAC is endowed with an iteration operator called dagger where given $f: X \to X\uplus Y$, $\dagger(f): X \to Y$ is given by $\sum_n f_2f_1^n$ where the sum is a partially defined infinitary operation on the homsets. I will look at these again in the light of your posts. –  Esfan Haghverdi Jan 21 '12 at 14:53

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...

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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).

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